Mathematical Models in Biology II

Christina Kuttler

July 21, 2009

Contents

1 Some problems from Ecology 31.1 System of May and Leonard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Food chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Metapopulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1 Levin’s Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.2 Spatial structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.3 Rescue and Allee effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.4 Real world data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Routh-Hurwitz Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.5 Predator prey model with time delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.5.1 Delay models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5.2 The classical Lotka-Volterra model . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.5.3 May’s Predator-Prey model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Diffusion processes 222.1 Random walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1.1 Random Motion and the Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . 222.2 Correlated random walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.1 Generalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Porous media equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4 Another type of diffusion: Biofilm model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Reaction-Diffusion equations 303.1 Travelling waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Spatial spread of an epidemic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.1 Simple model for the spatial spread . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 Facilitated Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.4 Morphogenesis / Pattern formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Evolutionary dynamics / Populations genetics 404.1 Replicator equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.1.1 Some basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.1.2 Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1.3 Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 Evolutionary stable strategies (ESS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2.1 Two-player games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2.2 Nash equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2.3 Evolutionarily stable strategy (ESS) . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2.4 More than two strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2.5 Replicator dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Hardy-Weinberg law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.4 Fisher-Wright-Haldane model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

1

5 Reaction kinetics 535.1 Enzyme kinetics: Michaelis-Menten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2 Oscillations in chemical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.3 Belousov-Zhabotinsky Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.3.1 Chemical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.3.2 The ODE system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.3.3 Spatial pattern formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.4 Metabolic pathways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.4.1 Glycolysis model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.5 Gene regulatory networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.5.1 Clock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.5.2 Toggle-Switch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.5.3 Positive Feedback loop / Modelling Quorum sensing . . . . . . . . . . . . . . . . . 72

6 Modelling of diseases / Medical applications 816.1 Vector-borne diseases / Dynamics of Malaria spread . . . . . . . . . . . . . . . . . . . . . 816.2 Diabetes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.2.1 Model for glucose and insulin dynamics, including β-cell mass dynamics . . . . . . 876.3 HIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.3.1 Anderson’s First Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.3.2 Anderson’s improved HIV model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.3.3 Modelling Combination Drug Therapy . . . . . . . . . . . . . . . . . . . . . . . . . 946.3.4 Delay model for HIV infection with drug therapy . . . . . . . . . . . . . . . . . . . 97

6.4 Tumour growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.4.1 Spherical tumour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7 Cellular Automata 1027.1 Basics of Cellular Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.1.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.1.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.1.3 2D case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.1.4 Wolfram Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.2 Greenberg-Hastings Automaton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057.3 Game of life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.4 Generalised Cellular Automata and related models . . . . . . . . . . . . . . . . . . . . . . 108

Literature 108

Index 111

2

Chapter 1

Some problems from Ecology

Some typical problems from ecology were already used in Part I of the lecture, like simple predator-preysystems or competition between two species. Here, we will consider more sophisticated models, dealingwith special situations in ecology. There may be more connections for the populations to the rest of theworld, e.g. by taking into account more than two interacting species, or by spatial effects.

1.1 System of May and Leonard

Literature: May & Leonard [23]

The system of May and Leonard was developed in 1975 in order to show that competitive interaction ofthree or more species can lead to a unexpected dynamical behaviour. On the other hand we know fromthe theory of quasimonotonic systems that a system for three competing species behaves essentially likea two-dimensional system. Therefore, we examine here a concrete system:

x1 = x1(1 − x1 − αx2 − βx3)

x2 = x2(1 − x2 − αx3 − βx1) (1.1)

x3 = x3(1 − x3 − αx1 − βx2)

where xi denotes the population density of species i. The system is considered in the cone

C = x = (x1, x2, x3) : xi ≥ 0, i = 1, 2, 3

On each rim of the cone the system behaves in the same way; hence it is sufficient to consider the sidex3 = 0. There, the system is given by

x1 = x1(1 − x1 − αx2)

x2 = x2(1 − x2 − βx1) (1.2)

(i.e., just the 2D competition model from lecture part I).In the interior of the cone C there is a stationary point

x =1

1 + α + βe,

where e = (1, 1, 1)T . In vector notation the system reads

x = X(e − Ax),

where

A =

1 α ββ 1 αα β 1

and X = (xiδij) denotes the diagonal matrix which belongs to the vector x. (Later, diagonal matrices Uand Y are introduced in the same way). Thus, the Jacobian at position x, applied on y, is

Jy = Y (e − Ax) − XAy.

3

We introduce the relative frequencies u = x/eT x and the population size s = eT x as new variables. Asimple computation shows that u and s satisfy the following equations:

u = −s(UAu − uT Au · u)

s = s(1 − φ(u)s),

whereφ(u) = uT Au.

In the set C \ 0 a new time variable along the trajectories can be introduced: τ = s · t ⇔ t = τs . Thus,

the vector of the relative frequencies behaves like the solution of the replicator equation

u = −(UAu − uT Au · u) (1.3)

(due to dudτ = du

dtdtdτ = du

dt1s ).

This system is considered on

S = u = (u1, u2, u3) : ui ≥ 0, i = 1, 2, 3; eT u = 1.

For this system we introduceV (u) = u1u2u3.

The following proposition shows that V is a Lyapunov function for system (1.3) (remember: V ∈ C1,V ≥ 0, V ≤ 0).

Proposition 1 It is

(α + β − 2)dV (u(t))

dt≤ 0.

Proof: In coordinate notation, system (1.3) reads

u1 = −u1(u1 + αu2 + βu3 − φ(u))

u2 = −u2(u2 + βu1 + αu3 − φ(u))

u3 = −u3(u3 + αu1 + βu2 − φ(u)),

where

φ(u) = u21 + u2

2 + u23 + (α + β)(u1u2 + u2u3 + u3u1)

= 1 + (α + β − 2)(u1u2 + u2u3 + u3u1).

(the last two transformation are according to the definition of A and

u1 + u2 + u3 = 1

; 1 = (u1 + u2 + u3)2 = u2

1 + u22 + u3

3 + 2u1u2 + 2u2u3 + 2u3u1

; u21 + u2

2 + u23 = 1 − 2u1u2 − 2u2u3 − 2u3u1

)Now we get

V = u2u3u1 + u1u3u2 + u2u1u3

= −u1u2u3[u1 + αu2 + βu3 − φ(u) + u2 + βu1 + αu3 − φ(u) + u3 + αu1 + βu2 − φ(u)]

= −V ((1 + α + β) − 3φ(u)).

Hence the inequality V < 0 holds if and only if

1 + α + β − 3φ(u) > 0

⇔ 1 + α + β − 3(1 + (α + β − 2)(u1u2 + u2u3 + u3u1)) > 0

(α + β − 2)(1 − 3(u1u2 + u2u3 + u3u1)) > 0.

Trivially, it is

u1u2 + u2u3 + u3u1 ≤ 1

3

4

with equality if and only if u1 = u2 = u3 = 13 . (Obviously, for u1 = u2 = u3 = 1

3 we have equality.Consider the “deviation” from that:

(1

3+ a)(

1

3+ b) + (

1

3+ b)(

1

3− a − b) + (

1

3− a − b)(

1

3+ a)

=1

9+

a

3+

b

3+ ab +

1

9− a

3− b

3+

b

3− ab − b2 +

1

9− a

3− b

3+

a

3− a2 − ab

=1

3− a2 − ab − b2.

If a, b > 0 or a, b < 0, - done; if (without loss of generality) a > 0, b < 0, then transform the last equationinto 1

3 − (a + b)2 + ab - also done.)

2

Hence it was shown: The solutions of (1.1), except for x = 0, are limited and stay away from x = 0. Eachtrajectory has a one-to-one projection onto the two-dimensional set S. (In the theory of quasi-monotonesystems this property is only shown for limit sets).If α + β < 2, then u = e/(α + β + 1) is global attractive in the set S.If α+β > 2, then the limit set of each trajectory is contained in a certain compact subset of the boundary∂C. Since we know the behaviour of system (1.1) on the rims of C, we have all the information which isnecessary to describe the behaviour of system (1.1). Please note that in the case α + β > 2 the limit setof each trajectory of (1.1) is a invariant subset of ∂C. Each rim of ∂C is invariant, thus the limit set isa union of invariant sets in these rims. Now we distinguish four cases:

Case 1: α < 1, β < 1: Then α+β < 2, the point e/(α+β+1) is a global attractor, in each rim there is acoexistence point for two species which is a saddle point with a one-dimensional unstable manifold.

Case 2: α > 1, β > 1: Then α+β > 2 and the plot of case 1 is completely reverse. The point e/(α+β+1)is a repellor, the coexistence points in the rims are saddle points with a one-dimensional stablemanifold. Only the three points (1, 0, 0), (0, 1, 0) and (0, 0, 1) have open domains of attraction.

Case 3: α > 1, β < 1, α + β < 2: Then there are no coexistence points in the rims, but the pointe/(α + β + 1) is stable.This case is biologically interesting. Each of the three species can survive on its own. Two of themcannot coexist, but all three can!

Case 4: α > 1, β < 1, α + β > 2: Then the point e/(α + β + 1) is unstable. The ω-limit sets oftrajectories in C lie in ∂C. Since the points (1, 0, 0), (0, 1, 0), (0, 0, 1) are saddles, the ω-limit setsof each such a trajectory is a union of the three stationary points (1, 0, 0), (0, 1, 0), (0, 0, 1) and theconnecting trajectories in the boundary of C (which are not the direct connecting lines, but thesaddle-node connections of the two-dimensional system (1.2)). For large α, they come near to theoffspring.

1.2 Food chains

In lecture part I, we already considered e.g. the predator-prey model of Lotka-Volterra (and others),where one species lives on another one. In some sense, it is a quite artificial situation, to consider justtwo species; in natural systems, many species are interacting. Also competition plays a role, of course.Such systems can be visualised in graphs and described as so-called food webs. Remark, that there aresome problems:

• These graphs only give very rough information about the connections between the species, i.e. nodetails about functional responses or the corresponding differential equations

• Modelling of larger, i.e. complex natural ecosystems by food webs is “difficult”; e.g., in most cases,there is not enough information available concerning parameter values, or the relationships betweenthe species, or sometimes even not about the involved species at all. So, the modelling must berestricted to only few variables.

The species in such a food web need not necessarily to be a biological species. For example, in aquaticsystems, one can collect all “small crabs” or “fishes” in one variable, each.A simple special case of a food web is a so-called food chain. There is just a chain of species, where

5

one eats up the preceding one. So, a classical predator-prey model can be interpreted as a food chain oflength two.In the next step, we consider a food chain of length three, e.g. a coastal biotope, including substrate (x1),phytoplankton (x2) and zooplankton (x3). Further assumptions: The substrate is just introduced withconstant rate σ (no self-reproduction), phytoplankton lives on the substrate (with saturation, Michaelis-Menten kinetics, which we will consider later), and the zooplankton lives on the phytoplankton (also withthe same saturation law). Additionally, there is a washout rate Q (the same for all three species). Thisyields the following model approach:

x1 = σ − x1x2

A + x1− Qx1

x2 =x1x2

A + x1− x2x3

A + x2− Qx2

x3 =x2x3

A + x2− Qx3

(Obviously, here we do not have conservation of substrate, as we had in the simple plankton model inlecture part I).Let us study some properties of this model system. First observation: A solution which starts somewherein R

3+, will stay in that set. Now, we introduce a new magnitude, x = x1 +x2 +x3. Obviously, it satisfies

the equation˙x = σ − Qx.

For x there is exactly one stationary point, x = σ/Q, which is furthermore stable. The solutions cannotescape to infinity (they are bounded), so they tend to σ/Q, respectively to the set

S = (x1, x2, x3) : x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x1 + x2 + x3 = σ/Q.

This set (a triangle) is invariant, all ω limit sets are included in S. So, if there are any stationary pointsin R

3+, they are restricted to be of the form (x1, 0, 0), or (x1, x2, 0), or (x1, x2, x3), where xi are positive,

i = 1, 2, 3.Next step: Analysis of the system. For that purpose, we use a transformation, to get rid of the parameterA. We choose: xi = Axi and σ = Aσ. Using again the variables notated without the hat (for reasons ofsimplicity) yields the equations

x1 = σ − x1x2

1 + x1− Qx1 (1.4)

x2 =x1x2

1 + x1− x2x3

1 + x2− Qx2 (1.5)

x3 =x2x3

1 + x2− Qx3. (1.6)

So, there are just two parameters left in the system, σ and Q.In the following, we only consider the case of 0 < Q < 1 (the case of Q ≥ 1 is not so interesting - shortexercise: why?)First, we look for the existence of stationary points.In case of x3 > 0, we need

x2

1 + x2= Q ⇔ x2 =

Q

1 − Q

(in order to satisfy the stationarity for (1.6)), thus we get (from (1.4)):

σ =x1x2

1 + x1+ Qx1 =

x1

1 + x1· Q

1 − Q+ Qx1.

Increasing x1 from 0 to ∞, then the right hand side goes monotonely from 0 to ∞. Thus, there is exactlyone x1 > 0, such that this equation is satisfied; it can be computed from a quadratic equation. Thecorresponding x3 satisfies then the equation

x3 =

(x1

1 + x1− Q

)

· 1

1 − Q.

6

Obviously, x3 is positive if and only if x1

1+x1> Q. Rewriting that condition, we find x1 < Q

1−Q ; byinserting that into the equation for σ, we get

σ =x1

1 + x1· Q

1 − Q+ Qx1

>Q2

1 − Q+

Q2

1 − Q=

2Q2

1 − Q. (1.7)

Taken together, we have shown now:

Lemma 1 There exists a stationary point (x1, x2, x3), where x3 > 0 if and only if condition (1.7) issatisfied.

Next case: If there is a stationary point with x2 > 0 and x3 = 0, then we need:

σ =x1x2

1 + x1+ Qx1 and

x1

1 + x1= Q,

thus

x1 =Q

1 − Q

x2 =σ

Q− Q

1 − Q.

Obviously, x2 is positive if and only if

σ(1 − Q) > Q2 ⇔ σ >Q2

1 − Q. (1.8)

This yields the following

Lemma 2 There exists a stationary point (x1, x2, 0) with x2 > 0 if and and only if the condition (1.8)is satisfied.

The last remaining possibility: If there is a stationary point of the form (x1, 0, 0), then x1 = σ/Q - thispoint exists in each case!Thus, we can distinguish three different situations in the parameter space:

• For a very small σ, only the substrate x1 can exist (in a stationary state)

• If condition (1.8) is satisfied, then an equilibrium between substrate and phytoplankton exists.

• If condition (1.7) is satisfied, then an equilibrium, including also the zooplankton, exists.

We already remarked that all ω limit sets are contained in

S = (x1, x2, x3) : x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x1 + x2 + x3 = σ/Q.

In the next step, we project that triangle into the x1, x2 plane, i.e. S corresponds to

S = (x1, x2) : x1 ≥ 0, x2 ≥ 0, x1 + x2 ≤ σ/Q.

(x3 can be replaced by σ/Q − x1 − x2). The corresponding 2D system reads:

x1 = σ − x1x2

1 + x1− Qx1 (1.9)

x2 =x1x2

1 + x1− x2

1 + x2

(σ

Q− x1 − x2

)

− Qx2. (1.10)

We have to check if S is positively invariant:

• on x1 + x2 = σ/Q:

˙(x1 + x2) = σ − Qx1 −x2

1 + x2

(σ

Q− x1 − x2

)

︸ ︷︷ ︸

=0

−Qx2

= σ − Q(x1 + x2)

= 0

thus, x1 + x2 = σ/Q is invariant.

7

• on x2 = 0:x2 = 0

; x2 = 0 is invariant

• on x1 = 0:x1 = σ

; on x1 = 0, the arrows point inwards ; S is indeed positively invariant.

Next, we check the stability of the stationary points. The general Jacobian matrix reads:

J(x1, x2) =

(

− x2

(1+x1)2− Q − x1

1+x1x2

(1+x1)2+ x2

1+x2

x1

1+x1− 1

(1+x2)2( σ

Q − x1 − x2) + x2

1+x2− Q

)

.

Let us first consider the stationary point ( σQ , 0), there, the Jacobian matrix reads:

J =

(

−Q − σ/Q1+σ/Q

0 σ/Q1+σ/Q − Q

)

with

det J = −Q

(σ/Q

1 + σ/Q− Q + σ

1 + σ/Q

)

tr J = −2Q +σ/Q

1 + σ/Q.

In case of 0 < σ < Q2

1−Q we find:

(1 − Q)σ < Q2 ⇔ σ − σQ < Q2 ⇔ σ/Q − Q − σ < 0,

thus the determinant is > 0 and the trace is < 0, which says that ( σQ , 0) is stable.

In the same way we can consider the stationary point(

Q1−Q , σ

Q − Q1−Q

)

and find (by larger computations)

that it is stable in case of Q2/(1 − Q) < σ < 2Q2/(1 − Q).For the coexistence point, everything is a little bit more complicated. By using the conditions, that inthe coexistence point, we have

σ =x1

1 + x1

Q

1 − Q− Qx1

and

x2 =Q

1 − Q.

Then, the trace can be computed to be

tr J = − Q

1 − Q

1

(1 + x1)2+ Q

x1

1 + x1− Q2.

Obviously, this expression is negative for x1 near 0, but positive for very large x1 (remark that Q < 1).We can compute where the sign changes:

1

1 − Q− x(1 + 1) + Q(1 + x)2 = 0

⇔ 1

1 − Q+ Q + x(2Q − 1) + x2(Q − 1) = 0

⇔ x2 − x2Q − 1

1 − Q− 1 + Q − Q2

(1 − Q)2= 0.

Since the absolute term is negative, there is just one positive root,

x =2Q − 1

2(1 − Q)+

√

(2Q − 1)2

4(1 − Q)2+

1 + Q − Q2

(1 − Q)2.

8

This can be simplified, then

x =1

2(1 − Q)(2Q − 1 +

√5).

If we choose this value as x1 and compute the corresponding σ, we get

σ =1

2

Q

1 − Q(Q(1 +

√5) + 2),

called the critical value for σ. It means: For σ > σ, the coexistence point is unstable. Thus, the theoremof Poincare-Bendixson yields, that a periodic solution exists.

1.3 Metapopulations

- mainly taken from the lecture script “Mathematical models in Biology II” from Johannes Muller [25] -some additions can be found in [11]

Most populations do not just live in one habitat, but are spread over several habitats. Even thoughif such “small populations” may go extinct locally, the total population survives in many cases sinceindividuals from neighbouring surviving habitats can invade and thus reoccupy it again. This type ofproblems is considered in the metapopulation theory. Typical questions are:

• Is a certain patchwork of local habitats able to support a population for a long time?

• Which factors play a role?

Metapopulation models mainly start with the following idea: If a habitat is occupied, then the probabilityfor extinction per time interval is relatively small. If the population breaks down, however, it happens(approximately) at one moment in time. Vice versa, if a population invades a patch, its local density willreach very soon the carrying capacity by logistic growth.

1.3.1 Levin’s Basic Model

The first metapopulation model was introduced by Levin in 1969. For that, p(t) denotes the probability,that a patch is occupied, and e is the rate by which a patch crashes. Neglecting re-occupation for themoment, we would thus get:

p = −ep.

But of course, re-occupation happens and is important. For the re-occupation, we need an empty patch(the probability for that is 1−p) and an individual which arrives from another patch. Since the number ofindividuals that may arrive is proportional to p, also the arrival rate is proportional to p. Taken together,the re-occupation rate is cp(1 − p) and the resulting equation reads

p = −ep + cp(1 − p).

Obviously, this is a logistic equation, which shows up the typical threshold behaviour. For the persistenceof the metapopulation, it is sufficient and necessary to have

e < c.

This model is the basis for all metapopulation models. Many assumptions were used for it, e.g.

• Homogeneous patches are considered (same size, isolation, habitat quality etc. for each populationsite)

• No spatial structure is taken into account: the “neighbourhood” is neglected, which means that theexchange between different population sites is always the same.

• No time lags, instantaneous changes are assumed

• c and e are taken to be constant, independent of time

• We deal with a large number of patches, so demographic stochasticity can be neglected

Other models take this idea for refined approaches and by that explore the effect of the consideredmechanisms.

9

1.3.2 Spatial structure

For real world application, it is urgent to include spatial structure in a more explicit manner. Of course,there are several possibilities, how to do that.

Static habitat loss

A model that is quite close to Levin’s model is the one which incorporates habitat loss. We consider aspatial structure, e.g. a (finite) grid. Each site can assume one of the following three states:

1. State “1”: suitable site, unoccupied

2. State “2”: suitable site, occupied

3. State “3”: unsuitable site

Mainly two processes play a role in the metapopulation model:

1. A suitable occupied site can crash. The probability for a local extinction is δ per time step.

2. A suitable, but empty site can be re-occupied. The probability of reoccupation is c.Question: How can we determine c? We assume that a mean number M of individuals arrives at apatch per time step and ask for the probability that no one arrives during a time step. Due to the(plausible) assumption of a Poisson distribution, we have

P (i arrivals) =1

i!M ie−M ,

thus the probability of no arrivals reads

P (no arrivals) = e−M .

Obviously, M is proportional to the number of available, occupied patches, thus

M = dx2,

where xi denote the probability to find a patch in state i. Hence, the probability of re-occupationof a suitable, empty patch is 1 − e−dx2 .

This yields the model equations:

x1(t + 1) = e−dx2x1(t) + δx2(t)

x2(t + 1) = (1 − e−dx2)x1 + (1 − δ)x2

x3(t + 1) = x3.

(remark, that this is a discrete model, for discrete time-steps!).When initialising the model, the fraction 1 − h of sites is declared to be in state “3”, i.e. unsuitable,cannot be occupied. In the “static habitat loss” no process deals with state “3”, so the fraction staysunchanged. Of course, one can also consider a dynamics of unsuitable habitats, which means that theycould be destroyed or restored at a certain rate. For the moment we do not deal with that problem. Dueto

x1 + x2 + x3 = 1 and x3 = 1 − h

by definition, we can find

x1 = 1 − x3 − x2 = 1 − (1 − h) − x2 = h − x2

and thusx2(t + 1) = (1 − e−dx2)(h − x2) + (1 − δ)x2 = f(x2).

In the next step, we want to analyse the dynamics of that equation (which is a nonlinear differenceequation). As usual, we first look for a fixed point. Obviously, x2 = 0 is always possible (called trivialfixed point). The first and the second derivative of f read:

f ′(x2) = de−dx2(h − x2) − (1 − e−dx2) + (1 − δ)

f ′′(x2) = −d2e−dx2(h − x2) − de−dx2 − de−dx2 ≤ 0

Fixed points can be interpreted graphically as intersection points of the right hand side function f andthe first bisection line. As the second derivative is smaller or equal zero, there can at most exist onefurther fixed point, denoted by x∗

2 > 0 (compare it also to the figure below):

10

bisect. line

f

f(x)

x

The only possibility for x∗2 is that f ′(x∗

2) < 1, because the function f(x) has to cross the first bisectionline from upper left side to the lower right side. Vice versa, we find that f ′(x∗

2) > −1+(1−δ) = −δ > −1.Taken together, the stationary point x∗

2 is locally asymptotically stable.Furthermore, it is easy to see, that f ′(0) gives us information if there is a persisting fixed point or not:In case of f ′(0) > 1, there exists this further fixed point; in case of f ′(0) < 1, it doesn’t exist. Due to

f ′(0) = dh + (1 − δ),

we can reformulate the condition for persistence to

dh + 1 − δ > 1 ⇔ h >δ

d.

In the next step, we analyse, if we can find something about the dependency of the value of the fixedpoint x∗

2 on the parameter h. The fixed point equation reads

x∗2 = (1 − e−dx∗

2 )(h − x∗2) + (1 − δ)x∗

2,

Solving it up for h yieldsδx∗

2 + (1 − e−dx2)x∗2 = h(1 − e−dx∗

2 )

; h(x∗2) =

x∗2

1 − e−dx∗

2

(

δ + (1 − e−dx∗

2 ))

= x∗2

(

δ/(1 − e−dx∗

2 ) + 1)

.

Hence, for x∗2 → 0 we find

h(0) =δ

d,

which accords to the threshold condition; furthermore, it is monotone increasing.It is also possible to consider the relative part of suitable habitats which are occupied, i.e.

p∗ = x∗2/h.

For that, the fixed point equation reads

hp∗ = (1 − e−dhp∗

)(h − hp∗) + (1 − δ)hp∗

; p∗ = (1 − e−dhp∗

)(1 − p∗) + (1 − δ)p∗.

Solving that equation for h yields:

p∗ = (1 − e−dhp∗

)(1 − p∗) + (1 − δ)p∗

⇔ δp∗ = (1 − e−dhp∗

)(1 − p∗)

⇔ δp∗

(1 − p∗)= 1 − e−dhp∗

⇔ e−dhp∗

= 1 − δp∗

1 − p∗

⇔ −dhp∗ = ln

(

1 − δp∗

1 − p∗

)

⇔ h = h(p∗) =1

p∗dln

(1 − p∗

1 − p∗ − δp∗

)

Obviously, we get again limp∗→0 h(p∗) = δ/d, and limp∗→1/(1+δ) h(p∗) → ∞. By that, we can sketch thefunctions h(p∗) and p∗(h):

11

δ/d

δ/d

h(p*)

p*

p*(h)

h

1/(1+δ)

1/(1+δ)

Inhomogeneous location- and size structure

Of course, for real world data, things are not so simple as in our toy models up to now - the patches mayhave different sizes or other properties, the distances between patches may play a role etc. One approachinto that direction works as follows:We consider n different patches, which are localised at x1, . . . , xn and the area A1, . . . , An (the area istaken instead of the carrying capacity, because the latter one is more difficult to estimate). Then thefollowing notations / assumptions are used:

• Extinction rate: ei = e/Ai

• Immigration rate from patch j into patch i: e−α‖xi−xj‖Aj

• Total immigration rate into patch i:∑

j 6=i e−α‖xi−xj‖Aj

Let pi(t) denote the probability that patch i is occupied at time t, then the ODE for it reads:

pi =

∑

j 6=i

e−α‖xi−xj‖Ajpj(t)

(1 − pi) − eipi.

Let p = (p1, . . . , pn)T . The system can be linearised at the trivial equilibrium pi = 0, which yields:

p = (B − D)p,

where

B = ((bi,j))ij , bi,j =

e−α‖xi−xj‖Aj for i 6= j0 else

, D = diage1, . . . , en.

Obviously, B − D is a M-Matrix (i.e.: the off-diagonal elements are ≥ 0). One can show: The leadingeigenvalue is real; it depends on its sign, if the trivial stationary state is stable or not. In case ofits instability, it is possible to show by means of the theory of cooperative systems and monotonicityarguments, that there exists a stable nontrivial stationary state.

1.3.3 Rescue and Allee effect

Coming back to our toy models, we want to consider the extinction and re-occupation process in greaterdetail.

Rescue effect

Levin’s model assumes a constant extinction rate. But if a reasonable fraction of patches is occupied,there may be a constant immigration rate from these occupied patches into the given patch (even if it isnot empty yet). So, it might happen that these immigrants rescue this patch (before it goes extinct atall) and by that, the patch “survives”. We can take this effect into our model by decreasing the extinctionrate if p is high, i.e. e is replaced by e/(1 +

√ζp) and yields

p = c(1 − p)p − ep

1 +√

ζp= p

[

c(1 − p) − e/(1 +√

ζp]

.

There is always one stationary state:p = 0. For the other stationary state, we need

c = c(p) =e

(1 +√

ζp)(1 − p).

We can see: c(0) = e, that means that the nontrivial line of stationary points hits the line p = 0 at thesame position as Levin’s model does. For a sketch of the graph see the figure below (including also theAllee effect).

12

Allee effect

Considering sexually reproducing species, it is clear, that they need more than one individual to survive(others may need that, too). So, it does not help much if just one individual enters an empty patch:at least two individuals (a male and a female) are needed to start a new population. This is calledthe “Allee effect”. If the probability that one individual enters the considered patch in a given timeinterval is proportional to p, then the probability that two individuals enter it in the given time intervalis proportional to p2. Hence, the modified model reads

p = c(1 − p)p2 − ep.

In the same way as for the rescue effect, the lines of stationary points can be determined: the trivial line(p = 0) and the nontrivial line c(p) = e/p(1− p). In the case of the Allee effect, the nontrivial line neverintersects the trivial line; but one can see: All points above a critical value for c are bistable. The curvecan be seen in the following figure:

Levin

Rescue

Allee

p(c)

ce

(the nontrivial lines of stationary points - the trivial one p = 0 is left out in the figure)

One can see: Both, the Rescue effect and the Allee effect show up a bistable region. This leads to aninteresting behaviour: The immigration parameter c does in some sense reflect e.g. the habitat density).Only Levin’s Basic Model shows a continuous dependency on the population size. Due to the “blue skybifurcation” and their bistable behaviour, the rescue- as well as the allee effect model may sustain thepopulation size at a relatively high level, until they crash down suddenly. From a biological point ofview, it can be interpreted as follows: Also a relatively high (meta)population level does not necessarilyreflect a “safe situation”, where no risk for (global) extinction of the metapopulation is present. Such abehaviour can also be found in some experimental data.

1.3.4 Real world data

We just observed that the Allee effect and the Rescue effect predict a bistable behaviour. It is an inter-esting question to check, if that bistability can also be observed in empirical data? In Hanski et al., alarge patchwork of habitats for a species of butterflies Militaea cinxia was investigated in the southwestof Finland. 1530 habitats were located (small spots of 12 m2 up to large habitats with 7.3 ha). Thesehabitats were distributed over an area of 3500 km2. In each of the habitats, the population size wasmeasured in 1993 and 1994.For analysing the data with respect to factors which influence the probability of extinction respectivelythe probability for (re-)colonisation, a logit model was used. (A logit model is the straight generalisationof linear models suited for normal data to binary data). Goal is to estimate the influence of a certainfactor on a probability; and to predict the probability for given factors.

The logit model

The logit model is a statistical model for the influence of certain factors on a probability. The idea forthe logit model is the following: Considering a population of e.g. birds which undergo the molt, theprobability for a bird that it already has been molting at a given point of time has to be predicted. At anvery early time point, the probability will be almost zero; vice versa, for a very late time, the probabilitywill be high (close to one). The time in-between is the interesting one: The idea is that the birds behaveapproximately alike. If very few birds have been molted, then our bird of interest is very likely not to bemolting and the rate of change of the probability is small. Vice versa, if most of the birds already havebeen molting, the chances are high, that also our bird has been molting; again the rate of change of ourprobability is small. The assumption is, that in between, when about half of the birds have been molted,

13

the other half not, the rate of change of the probability should be high. This leads to the followingapproach:

d

dtp(t) = bp(t)(1 − p(t)). (1.11)

(Remark: Again the logistic equation!). We can use the so-called “logit transformed” of the probability.First, an auxiliary computation, we find:

(

ln

(p(t)

1 − p(t)

))′=

1p(t)

1−p(t)

· p′(t)(1 − p(t)) + p(t)p′(t)

(1 − p(t))2

=p′(t)

p(t) · (1 − p(t))

= b.

So, by taking the integral over dt of equation (1.11), we find:

logit(p(t)) = ln

(p(t)

1 − p(t)

)

= a + bt.

(a as a integration constant). This “logit transformed” of the probability is linear in time (time is thefactor which influences our probability). For the estimation of parameters, there are several possibilities.A simple, heuristic approach works as follows (and requires a relatively large sample for each value ofthe factor ti (time points, where measurements are available)): Let us assume that at time point ti, weinvestigated xi birds and yi birds are already molted. We require 0 < yi < xi, all time points with yi = 0or yi = 1 are dismissed. Then, we define

zi = logit(yi/xi)

and determine the linear regression for the data points (zi, ti). The slope of the linear regression curve isb and its intercept a.

A better, but numerically more involving approach is to use the maximum likelihood. For that, weassume that we have data for single individuals, taken at the time points t1, . . . , tn. Each data point xi

tells us, if the individual has not molted (xi = 0) or if it has molted (xi = 1). The likelihood of a datapoint (xi, ti) is given by

L(xi, ti; a, b) = p(ti)xi + (1 − p(ti))(1 − xi)

and the joint likelihood reads

L(a, b) =

n∏

i=1

L(xi, ti; a, b) =

n∏

i=1

p(ti)xi + (1 − p(ti))(1 − xi).

This function (respectively its logarithm) will be maximized using some numerical algorithms.This kind of model is not restricted to one factor only; the right hand sind may be composed by severalfactors like the influence of weight of the influence of the sex (this can e.g. be coded by -1 resp. 1 formale resp. female), the ansatz reads

logit(p(t,weight, sex)) = ln

(p

1 − p

)

= a + b1t + b2weight + b3sex.

The coefficients bi estimate more or less the “pure” effect. If we e.g. include weight and sex, then theeffect that e.g. male birds may have a higher weight than the female birds does not influence any morethe coefficient of the weight. Different effects can be considered independently of each other. However,the linear expression on the right hand side is most likely a linear approximation of a more complexfunction,

logit(p(t,weight, sex)) = f(t,weight, sex) ≈ a + b1t + b2weight + b3sex.

In principle, it is possible to include also “interaction terms”, i.e. higher order terms like

logit(p(t,weight, sex)) = a + b1t + b2weight + b3sex + b1,2tweight.

So, the model will return point estimates for a and bi. These point estimates indicate if the probabilityis increased resp. decreased by a certain factor (bi > 0 or bi < 0). Usually, a test is performed on the

14

significance of bi 6= 0 versus the zero hypothesis bi = 0. So, the significance level is given and allows tovalidate the result.

Let us return to the experiment of Hanski et al. A similar analysis can be done for the butterflies,looking for the extinction / re-colonisation probabilities. There are data available for two years; manypatches are colonised in the first year. Using the data from the second year, we can check, which coloniesare still colonised and which are not; the logit-model can help to estimate the various influences on thisprobabilities. Possible influencing factors could be:

• patch size and patch quality

• size of the population in the first year

• pooled size of neighbouring patches Nneigh

• the “trend” in the neighbouring populations (trend means: difference of pooled population size inthe two years; it is used as an “estimate” for global effects like weather conditions)

In a similar way, the immigration has been considered. Hanski et al. included the following variables intheir analysis:

Name VariableC Interceptlog(N1993) Size of the population in patch under consideration at 1993Ntrend Difference in pooled neighbouring population sizes of 1993 and 1994Nneigh pooled neighbouring population sizes

(Of course, further factors, e.g. describing patch quality (area, grazing cattles present etc.) were takeninto account, but we leave them out here, since we are mainly interested in the influence of neighbouringpopulations). The results for the interesting factors read:

Factor Extinct. estimate Extinct. p-value Colonis. estimate Colonis. p-valueC 2.926 < 0.0001 -3.524 < 0.0001log(N1993) -2.076 < 0.0001 -Ntrend -2.426 < 0.0001 1.474 < 0.0001Nneigh -0.805 0.0009 1.076 < 0.0001

Obviously, the colonisation population depends heavily on the neighbouring populations. Also the proba-bility for extinction is significantly influenced by this factor. Hence, the data hint on a pronounced rescueeffect - and we expect a bistable region. Hanski et al. tried to reveal this region. For that, the datasset was separated into 65 semi-independent subnetworks (well separated from each other, by distanceand/or natural barriers like a forest etc.). For each of these subnetworks, the fraction of occupied patches(i.e.: the area of occupied patches divided by the total area) was determined. Further, a measure for thecolonisation parameter was calculated:

∑

i,j

e−di,j√

Aj/n

1/2

,

where di,j denotes the distance of patch i and j, Aj the patch size, n the number of patches. By thatapproach, it is possible to compare the expected and the measured structure (see the Figures in the lectureor in Hanski et al. [13]). One can see (more or less) the bistable region and the predicted bifurcationdiagram!

1.4 Routh-Hurwitz Criteria

Literature: Pielou [32]

We are looking for a simple possibility to check the stability of stationary points in higher-dimensionalsystems, where it may not be so easy, to compute the eigenvalues explicitely.Let’s start with the characteristic polynomial (which comes e.g. from a Jacobian matrix),

λk + a1λk−1 + a2λ

k−2 + . . . + ak = 0.

15

The Routh-Hurwitz Criteria do not yield information about the absolute values of the eigenvalues, butabout their signs. For checking for stability, it is sufficient to know the sign of the real parts.

For k > 2 the criteria can be formulated in the following way:The following “auxiliary” matrices are defined:

H1 = (a1), H2 =

(a1 1a3 a2

)

, H3 =

a1 1 0a3 a2 a1

a5 a4 a3

. . .

Hj =

a1 1 0 0 . . . 0a3 a2 a1 1 . . . 0a5 a4 a3 a2 . . . 0...

......

......

a2j−1 a2j−2 a2j−3 a2j−4 . . . aj

...Hk =

a1 1 0 . . . 0a3 a2 a1 . . . 0...

......

...0 0 . . . ak

,

i.e. the (l,m) term in the matrix Hj is given by

a2j−m for 0 < 2l − m ≤ k1 for 2l = m0 for 2l < m or 2l > k + m.

All eigenvalues have negative real parts if and only if the determinants of Hj (called Hurwitz matrices)are positive, i.e.

detHj > 0, j = 1, 2 . . . , k.

By May, some “simpler” stability conditions were formulated (by using the general conditions) for some

special cases of k:

k = 2 : a1 > 0, a2 > 0

k = 3 : a1 > 0, a3 > 0, a1a2 > a3

k = 4 : a1 > 0, a3 > 0, a4 > 0, a1a2a3 > a23 + a2

1a4

k = 5 : ai > 0 (i = 1..5), a1a2a3 > a23 + a2

1a4, (a1a4 − a5)(a1a2a3 − a23 − a2

1a4) > a5(a1a2 − a3)2 + a1a

25.

These conditions are really easy to check in many cases!

Small remark, for those being interested in the history of mathematics: Adolf Hurwitz, 1859-1919,who contributed to these criteria, started to study mathematics at the “Koniglich Bayerische Technis-che Hochschule Munchen” - which is called today Technische Universitat Munchen - in 1877.

1.5 Predator prey model with time delay

1.5.1 Delay models

Literature: [27]

Considering a single population model as a simple example, even there a problem appears: The birthrate act instantaneously, but - realistically - it may take some time to reach maturity etc., i.e. a time de-laycan appear. For the mathematical model, we can formulate that delay by a so-called delay differentialequation (DDE) model:

N(t) = f(N(t), N(t − T )),

here T > 0 (the delay) is a parameter. More concretely, the extension of the logistic growth model, lookslike

N = rN(t)

(

1 − N(t − T )

K

)

. (1.12)

In this approach (which was introduced by an ecologist, Hutchinson), the regulatory effect just dependson the population at time t− T instead of t. It is also possible and useful to consider delay effects which

16

“average” in some sense over past populations. This idea results in an integrodifferential equation, whichcan look e.g. in the following way:

N = rN(t)

(

1 − 1

K

∫ t

−∞w(t − s)N(s) ds

)

.

w(t) weights the influence of the population size at earlier times; so, for s → −∞, it will tend to zero. Intypical examples, it may have a maximum at a representative time T , i.e. the graph looks as follows:

Tt

w(t)

In the limit, w(t) could be taken as Dirac function δ(t − T ), i.e.

∫ ∞

−∞δ(t − T )f(t) dt = f(T ).

In that case, the population growth model reads

∫ t

−∞δ(t − T − s)N(s) ds = N(t − T ).

Remark: Instead of the initial value in case of an ODE, N(t) needs to be given for all −T ≤ t ≤ 0, whichis called “history function” in that context.Let us consider the following situation: For some t = t1 let N(t1) = K, and for t < t1 let N(t) < K.From the delay equation (1.12) we find:

1 − N(t − T )

K> 0 ; N(t) > 0, for t1 ≤ t < t1 + T

thus, N(t) is still increasing at t1. When t = t1 + T , then

N(t − T ) = N(t1) = K ; N(t) = 0.

In the same way, for t1 + T < t < t2:

N(t − T ) > K ; N(t) > K,

thus, N(t) decreases, until t = t2 + T (we choose t2 in such a way, that N(t2 + T − T ) = N(t2) = K,then there is N(t) = 0 again.This observation shows, that in principle, oscillatory behaviour of such a delay system is possible.

tt1 t1+T t2 t2+T

K

N(t)

17

As a simple example, consider the simple linear delay equation

N = − π

2TN(t − T ).

Obviously, it has the solution

N(t) = A cosπt

2T,

(A constant), which is periodic in time.Indeed, the solutions of (1.12) can show up stable limit cycle periodic solutions (details are left out here).Stable limit cycle solution means that after a perturbation, the solution turns back to it in the long timerun (t → ∞) - the only thing that may have changed is a phase shift.Even though the model is quite simple, it can be applied to real world data. A nice example is the “Aus-tralian sheep-blowfly” - under laboratory conditions, their population size shows up the same oscillations- the model (1.12) can be fitted very nicely to the data.

Remark: There are, as always, different approaches for such a population growth. E.g., one can alsoinclude a kind of “delay in maturation”, which leads to the so-called blowfly equation:

N = b(N(t − T ))N(t − T ) − µ(N(t))N(t).

The mathematical properties of this equation are quite different to Hutchinson’s model introduced above(e.g., the blowfly equation can be derived from age-structured models; and also the oscillatory behaviouris somewhat different).

Literature for the next subsections: [20, 21]

1.5.2 The classical Lotka-Volterra model

From lecture part I we already know the classical predator-prey model of Lotka-Volterra:

N1 = ε1N1 − αN1N2

N2 = −ε2N2 + βN1N2,

where N1 denotes the prey and N2 the predators. Except for (0, 0), there is the equilibrium point ( ε2

β , ε1

α ),from which we know that it is a “centre”, with closed orbits around. Such oscillations can also be observedin natural systems. But the closed orbits in the Lotka-Volterra model can be easily “disturbed” - thenthe solution stays on another closed orbit. It is also known (e.g. from lecture part I), that modifiedpredator prey models show a different behaviour, e.g. if we consider

N1 = ε1N1

(

1 − N1

K

)

− αN1N2

N2 = −ε2N2 + βN1N2,

then the coexistence point is a stable spiral. Goal is to find a model which has a limit cycle.From a biological point of view, it is realistic to assume that N1 is reduced instantaneously by N2, but theincrease of N2 by taking up food (N1) should be a delayed effect. So, it would be better to incorporatevalues of N1 at earlier times. This can be done by a delay equation. Volterra introduced the followingapproach for that problem:

N1 = ε1N1 − αN1N2

N2 = −ε2N2 + βN2

∫ t

−∞F (t − τ)N1(τ) dτ.

One can of course also reformulate the second equation as

N2 = −ε2N2 + βN2

∫ ∞

0

F (z)N1(t − z) dz.

If we normalise F in such a way that∫ ∞

0

F (z) dz = 1,

18

then ( ε2

β , ε1

α ) is still a stationary point (with coexistence).

In the next step, we try to reduce that system (with the delay) to a system of (coupled) ordinarydifferential equations. In case of a simple form of the time delay function F (z), this can be done exactly.Assumption:

F (z) = a · exp(−az)

Let

N3 =

∫ t

−∞F (t − τ)N1(τ) dτ.

Obviously, that N3 satisfies

N3 = aN1 − a

∫ t

−∞a · exp(−a(t − τ))N1(τ) dτ

= a(N1 − N3).

So, we can reformulate our original system:

N1 = ε1N1 − αN1N2

N2 = −ε2N2 + βN2N3 (1.13)

N3 = a(N1 − N3).

The (nonzero) stationary solution of (1.13) is ( ε2

β , ε1

α , ε2

β ).For checking the stability, we consider the linearisation, e.g. the Jacobian matrix, as usual. In its generalform, it reads

J =

ε1 − αN2 −αN1 00 −ε2 + βN3 βN2

a 0 −a

Using the coordinates of the stationary point, we get:

J =

0 −α ε2

β 0

0 0 β ε1

αa 0 −a

.

We can use now the Routh-Hurwitz criteria to check if the coexistence point is stable or not, thus, weneed the characteristic polynomial:

det

−λ −α ε2

β 0

0 −λ β ε1

αa 0 −λ − a

= 0 ⇔ λ2(−λ − a) − ε1ε2a = 0

⇔ −λ3 − aλ2 − ε1ε2a = 0

⇔ λ3 + aλ2 + ε1ε2a = 0.

We can take the special case k = 3, then the stability conditions are: a > 0 (satisfied), ε1ε2a > 0(satisfied) and 0 > ε1ε2a (not satisfied). Thus, the coexistence point is unstable, as desired.The same method can be used also in case of a slightly more complicated function F (and will then leadto larger sets of these equations).Another method is based on the following idea: Instead of taking a specified delay function F (z), we onlyuse its moments:

γ =

∫ ∞

0

F (z)z dz

δ =

∫ ∞

0

F (z)z2

2dz.

Of course, we will haven then only an approximate method which is valid for small values of γ, thatcorresponds to short time delays. We use the notation

N2 = −ε2N2 + βN2

∫ ∞

0

F (z)N1(t − z) dz.

19

For N1(t − z), the Taylor expansion about z = 0 is used, thus

N2 ≈ −ε2N2 + βN2

∫ ∞

0

F (z)[N1(t) − N1(t) · z] dz

= −ε2N2 + βN2N1 − βN2 · N1 ·∫ ∞

0

F (z)z dz

= −ε2N2 + βN1N2 − βγN2 · N1

as first order approximation. Substituting N1 yields then as complete system:

N1 = ε1N1 − αN1N2

N2 = −ε2N2 + β(1 + γε1)N1N2 + αβγN1N22

Obviously, we can regain the instantaneous model by letting γ → 0. The (non-zero) stationary pointstays at ( ε2

β , ε1

α ).

1.5.3 May’s Predator-Prey model

Another approach for a predator-prey model, taking into account a limited prey population and a limitedpredator appetite, was introduced by May and looks as follows :

N1 = ε1N1

(

1 − N1

K

)

− AN1N2

N1 + B(1.14)

N2 = ε2N2

(

1 − N2

CN1

)

. (1.15)

The equilibrium point can be computed to be

N∗1 = B(1 − α − β + R)/(2β), N∗

2 = CN∗1 ,

whereα = AC/ε1, β = B/K, R =

√

(1 − α − β)2 + 4β.

As usual, one can consider the corresponding linearised system (the Jacobian matrix) and determine thecharacteristic equation for the eigenvalues. The Jacobian matrix (in the equilibrium point) reads

J =

(

− ε1N∗

1

K +AN∗

1 N∗

2

(N∗

1+B)2 − AN∗

1

N∗

1+B

Cε2 −ε2

)

=:

(A11 A12

A21 A22

)

,

resulting in the characteristic equation:

λ2 − (A11 + A22)λ + (A11A22 − A12A21) = 0,

where A12 < 0, A21 > 0, A22 < 0, and A11A22 − A12A21 > 0 (for details, see May [22] - just standardcomputations). Referring again to the Routh-Hurwitz criteria, the equilibrium point is stable if and onlyif A11 + A22 < 0. Using again an exponential time delay (F (z) = a · exp(−az)), i.e. choosing

N3 =

∫ t

−∞F (t − τ)

1

N1(τ)dτ

; N3 = a1

N1− a

∫ t

−∞a exp(−a(t − τ))

1

N1(τ)dτ

= a

(1

N1− N3

)

yields the following two equations, instead of equation (1.15):

N2 = ε2N2

(

1 − N2N3

C

)

,

N3 = a

(1

N1− N3

)

.

20

For that system, the corresponding linearised system yields the following characteristic equation:

λ3 − λ2(A11 + A22 − a) + λ(A11A22 − a(A11 + A22)) + a(A11A22 − A12A21) = 0. (1.16)

In the limit a → ∞, the characteristic equation of the “originally delayed” model goes over to that of theoriginal ODE model. Vice versa, considering the limit a → 0, the characteristic equation (1.16) tends to

λ2 − λ(A11 + A22) + A11A22 = 0.

(N3 is just a constant here). In that special case, we find stability in case of

A11 + A22 < 0 and A11A22 > 0.

Under the already given property (A22 < 0), we thus need further that A11 < 0.Next step: Consider the characteristic equation (1.16) and try to apply the Routh-Hurwitz conditions.We find:

a1 = −A11 − A22 + a

a2 = A11A22 − a(A11 + A22)

a3 = a(A11A22 − A12A21).

Thus, the Routh-Hurwitz conditions read:

• Condition 1: a1 = −A11 − A22 + a > 0

• Condition 2: a3 = a(A11A22 − A12A21) > 0 (which is always satisfied, see above)

• Condition 3: a1a2 > a3 ⇔ (−A11 − A22 + a)(A11A22 − a(A11 + A22)) > a(A11A22 − A12A21)

So, the stability behaviour can be different, according to the actual parameters. From the given conditions,one can get a quadratic equation in a, the roots (and their position) of this quadratic equation influencethe stability behaviour. Possible cases are:

• No real roots ; unstable equilibrium point for all a

• No real roots ; stable equilibrium point for all a

• One real root, ; stable equilibrium point for a → ∞, unstable equilibrium point for a = 0, i.e. inthis case, the time delay induces instability, as desired!

• Two real roots, with an interval of instability for a

21

Chapter 2

Diffusion processes

2.1 Random walk

2.1.1 Random Motion and the Diffusion Equation

Literature: [6, 41]

The Diffusion equation can also be derived from the so-called Random walk / the “Brownian motion”.Here, we consider the following 1D situation (a particle on a 1D grid):

x x+ xx− x∆ ∆

λ rlλ

For the Brownian motion: Per time unit τ , the particles move left or right with an average step length∆x, starting from some x. Which direction they choose, is determined randomly, there is no “connection”between the steps (thus, it is also called “uncorrelated random walk”. Here, we assume equal probabilitiesof moving left (λl) or right (λr), i.e. λl = λr = 1

2 .Let ξk ∈ −∆x,∆x be the shift in the time interval [(k − 1)∆t, k∆t], 1 ≤ k ≤ n and (without loss ofgenerality) x = 0 the starting point, then the complete shift after time t = n · ∆t is xn =

∑nk=1 ξk. If r

steps are made per time unit, then each step of length ∆x needs ∆t = 1r of the time unit and n steps

need n · ∆t = nr of the time unit. The position of the particle, xn at time t = n · ∆t can be interpreted

as a random variable.Let u(x, t) = P (xn = x) at time t = n ·∆t be the probability, that the particle is at position x at time t.It can be given explicitely by the binomial distribution. The probabilities for both directions are equal.x can be displayed as x = m · ∆x, where m ∈ Z (i.e., it is positioned on the grid). Let nl the number ofsteps to the left and nr the number of the steps to the right, performed by the particle. Obviously,

n = nl + nr.

Assuming that the particle is at m · ∆x after n steps, a further condition reads

nr − nl = m.

Taken together, it yields:

2nr = n + m ⇔ nr =n + m

2.

Then we can apply the probability function which belongs to the binomial distribution:

P (xn = m · ∆x) =

(n

nr

) (1

2

)nr

·(

1

2

)n−nr

=

(1

2

)n

·(

nn+m

2

)

=1

2n·(

nn+m

2

)

.

22

The expected value E(xn) and the variance V (xn) can be gained from the binomial distribution ordetermined directly:

E(xn) = E

(n∑

k=1

ξk

)

=

n∑

k=1

E(ξk)

=

n∑

k=1

(−∆x) · P (ξk = −∆x)

︸ ︷︷ ︸

= 12

+(∆x) · P (ξk = ∆x)︸ ︷︷ ︸

= 12

= 0,

and

V (xn) = V

(n∑

k=1

ξk

)

=n∑

k=1

V (ξk)

=

n∑

k=1

(−∆x)2 · P (ξk = −∆x)

︸ ︷︷ ︸

= 12

+(∆x)2 · P (ξk = ∆x)︸ ︷︷ ︸

= 12

= (∆x)2 · n= (∆x)2 · n

t· t

= (∆x)2 · r · t.

Thus, we find√

V (xn) ∼√

t:

~ sqrt(t)

t

This means: The particle is always “expected” to be in the starting point (due to the “symmetry” of thechoice of direction), but for a growing t, the probability is decreasing (but slower than t) that the particleis really at its starting point.

Now, we want to leave that discrete model and go to a continuous model for the Brownian motion,by letting the step size ∆x tend to zero and the number of steps per time unit r → ∞, in such a waythat lim∆x→0,r→∞(∆x)2r = 2D, D 6= 0.The number of particles in [x, x + ∆x] at time t is described by u(x, t)∆x. The corresponding discreteequation reads:

u(x, t + ∆t) = u(x, t) + λru(x − ∆x, t) − λru(x, t) + λlu(x + ∆x, t) − λlu(x, t). (2.1)

Generally, the Taylor-series expansions hold true:

u(x, t + ∆t) = u(x, t) +∂u

∂t∆t +

1

2

∂2u

∂t2∆t2 + . . . (2.2)

u(x ± ∆x, t) = u(x, t) ± ∂u

∂x∆x +

1

2

∂2u

∂x2∆x2 ± . . . (2.3)

23

Inserting (2.2) and (2.3) into (2.1) and using λl = λr = 12 yields

u(x, t) +∂u

∂t∆t +

1

2

∂2u

∂t2∆t2 + . . . =

1

2

[

u(x, t) − ∂u

∂x∆x +

1

2

∂2u

∂x2∆x2 + . . .

]

+1

2

[

u(x, t) +∂u

∂x∆x +

1

2

∂2u

∂x2∆x2 + . . .

]

⇔ ∂u

∂t∆t +

1

2

∂2u

∂t2∆t2 + . . . =

1

2

∂2u

∂x2∆x2 +

1

4

∂4u

∂x4∆x4 + . . .

This equation is divided through by ∆t. Consider the limit ∆t → 0, ∆x → 0 in such a way that

(∆x)2

2∆t= D = const.

This leads us to∂u

∂t=

(∆x)2

2∆t

∂2u

∂x2= D

∂2u

∂x2, (2.4)

which corresponds to the Diffusion equation. From lecture part I, we already know: The solution of theinitial value problem (2.4) with the initial condition

u(x, 0) = δ(x)

(the Dirac δ “function”) reads

u(x, t) =1√

4πDte−

x2

4πDt . (2.5)

Remark: The assumption that (∆x)2

2∆t tends to a finite limit D 6= 0, if ∆x and ∆t tend to zero, implicates,

that the same limit yields ∆x∆t → ∞. This means: The velocity of a particle which performs Brownian

motion, is infinitely large.This fact also follows from the solution of the initial value problem, (2.5): It is the (Gaussian) normaldistribution, with expected value 0 and the variance 2Dt. Obviously, for all x ∈ R and t > 0, it isu(x, t) > 0, thus, there is a positive probability that the particle is located in a neighbourhood of aposition x, which may be located arbitrarily far away from the origin, as soon as t > 0. Already Einstein,who examined the connection between Brownian motion and diffusion equation first, recognised that thediffusion equation yields a valid model only for large t.

2.2 Correlated random walk

In the following, u(x, t, s) will be interpreted as particle density. The state of a particle is given by itsspatial position x ∈ R and its velocity s ∈ R. Assumed that the velocity does not change, then thedevelopment of the particle density can be described by the following equation:

∂u

∂t+ s · ∂u

∂x= 0. (2.6)

New assumption: The particles stop at a randomly determined time point and choose a new velocity.This time point is chosen by a Poisson process, which means for the probability of a change of velocityduring the time interval [t, t + ∆t]:

P (Change in [t + ∆t]) = µ · ∆t + o(∆t)

with parameter µ.Compared to the Brownian motion this means: There is a positive correlation between two adjacentsteps; the particles prefer to keep the old direction. This model is thus called “correlated random walk”.Let K(·, s) the density of the new velocity, s the former velocity. K has the following properties: K(s, s) ≥0 and

∫ ∞−∞ K(s, s) ds = 1. Then, the equation reads:

ut + s · ux = −µu + µ

∫ ∞

−∞K(s, s) · u(t, x, s) ds. (2.7)

Now we consider a special case: The absolute value of the velocity is constant, γ, only the direction rcan change, r ∈ −1, 1. Thus, the velocity s can be described by s = γ · r. Similar to above, a particlechanges its direction of velocity at a time point which is determined by a Poisson process. A possiblepath of a particle looks as follows:

24

γ

−γ

γ−γ

γ

t

x

Then, the particle density can be written dependent on the direction instead of the velocity: u(x, t, r).Let u+(x, t) = u(x, t, 1) be the particle density of the particles running to the right; analogously letu−(x, t) = u(x, t,−1) be the particle density of the particles running to the left. By definition, for allt ∈ R and x ∈ R it is

u+(x, t) ≥ 0 and u−(x, t),

and for all t ∈ R: ∫ ∞

−∞

(u+(x, t) + u−(x, t)

)dx = 1.

Due to (2.6), the wave running to the right, u+(x, t) satisfies

u+t + γu+

x = 0,

analogously for the wave running to the left, u−(x, t):

u−t − γu−

x = 0.

u+(x, t) can be denoted as “wave running to the right”, since a wave can be written in the form ofφ(x − γt) and satisfies thus exactly the differential equation:

∂φ(x − γt)

∂t+ γ · ∂φ(x − γt)

∂x= −γ · φ′(x − γt) + γ · φ′(x − γt) = 0.

In the same way, the wave running to the left satisfies:

∂φ(x + γt)

∂t− γ · ∂φ(x + γt)

∂x= γ · φ′(x + γt) − γ · φ′(x + γt) = 0,

i.e. u−(t, x) can be called running to the left.In the considered special case, equation (2.7) simplifies to

u+t + γu+

x = −µu+ + µu− (2.8)

u−t − γu−

x = −µu− + µu+. (2.9)

One can consider several special cases; they may e.g. yield again the Brownian motion, or the waveequation (for some details, see the exercises).

2.2.1 Generalisation

In the case of an uncorrelated random walk, one gets the diffusion equation. System (2.8),(2.9) (or acorresponding system with transformed variables) replaces that in the case of a correlated random walk.We are looking for meaningful generalisations of the scalar reaction-diffusion equation ut = Duxx + f(u).Assuming symmetry, the system should be of the form

u+t + γu+

x = µ(u− − u+) + F (u+, u−)

u−t − γu−

x = µ(u+ − u−) + F (u−, u+).

25

If the production/deletion (i.e. reaction) does not depend on the direction of motion, then F (u+, u−) +F (u−, u+) may be displayed as a function f(u) on u = u+ + u−. If that term is distributed on bothdirection, the resulting simple system reads

u+t + γu+

x = µ(u− − u+) +1

2f(u)

u−t − γu−

x = µ(u+ − u−) +1

2f(u).

2.2.2 Boundary conditions

We consider again the 1D area G = [0, l]. Boundary conditions can be prescribed only for ingoingparticles (this can be seen by using the corresponding telegraphers equation - see the exercises - andtheir characteristic curves; so originally, the boundary conditions can be prescribed in direction of thecharacteristic curves, but these correspond here exactly to the u+ particle density on the left boundaryand the u− particle density on the right boundary), i.e.:

0 l

x

u+(0,t)

u−(l,t)

t

Some special cases:

• Homogeneous Dirichlet condition: All particles, which arrive at the boundary, are absorbed. Ob-viously, in x = 0 only particles of type u− and none of type u+ can appear. Analogously, in x = lonly particles of type u+ (and not u− - those would come from outside, which is not allowed) canappear.

u+(0, t) = 0, u−(l, t) = 0

• Homogeneous Neumann condition: Each particle, which arrives at the boundary, is reflected (i.e.it takes the same velocity, with the opposite direction):

u+(0, t) = u−(0, t), u−(l, t) = u+(l, t).

0 l

x

u+(0,t)

u−(l,t)

t

Neumann

2.3 Porous media equation

For some examples, the Diffusion equation has one important disadvantage: there appears a very fastspread of some particles. How to circumvent this problem?

26

Basic idea:√

αD describes the overall velocity of a population with growth rate α and diffusion constantD. Thus, a reduction of D also reduces its net velocity. Assume now that D = D(u) depends on theactual population density with very small values for small u (only a few particles spread very fast).Construction of the model: Again we use the first law of Fick as a starting point:

d

dtu(x, t) = −div J(x, t).

The flux may depend on the density:J = F (u).

In case of the diffusion equation, we derived F (u) = −D∇u. Now, we assume also

F (u) = −D∇u, but with D = D(u)

(diffusion dependent on u, where D(u) → 0 for u → 0). This leads to

ut = ∇(D(u)∇u) (2.10)

which is called porous media equation. Typical behaviour of D(u):

u

D(u)

• D(0) = 0

• D(u) strictly increasing

• D(u) globally bounded

Originally, the porous media equation was introduced to describe the spread of water in “porous media”,i.e. in material with many very small caverns. There, only high pressure (corresponding to a high localdensity) pushes the water forward, otherwise it stays more or less.It can be shown (under appropriate assumptions), that the total mass is conserved, i.e.

d

dt

∫

u(x, t) dx =

∫

∇(D(u(x, t)∇u(x, t)) dx = 0.

Special example:

Literature: [27]

Here, we consider the following function for D(u):

D(u) = D0

(u

u0

)m

.

(where m > 0, D0 and u0 are positive constants). Although this function is not bounded (as requiredabove), it yields a reasonable model equation, since the main effect we are looking for should depend onthe behaviour of D(u) for small densities u.

Possible application in mathematical biology: Spread of insects, e.g. for a special type of mosquitoswarm with m = 1/2 (see [31]). Just the dispersal, without any growth or death term is considered.In 1D, the equation

ut(x, t) = (D(u)ux))x

27

with the initial condition u(x, 0) = Q · δ(x) (this means, that the insects are released at the origin at timet = 0) can be shown to have the solution

u(x, t) =

u0

λ(t)

[

1 −(

xr0λ(t)

)2]1/m

for |x| ≤ r0λ(t)

0 else ,

where

λ(t) =

(t

t0

)1/2m

r0 =QΓ(1/m + 3/2)√πu0Γ(1/m + 1)

t0 =r20m

2D0(m + 2);

By definition, Q is the initial number of insects. The Γ-function is defined as

Γ(z) =

∫ ∞

0

tz−1e−t dt for z ∈ C with Re(z) > 0

(This definition can be extended to the whole complex plane, except for the non-positive integers; forpositive integers n, it is Γ(n) = (n − 1)!)r0 can also be determined in the following way:

Q =

∫ ∞

−∞u(x, t) dx = r0

∫ r0λ(t)

−r0λ(t)

u0

r0λ(t)

[

1 −(

x

r0λ(t)

)2]1/m

dx = r0

∫ 1

−1

u0[1 − x2]1/m dx.

The time course of the solution looks like

u

x

t1

t2>t1

wave front

Obviously, the solution function is not differentiable in x = ±r0λ(t) (the wave front, i.e. where thesolution meets the x-axis). Since also D(u) becomes zero there, the equation is satisfied formally, but ingeneral, it is necessary to introduce definitions of weak solutions.Properties of this solution: As it is typical for solutions of the porous media equation, it has a finite sup-port, which disperses with a finite maximal velocity. But analogously to the random walk (as introducedabove), there is a stochastic process which corresponds to the porous media equation - the particles theremay have locally an arbitrary high speed within the support.

Also the radially symmetric 2D problem can be easily formulated:

ut =

(D0

r

)∂

∂r

[

r

(u

u0

)m∂u

∂r

]

and the explicit solution can be computed to be:

u(r, t) =

u0

λ2(t)

[

1 −(

rr0λ(t)

)2]1/m

for r ≤ r0λ(t)

0 else ,

,

28

where

λ(t) =

(t

t0

)1/2(m+1)

,

t0 =r20m

4D0(m + 1)

r20 =

Q

πn0

(

1 +1

m

)

.

It can be shown: for m → 0 (which means D(u) → D0), also the solution tends to the usual solution forthe problem with constant solution.

2.4 Another type of diffusion: Biofilm model

Literature: [5, 4]

Biofilms consist of a “matrix”, i.e. a kind a slime, where microorganisms as bacteria are embedded.It is well-known from very different places, where it occurs:

• on rocks in rivers and pools

• in the shower at home

• the dental plaque on the teeth

• in the lung - e.g. by pathogenic bacteria (difficult for antibiotic therapy)

• on the surface of roots of plants ...

The biofilm itself is produced by the microorganisms, mainly in aquaeous systems and adherent to asurface. Its shape is not just a homogeneous film-like layer, but can also form structures like “mushrooms”(influenced by the availability of nutrients etc.).The model for the biofilm formation consists of a quasi-linear diffusion equation, for the development of aspatially structured biofilm. The variable u describes the biomass density. The diffusion coefficient D(u)is density-dependent, with two “degeneracies”:

1. D(0) = 0 (as in the porous media equation)

2. limu→1 D(u) → ∞, i.e. before the biofilm density approaches is maximum (u = 1; we assume u tobe normalised with respect to the maximum biomass density), it is forced to spread.

I.e. the graph of D(u) looks qualitatively as follows:

D(u)

u

1

The domain Ω ⊂ Rd, d = 1, 2, 3 is splitted up into two parts:

Ω1(t) := x ∈ Ω : u(x, t) = 0, the surrounding aquatic environment, where no biomass is presentΩ2(t) := x ∈ Ω : u(x, t) > 0, where biofilm is presentThe evolution of the biofilm is described by the following equation:

ut = ∇(D(u)∇u) + ku,

where

D(u) = δub

(1 − u)a, a, b ≥ 1 ≫ δ > 0.

k describes the production rate of the biofilm. In case of non-limited nutrients, k can be taken as apositive constant (leading to a homogeneous biofilm morphology); in case of limited nutrients, it maydepend on time and space, i.e. k(x, t).

29

Chapter 3

Reaction-Diffusion equations

3.1 Travelling waves

Literature: [6, 27]

Question: Is it possible for a species to invade into new habitats and how does this work?

Approach: Travelling wave solutions of a reaction-diffusion equation.

The goal is to find solutions of the following form,

u(x,t)

x

c

which can describe such an invasion into a new habitat, where this move appears with a constant speedc (the so-called wave speed), i.e.

u(x, t) = φ(x − ct).

The new variable z := x− ct denotes the wave variable and φ(z) is the wave profile. The travelling waveansatz includes conditions at ±∞ instead of “classical” boundary conditions:

u(x, t) = φ(x − ct), φ(−∞) = 1, φ(+∞) = 0, (3.1)

which means that the population has reached its capacity for x → −∞ (normalised to 1), and nopopulation has arrived yet for x → +∞. (3.1) leads to:

∂

∂tu(x, t) = −cφ′,

∂2

∂x2u(x, t) = φ′′.

As an example for a reaction-diffusion equation, we take here again the Fisher equation

ut = Duxx + µu(1 − u);

together with the travelling wave ansatz, this yields

−cφ′ = Dφ′′ + µφ(1 − φ),

which corresponds to a second order ODE and we can transform it into a 2D system of first order ODEsby introducing a new variable ψ := φ′:

φ′ = ψ

ψ′ = − c

Dψ − µ

Dφ(1 − φ). (3.2)

30

The stationary states here are P1 = (0, 0) and P2 = (1, 0). The general Jacobian reads

J =

(0 1

− µD + 2 µ

Dφ − cD

)

.

Applied on the stationary states, we get

• J(0, 0) =

(0 1

− µD − c

D

)

, i.e. det = µD > 0 and tr = − c

D , so at least, (0, 0) is stable. The

discriminant is tr2−4det = c2

D2 −4 µD , thus, for c < 2

√Dµ we have a stable spiral and for c > 2

√Dµ

a stable node.

• J(1, 0) =

(0 1µD − c

D

)

, i.e. det = − µD < 0, so, there is always a saddle in (1, 0).

For the wave profile, we need φ(−∞) = 1 and φ(+∞) = 0, and also ψ(−∞) = ψ(+∞) = 0. Consideringthe phase portrait of (3.2), we are looking for a connection from the stationary point (1, 0) to (0, 0).Since φ describes a population density, negative values are not biologically meaningful. In the case ofc < 2

√Dµ, (0, 0) is a stable spiral:

φ

ψ

obviously leading to negative values for φ (this case corresponds to a so-called oscillating front). So,c∗ = 2

√Dµ is the minimal wave speed for the existence of a wavefront solution; for c > c∗, (0, 0) is a

stable node.

φ

ψ

A proof can be found in [15].

Until now, we derived only a criterion which gives information about the possible wave speeds, butnothing about the appearance of propagation speeds in the reality. It could be shown under reasonableassumptions that a solution of the Fisher equation will evolve into a travelling wave with minimal speedc∗ (details are left out here).

Remark 1 For linear parabolic equations like the standard diffusion equation, there are no physicallyrealistic travelling wave solutions; for that purpose it is necessary to “add” a nonlinear term; e.g. theconsidered reaction-diffusion equations can exhibit such solutions. Of course, their form depends on theadditional term f(u).

3.2 Spatial spread of an epidemic

Literature: Murray II [28]

For an better understanding and control of diseases and epidemics, it is not sufficient to study thetemporal development only, but also the geographic spread may play a role.Question: How can we include and quantify spatial effects to epidemic models?

31

3.2.1 Simple model for the spatial spread

Assumption: We consider here a population which consists only of susceptible (S(x, t)) and infective(I(x, t)) individuals (see also the basic SIR-model in the first part of the lecture). Since we want to studyalso spatial effects, S and I depend on time t (as usual) and on a space variable x.The spatial dispersal is modelled by simple diffusion; and for the moment we assume that both, S andI have the same diffusion coefficient D. Further assumptions concern the mortality: There is a disease-induced mortality rate aI, thus, 1/a is the life expectancy of an infective. No recovery is assumed in thissimple approach. This yields as a basic model:

∂S

∂t= −rIS + D∆S, (3.3)

∂I

∂t= rIS − aI + D∆I. (3.4)

In the following, we are interested in the following problem: A certain number of infected individuals is“introduced” into a population. For that population, an initial homogeneous susceptible density S0 isassumed; we want to determine the “geotemporal” spread of the disease.For that purpose, we shrink our problem down to a 1D situation (for the moment). As usual, we cannondimensionalise our original system (3.3, 3.4) by using:

I∗ =I

S0, S∗ =

S

S0, x∗ =

(rS0

D

)1/2

x, t∗ = rS0t, λ =a

rS0

which leads to

∂S

∂t= −IS +

∂2S

∂x2, (3.5)

∂I

∂t= IS − λI +

∂2I

∂x2. (3.6)

(The asterisk are dropped again, for simplicity). Remark that there is just one parameter left, λ.Next step: We are looking for a travelling wave, i.e. existence and speed of propagation. The usualansatz for a travelling wave solution (a wave with constant shape which travels in positive x-direction)reads

I(x, t) = I(z), S(x, t) = S(z), where z = x − ct;

c is the wave speed, which has to be determined. This ansatz is substituted into (3.5), (3.6) and yieldsthe following ODE system:

I ′′ + cI ′ + I(S − λ) = 0

S′′ + cS′ − IS = 0.

(′: differentiation with respect to z).The adequate conditions in ±∞ read

I(−∞) = I(∞) = 0, 0 ≤ S(−∞) < S(∞) = 1;

for I, it corresponds to a pulse wave of infectives, and this wave propagates into the population ofsusceptibles. I.e., the graph looks like

z

I

S

1

Epidemic wave

We consider now the equation for I,

I ′′ + cI ′ + I(S − λ) = 0

Near the leading edge of the wave (there, it is S → 1, I → 0):

I ′′ + cI ′ + (1 − λ)I ≈ 0.

32

For the solution, we take the following ansatz:

I(z) = exp[(

−c ± c2 − 4(1 − λ)1/2) z

2

]

.

Indeed, this function is a solution of the given ODE, as can be seen by computing

I ′(z) =1

2

(

−c ± c2 − 4(1 − λ)1/2)

I(z)

I ′′(z) =1

4

(

−c ± c2 − 4(1 − λ)1/2)2

I(z)

=1

4

(

−c ± 2cc2 − 4(1 − λ)1/2 + c2 − 4(1 − λ))

I(z)

=1

2

(

c2 ± cc2 − 4(1 − λ)1/2 − 2(1 − λ))

I(z).

We require I(z) → 0 with I(z) > 0, so the solution shouldn’t oscillate around I = 0 (otherwise, therewould be I(z) < 0 for some z). To keep that property, we need c2 − 4(1 − λ) ≥ 0, thus c (wave speed)and λ must satisfy

c ≥ 2(1 − λ)1/2, λ < 1.

Obviously, if λ > 1, it is impossible to find a wave solution. In the original dimensions, this necessarythreshold condition for the propagation of an epidemic wave reads

λ =a

rS0< 1.

Remark: That condition corresponds exactly to the condition for the existence of an epidemic in thespatially homogeneous situation (see Lecture part I [18]). In our context, it can be interpreted in severalways: E.g., Sc = a

r can be interpreted as minimum critical population density, allowing an epidemic waveto occur. Or, vice versa, if a population size S0 and mortality rate a are given, then we can define acritical transmission coefficient rc = a/S0. As long as r < rc, the spread of the infection is prevented.Last possibility, threshold mortality ac = rS0; if a > ac, mortality is “too high” and by that prevents anepidemic.Coming back to the original dimensions, then the wave velocity, denoted by V , reads (using the minimumwave speed c = 2(1 − λ)1/2):

V =

(D

rS0

) 12

1rS0

· c = (rS0D)1/2c = 2(rS0D)1/2

(

1 − a

rS0

)1/2

,1

rS0< 1.

Example: Spread of the Black Death in Europe 1347-1350The so-called Black Death, is caused by Bacillus pestis and is transmitted by fleas, mainly from rats tohumans. The approximate chronological spread in Europe can be seen on a map in [28]. The epidemicstarted in Italy; there it was introduced via a ship from the East (where the disease was present alreadyfor years). From there, it spread through whole Europe. We want to apply the above-mentioned modelto that disease, for that purpose we have to estimate parameter values (as far as possible).Available data:

• around 85,000,000 people in Europe in 1347; mean population density S0 ≈ 50/miles2

• Diffusion coefficient: As computed in lecture part I [18], pure diffusion allows to cover a distance ofL miles in time O(L2/D). Rough assumption: News etc. travelled with approx. 100 miles / year(due to limited communication possibilities at that time)

; D ≈ 104 miles2

year

• Estimate for the “transmission efficiency” r by Noble (1974): r ≈ 0.4miles2

year

• Mortality rate: From the assumption of an average infectious period p ≈ 2 weeks, we get

a =ln 2

1/24year≈ 15/year ; λ =

a

rS0≈ 0.75

33

Using these estimates, we get for the speed of propagation:

V = 2(rS0D)1/2

(

1 − a

rS0

)1/2

≈ 140miles/year.

Comparing it to the historic observations, which yield a speed of 200.400 miles/year, the estimate is nottoo bad (even though the very rough assumptions).

3.3 Facilitated Diffusion

Literature: [36]

We consider the following biological problem: Oxygen (O2) has to be transported into the muscle fi-bres, and the protein Myoglobin (Mb) helps in this process. Chemically, the process just consists ofbinding and unbinding of the O2 to the Mb:

O2 + Mbk+

k−

MbO2

The “reaction product” is Oxymyoglobin (MbO2).The following variables are introduced:

s = O2, e = Mb, c = MbO2.

For the modelling approach, the muscle fibre is assumed to be 1D (in the interval [0, L]), Mb and MbO2

cannot leave it. Only the O2 itself can pass the boundaries. So the corresponding boundary conditionsread:

s(0) = s0, s(L) = sL (≪ s0)

∂e

∂x(0) = 0,

∂e

∂x(L) = 0

∂c

∂x(0) = 0,

∂c

∂x(L) = 0

(the boundary conditions are constant in time).Inside the muscle fibre, all three components can diffuse, and of course the above-mentioned reactiontakes place. This yields the following system of reaction-diffusion equations:

∂s

∂t= Ds

∂2s

∂x2+ k−c − k+se

∂e

∂t= De

∂2e

∂x2+ k−c − k+se

∂c

∂t= Dc

∂2c

∂x2− k−c + k+se.

Mb and MbO2 have more or less the same size, so it is reasonable to assume De = Ec. Let us considerthe steady-state problem:

Dssxx + k−c − k+se = 0 (3.7)

Deexx + k−c − k+se = 0 (3.8)

Dccxx − k−c + k+se = 0. (3.9)

Due to De = Dc, we have obviously (e + c)xx = 0, thus e + c is a linear function in x. The boundaryconditions prescribe the first derivative to be zero, so the only remaining possibility is that

e + c = const =: e0.

Adding up equations (3.7) and (3.9) yields:

(Dssx + Dccx)x = Dssxx + Dccxx = 0,

34

so Dssx + Dccx is also constant. We call this constant −J , i.e.

Dssx + Dccx = −J.

Remark: J is the total oxygen flux (including unbound O2 and bound in MbO2).Let f(x) = Dss(x) + Dcc(x). We already know from above:

f ′(x) = −J,

so it follows directly thatf(0) − f(L) = JL,

and this again yields

J =f(0) − f(L)

L=

Dss(0) + Dcc(0) − Dss(L) − Dcc(L)

L

=Ds

L(s0 − sL) +

Dc

L(c0 − cL).

In the next step, we assume that Ds is very small compared to the other magnitudes appearing in

Dssxx + k−c − k+s(e0 − c) = 0.

Thus, doing a quasi-steady state approximation leads to

c =k+se0

k− + k+s= e0

s

K + s,

where K = k−/k+. The advantage now is, that we can replace c0 in terms of s0 and cL in terms of sL.Then we can reformulate the flux equation as follows:

J =Ds

L(s0 − sL)

︸ ︷︷ ︸

“normal O2 flux”

+Dc

L

(s0

K + s0− sL

K + sL

)

e0

︸ ︷︷ ︸

“additional flux dependent on Mb diffusion”

.

Remark: sK+s is monotonely increasing in s, thus, due to s0 > sL, the second term is positive. This

means: The total oxygen flux with myoglobin is higher than without myoglobin, so there is really anadvantage of using myoglobin.In principle, it is possible to solve the system for c(x) and s(x) (left out here, but can be found inKeener-Sneyd [16]). The corresponding graphs look qualitatively in the following way:

x x

Oxy

gen

conc

entr

atio

n

Oxy

gen

flux

free

bound

total

bound oxygen flux

free oxygen flux

Remark: The bound oxygen flux and the free oxygen flux add up to the constant J !Interpretation: The binding of oxgen to myoglobin leads to a lower level of free oxygen near the leftendpoint. The boundary condition requires the concentration to be s0, but since the diffusion tends toequalise the concentration inside and outside, it forces a stronger flow into the cell (at the left end).At the other end, the opposite effect appears: Due to the MbO2, there is a continuous “additionaldelivery” of O2 and more can flow out (again the diffusion wants to equalise the levels, now with lowlevel of O2 outside).

35

3.4 Morphogenesis / Pattern formation

Literature: e.g. [12, 6, 33]

Developmental biology contains three fundamental aspects: The control of cell growth, the cellular dif-ferentiation and the morphogenesis. The latter one deals with the shape, pattern or form in an organism.The underlying processes are quite complex, since they cover many different levels - from molecular up tomulticellular processes within the individual. Apart from D’Arcy Thompson, mainly Alan Turing (well-known from the so-called Turing machine) started to introduce modelling approaches into that field. Hisbasic model describes, how patterns can evolve from a previously homogeneous structure. For that pur-pose, the presence of chemical signals and processes such as diffusion, activation and deactivation wereassumed (for a full understanding, it is necessary to consider (intracellular) molecular processes in a verydetailed way).

Modelling assumptions:

• We consider a chemical reaction with two reactants, which tends to a stable stationary state.

• The reactants may diffuse in medium (with different rates)

• both processes (chemical reaction and diffusion) tend to a stationary state (considered independentlyfrom each other), the diffusion cares for the spatial balancing.

Expected behaviour: The spatially constant distribution, which corresponds to the chemically stablestate in each point, should be stable. But Turing showed that this presumption is not valid generally.

We consider the reaction diffusion equation

ut = D∆u + f(u), (3.10)

on the spatial domain Ω ⊂ Rm, where f : R

m → Rm (continuously differentiable) describes the reaction

of m reactants according to u = f(u). D = (djδjk) is a diagonal matrix with positive entries dj , whichcorrespond to the diffusion coefficients of reactant j. Homogenous Neumann boundary conditions describea situation where no substance can pass through the boundary:

uν = 0 for x ∈ ∂Ω. (3.11)

In the following, we consider the 1D situation, i.e. ut = Duxx + f(u) and the boundary conditionsux(0, t) = ux(l, t) = 0.Let u ∈ R

m be a stationary state of the (well-stirred) reaction, i.e. f(u) = 0. Then, u(x, t) = uis a stationary solution of the equations (3.10), (3.11). The stability analysis is again done by a smallinhomogeneous perturbation of this steady state and a subsequent linearisation. Thus, u(x, t) = u+v(x, t)inserted into (3.10) yields

ut + vt = Duxx + Dvxx + f(u) + f ′(u)v + . . . (3.12)

Omitting higher order terms leads tovt = Dvxx + Av,

where A = f ′(u) means the Jacobian matrix of f at position u. In order to consider the stability of thezero solution of equation (3.12) we introduce a Fourier approach for the function v:

v(x, t) =∞∑

k=0

ukeikπx/l+λkt

with constant vectors uk (the so-called Fourier coefficients), l describes the length of the interval. The realpart of v corresponds to a real solution which is expanded with respect to Cosinus functions. Insertingthat approach into (3.12) and comparing terms of same order k yields

λkuk = Auk − k2π2

l2Duk, k = 0, 1, 2, . . .

There are countable infinitely many equations in Rm, one for each “frequence” or “mode” k. Inserting

µ =k2π2

l2≥ 0

36

and omitting the index k leads to the following matrix eigenvalue problem:

(A − µD)u = λu. (3.13)

For each µ, equation (3.13) exhibits m eigenvalues λ. Solution v ≡ 0 is stable if all these eigenvalues arecontained in the left halfspace. Due to the assumptions (the chemical reaction alone exhibits a stablestationary state) the eigenvalues of A are contained in the left halfspace. For m > 2, this problem is notsolvable satisfactory in general; here, we consider the case of m = 2 in detail. Thus,

A =

(a11 a12

a21 a22

)

, D =

(d1 00 d2

)

Let A be stable, i.e.

tr(A) = a11 + a22 < 0 and det(A) = a11a22 − a12a21 > 0. (3.14)

Let d1, d2 > 0. The matrix A − µD is stable, if

a11 − µd1 + a22 − µd2 < 0 and (a11 − µd1)(a22 − µd2) − a12a21 > 0.

Without loss of generality, it can be assumed that a11 > 0 and a22 < 0. (Otherwise, we do not find thedesired property, if a11 and a22 have the same sign.)One can show:

Proposition 2 Let the matrix A satisfy

a11 > 0, a22 < 0, a11 + a22 < 0, a11a22 − a12a21 > 0.

Then, there are diagonal matrices D = (djδjk), d1, d2 > 0 such that for certain µ the matrix A−µD haseigenvalues with positive real part. The set of all these diagonal matrices is described by the followinginequality:

√

d2

d1>

√

det(A)

a11+

1

a11

√−a11a22. (3.15)

Obviously, it is a11a22 < 0; since the determinant is positive, it is a12a21 < 0. Thus, there are two possibleconfigurations of sign for the matrix A:

A1 =

(+ −+ −

)

, A2 =

(+ +− −

)

. (3.16)

The pattern A1 corresponds to an activator-inhibitor dynamics (but also pattern A2 may lead to a pat-tern formation).

Let A and D satisfy the assumptions of the proposition above, and (µ1, µ2) be an interval such thatA − µD possesses a positive eigenvalue for µ ∈ (µ1, µ2).Firstly, fix k ∈ N and vary l. There is a positive eigenvalue for the chosen k, if

µ1 <

(kπ

l

)2

< µ2, (3.17)

i.e. if

l2 < l < l1, lj =kπ√

µj, j = 1, 2.

If we let l from 0 to +∞, then the k-th term (called mode) in the Fourier series becomes unstable, ifl exceeds l1, and it becomes stable again, if it exceeds l2. This means that the mode k fits only tocertain intervals which shouldn’t be too long or too short. A bifurcation is expected when l exceeds l1.The spatially constant solution of the reaction diffusion equation (3.10) looses its stability; a new stablesolution branches off.Fix l, then condition (3.17) leads to an interval

õ1l

π< k <

õ2l

π

37

for the parameter k. Of course, there are only finitely many k ∈ N (if there is any at all) which satisfythis condition. Concerning perturbations with the corresponding modes, the spatially constant stationarysolution is unstable.

This mechanisms explains, how a spatially homogeneous structure becomes unstable and how stablespatial patterns can show up. In the most simple case, the pattern consists of a polarity: If the modek = 1 becomes unstable, the interval exhibits a positive and a negative end. The Turing model givesthe hint that at least two substances have to react, both of which are diffusing, and the Jacobian matrixneeds a certain sign distribution,

a11 > 0, a22 < 0, a11 + a22 < 0, a11a22 − a12a21 > 0.

In this situation, substance u1 is called “activator”, u2 “inhibitor”. During absence of the inhibitor, theactivator grows according to

u1 = a11u1,

while the inhibitor is governed byu2 = a22u2,

while there is no activator available. But the interaction of both substances yields a stable stationarystate. Considering (3.15), the quotient of the diffusion rates d2/d1 has to be large (i.e. u2 has a broaderrange than u1). Shortly speaking: A destabilisation of a spatially constant distribution appears if awide-range inhibitor meets a short-range activator. Of course, the parameters have to satisfy certainquantitative conditions and the geometry (e.g. the interval length) has to be chosen suitably.

Example for Pattern formation: The slime mold Dictyostelium discoideum;

From F. Siegert and C. J. Weijer, J. Cell Sci. 93, 325-335 (1989); taken from :http://dictybase.org/Multimedia/morphogenesis/morphogenetic movements.htm

Gierer-Meinhardt model

The Gierer-Meinhardt model is one of the explicit systems that show up the phenomenon of Turinginstability. The model was developed in order to explain the exhibition of structures in embryonal tissues.By such models, it is possible, e.g. to explain the patterns on seashells, snails, snakes or butterflies.Basic idea: The activator u is produced by a rate above a certain threshold, which becomes very large forlow concentrations of the inhibitor v. It is degraded exponentially. While there are low concentrationsof the activator, the inhibitor is synthesised only slowly, but very fast for high concentrations of theactivator. It is also degraded exponentially. The diffusion rate of the activator is taken to be much lowerthan that one of the inhibitor. The Gierer-Meinhardt equations read:

ut = a + bu2

v− µu + d1uxx

vt = cu2 − νv + d2vxx,

where the constants a, b, c, µ, ν are positive and d2 ≫ d1 ≥ 0. A homogeneous stationary state satisfies

a + bu2

v= µu and cu2 = νv,

thus, there is only a single stationary state:

u =1

µc(bν + ac), v =

1

cνµ2(bν + ac)2.

38

The corresponding Jacobian reads

J =

(2buv − u2b

v2

2cu −ν

)

=

(

µ bν−acbν+ac − bν2µ2

(bν+ac)22µ (bν + ac) −ν

)

,

i.e.

det J = µν > 0, tr J = µbν − ac

bν + ac− ν.

The interesting case appears for a stable matrix J which can be destabilised by sufficiently different diffu-sion rates. Hence, we claim that the matrix has the above-mentioned sign pattern and the correspondinginequalities are satisfied. a11 is taken to be positive, thus

bν − ac > 0.

A negative trace corresponds toµ(bν − ac) − ν(ac + bν) < 0.

a21 > 0 and a12 < 0 are also satisfied, so matrix J exhibits the sign configuration A1.Concluding, if the quotient of the diffusion rates is large (in the sense of (3.15),

√

d2

d1>

√ν

µ

bν + ac

bν − ac

(

1 +

√

bν − ac

bν + ac

)

,

then the homogeneous stationary state is unstable for suitably chosen lengths of the spatial interval.

39

Chapter 4

Evolutionary dynamics /Populations genetics

There is mainly one famous person who is “connected” with the theory of evolution: Charles Darwin.But the concept of evolution was already known before (e.g. mentioned by Jean-Baptiste Lamarck), eventhough, the mechanism - how can species change and become other species - was introduced by CharlesDarwin (and in the same time by Alfred Russel Wallace).In this lecture, we will deal a little bit with some basic principles (like selection and mutation) of theevolution from a mathematical point of view. A lot of interesting questions arise, e.g. one can deal withevolutionary stable strategies. This can be done by approaches which come from the game theory, thencalled “evolutionary game theory”.Nevertheless, only Mendels experiments on plant heridity could explain the large diversity in a population- detected by Hardy and Weinberg in a simple mathematical approach.

4.1 Replicator equation

Literature: e.g. Nowak [30]

4.1.1 Some basics

Basically, there are three principles for evolutionary dynamics: Replication, selection, and mutation. Anybiological organisms is subject to these principles, independent on the particular details of the organism.In the next steps, we try to introduce very basic mathematical equations to describe replication, selection,and mutation.

Reproduction

Let us consider reproduction of just one species. As already known, there are several possibilities for amathematical description (at the moment, we assume an unlimited growth), e.g. by a discrete system(xn+1 = axn), or by the differential equation for exponential growth

x = rx with the solution x(t) = x0ert.

In that equation, it is assumed that each cell division occurs after an exponentially distributed time(around an average of 20 minutes); i.e. the probability that the cell division happens between time 0 andtime τ is given by 1 − e−rt.For including also cell death in the equation, we assume that the cells die with rate d (i.e. they have anexponentially distributed lifespan, average 1/d) . Thus, we get

x = (r − d)x,

the effective growth rate is then r − d. Obviously, r = d leads to a constant population size, but only an“unstable” one - already small deviations can lead to exponential expansion or decline. From here, we canalso take the definition of the so-called “basic reproductive ratio”, r/d, it describes the expected numberof offspring from one certain individual; the threshold between expansion or decline is then r/d = 1, using

40

that notation.Introducing a maximum carrying capacity into the system yields the logistic equation,

x = rx(1 − x/K), with the solution x(t) =Kx0e

rt

K + x0(ert − 1),

also well-known from part I of the lecture.So, we have an idea how to describe reproduction.

4.1.2 Selection

Now we consider a situation where different types of individuals are involved, reproducing at differentrates.Let us start with a system of two types, called A and B and denoted by the variables x(t) and y(t)(initial values x0 and y0) and with the reproduction rates a and b, respectively. The reproduction ratesare interpreted as the so-called fitness. Neglecting capacities for the moment gives the following equationsfor the two subpopulations A and B:

x = ax

y = by,

with the solutions

x(t) = x0eat

y(t) = y0ebt.

Obviously, the doubling time for subspecies A is ln 2/a and for B it is ln 2/b; the one with the smallerdoubling time corresponds to a faster reproduction, leading to a larger number of individuals in the longtime run.Let ρ(t) = x(t)/y(t) denote the ratio of A over B at time t. Thus, we get

ρ =xy − xy

y2= (a − b)ρ

and the solution (with initial condition ρ0 = x0/y0) reads:

ρ(t) = ρ0e(a−b)t.

Also here, we can see: ρ tends to ∞ if a > b or tends to 0 vice versa, leading to an outcompete of one ofthe species.

Now we change the situation in such a way that we want to keep the total population size on a constantlevel. This can be caused e.g. by a constant maximum carrying capacity of an ecosystem. So, in the fol-lowing let x(t) (y(t)) the relative abundance / frequency of A (B) at time t; obviously we have x+ y = 1.The reproduction rates are still assumed to be a and b, but we need something additional, which ensuresthat x + y = 1:

x = x(a − φ)

y = y(b − φ),

where φ = ax+by (φ is the average fitness of the population). The system can be reduced to one equationby replacing y by 1 − x:

x = x(1 − x)(a − b).

There are two equilibria: x = 0 and x = 1, both correspond to a situation where only one species ispresent, the other not. It depends on the sign of a − b, which equilibrium will be stable (and the otherone unstable). E.g., if a > b, then x > 0 for 0 < x < 1; so the fraction of A will increase up to 1 and thefraction of B converges to 0. This corresponds to the concept of “survival of the fitter”!

The same approach can be used to describe n different types. Let xi(t) denote the frequency and fi

the fitness of type i (the latter one describes the rate of reproduction, as before). The average fitness ofthe population can be computed by

φ =n∑

i=1

xifi.

41

111

1

11

1 11

1

2S

1S S3

Figure 4.1: Simplex Sn for n = 2, 3, 4

Analogously, the dynamics of selection can be formulated as

xi = xi(fi − φ), i = 1, . . . , n. (4.1)

Obviously, in case of the fitness of type i is larger than the average fitness of the population, the frequencyincreases, otherwise it declines. Next, we introduce the so-called simplex, which includes all points whosecoordinates are not negative and add up to one, i.e.

Sn = (x1, . . . , xn) |n∑

i=1

xi = 1 and xi ≥ 0 for i = 1, . . . , n.

It corresponds to an n−1 dimensional structure which is embedded in an n dimensional Euclidian space.As it can also be seen in Fig. 4.1, it has n so-called “faces”, where each of the faces consists of the simplexSn−1. The equation for the selection dynamics (4.1) contains just one globally stable equilibrium. Thus,starting from an initial condition in the interior of the simplex, the solution converges to a point withxk = 1 (for a certain k with 1 ≤ k ≤ n) and xi = 0 for 1 ≤ i ≤ n, i 6= k. The k is determined by havingthe largest fitness fk, i.e. fk > fi for all i 6= k.Hence, this systems shows up “competitive exclusion”, or in other words “survival of the fittest”.

This kind of approach can be generalised in such a way that the growth rates are not linear in thefrequencies anymore, i.e. the system reads

x = axc − φx (4.2)

y = byc − φy. (4.3)

This means: For c < 1 we have subexponential growth, for c > 1 superexponential. In the same way, φhas to be adapted to keep the constant population size x + y = 1. Therefore, we need φ = axc + byc andget the reduced equation from (4.2):

x = x(1 − x)f(x),

wheref(x) = axc−1 − b(1 − x)c−1.

It can be shown, that this equation has the fixed points x = 0 and x = 1, and an additional one,

x∗ =1

1 + c−1√

a/bfor c 6= 1.

The following proposition gives information about outcompetition in that situation.

Proposition 3 If c < 1, then the fixed points x = 0 and x = 1 are unstable, and the interior fixed pointx∗ is globally stable; thus, both species can survive and coexist.If c > 1, then the fixed points x = 0 and x = 1 are stable, and the interior fixed point x∗ is unstable.

For the proof, see the exercises!

The first situation of the proposition is be called “survival of all”.Especially the second situation of the proposition is interesting: The species which was present first willsurvive, even though the second one may be fitter. This behaviour is called the “survival of the first”;invasion e.g. by a mutant is impossible.

42

4.1.3 Mutation

During the replication of DNA or RNA “errors” (i.e. slightly modified sequences) may occur, which leadto new variants. This phenomenon is called mutation. We are looking for a very simple description formutation, considering two possible types, A and B.Let u1 denote the mutation rate from A to B, and u2 the mutation rate from B to A. As before, x and ydescribe the frequencies of A and B, respectively. Thus, the corresponding ODE system reads:

x = x(1 − u1) + yu2 − φx (4.4)

y = xu1 + y(1 − u2) − φy. (4.5)

(A and B are assumed to have the same fitness, i.e. a = b = 1). The average fitness is obviously φ = 1(constant). Due to x + y = 1, the ODE system (4.4), (4.5) can be reduced to a single ODE:

x = u2 − x(u1 + u2).

The dynamic behaviour is easy to see: x will converge to the stable equilibrium

x∗ =u2

u1 + u2.

This means: Here, mutation leads to coexistence; the relative proportion of the both species depends (asexpected) on the mutation rates: x∗/y∗ = u2/u1.There may be special cases, where one direction of mutation plays a much bigger role that the otherdirection. So, sometimes it can be useful to assume u2 = 0 (only one direction mutation, neglecting theother one). Then the equation reads:

x = −xu1

with the solutionsx(t) = x0e

−u1t and y(t) = 1 − (1 − y0)e−u1t.

It means: In the long time run, A will die out and the whole population will consist of type B. So, alsomutation can influence the survival (even if the reproductive rates are the same).

The same idea can be generalised for the case of n different types. For that, we introduce a so-called mu-tation matrix Q = [qij ]. qij describes the probability, that type i mutates to type j. Since

∑nj=1 qij = 1

(either the type i stays or mutates into another type), Q is a stochastic matrix. The dynamics of mutationcan then be described by the following ODE:

xi =

n∑

j=1

xjqji − φxi, i = 1, . . . , n

or in vector notation~x = ~xQ − φ~x.

Also in this case,, the average fitness is φ = 1. In this kind of notation, the (unique globally stable)equilibrium of the mutation dynamics corresponds to the left-hand eigenvector to the eigenvalue 1:

~x∗ = ~x∗Q.

4.2 Evolutionary stable strategies (ESS)

Literature: [30]

In the next step, we consider non-constant fitness of individual - instead of constant values, the fit-ness may depend on the frequencies of the phenotypes in a population. This is the basis for evolutionarygame theory. The connection between game theory (originally introduced by John von Neumann andOskar Morgenstern) and evolutionary biology was initialised in 1973 by John Maynard Smith and GeorgePrice. A population of players is assumed to interact in a game, with fixed strategies, but random inter-action with other individuals. The fitness (replacing the payoff in the classical game theory), added upof all “encounters” can be translated into reproduction success, i.e. good strategies reproduce faster andpoor strategies are outcompeted ; natural selection.

43

Let us start with a simple example: Two phenotypes, the red one (A) can move, the blue one (B)cannot. The ability to move “costs” something, but may bring an advantage. (In case of constant fitness,that may lead to an outcompete of B by A). Assumption: The advantage of mobility is only present incase of few others “on the road” - otherwise it can turn into a disadvantage.

fitness of A < fitness of Bfitness of A > fitness of B

Mathematically, the fitness of A is monotonely decreasing in A. More formally, we introduce xA and xb thefrequencies of A and B, i.e. the composition of the population can be given by the vector ~x = (xA, xB);fA(~x) and fB(~x) denote the fitness of A and B. The the frequency-dependent selection between thestrategies A and B can be formulated as follows:

xA = xA[fA(~x) − φ]

xB = xB[fB(~x) − φ],

whereφ = xAfA(~x) + xBfB(~x)

(the average fitness). Again, we can use the fact that xA + xB = 1 and replace one variable (and use thenotation x = xA, which is simpler) and get

x = x(1 − x)[fA(x) − fB(x)] =: g(x).

x = 0 and x = 1 are obvious equilibria; also those values x ∈ (0, 1) with fA(x) = fB(x) yield equilibriumpoints. Due to

g′(x) = (1 − x)[fA(x) − fB(x)] − x[fA(x) − fB(x)] + x(1 − x)[f ′A(x) − f ′

B(x)],

we find the following stability “conditions” for the equilibria:

• x = 0 is stable, if fA(0) < fB(0)

• x = 1 is stable, if fA(1) > fB(1)

• the “interior equilibrium” x∗ is stable if f ′A(x∗) < f ′

B(x∗)

The behaviour can also be “seen” in the following graphs:

0 1

0 1

0 1

fA(x

) −

fB(x

)

x

x

x

44

(the red line marks the difference of fitness between A and B; the blue arrow shows the selection dynam-ics, the filled circles denote stable equilibria and the empty circles denote unstable equilibria).

4.2.1 Two-player games

In “classical game theory”, the so-called payoff matrix describes a game, e.g. for two strategies A and B:

A BA a bB c d

It is used as follows:

• if A plays against A, it gets payoff a

• if B plays against B, it gets payoff d

• if A and B play against each other, A gets payoff b and B gets payoff c

In the context of evolutionary game theory, populations of A and B are considered, and the payoff isinterpreted as fitness. Thus, the expected “payoff” for A and B reads

fA = axA + bxB

fB = cxA + dxB .

(underlying assumption: players meet each other randomly, and the probability of interaction is equal tothe relative frequency of the “partner”).Using the same idea as above (x = xA), we get:

x = x(1 − x)[(a − b − c + d)x + b − d].

There are different possibilites of behaviour:

1. If a > c and b > d, then A dominates B:

A B

This means: A has always a higher average fitness than B, and selection favours thus A over B

2. If a < c and b < d, then B dominates A:

A B

This means: B has always a higher average fitness than A, and selection favours thus B over A

3. If a > c and b < d, then A and B are bistable:

A B

This means: The best thing is to play the same strategy as the other one. The equilibrium(x∗ = (d − b)/(a − b − c + d)), which is unstable, separates the “domain of attraction” - i.e.,it depends on the initial condition, where the system will converge to.

4. If a < c and b > d, then A and B coexist:

A B

This means: The equilibrium x∗ = (d − b)/(a − b − c + d) is stable. The best thing is to play theopposite strategy as the other one. The system will always converge to the coexistence equilibrium.

5. If a = c and b = d, then A and B are neutral:

A B

This means: The payoff is always the same, independent on the own choice; there is a “line ofequilibria” between A and B for the selection dynamics.

45

4.2.2 Nash equilibrium

The Nash equilibrium (which is a central concept in game theory, it was introduced by John Forbes NashJr. in 1950) is defined as follows: If both players play a strategy which is a Nash equilibrium, then nosingle player can increase his payoff by changing his own strategy (in a non-cooperative game).

Let us consider again the example of a general payoff matrix for the two strategies A and B,

A BA a bB c d

The following criteria hold true:

1. If a > c, then A is a strict Nash equilibrium.

2. If a ≥ c, then A is a Nash equilibrium.

3. If d > b, then B is a strict Nash equilibrium.

4. If d ≥ b, then B is a Nash equilibrium.

Examples:

•A B

A 3 0B 5 1

First, we check A: If both choose A (so both get 3), then if one player switches to B,

he will get 5 (because then he plays still against A, but when being B), i.e. he improves his payoffby switching. Thus, A is no Nash equilibrium.If both players choose B, then none can increase his payoff by switching. Thus, B is a Nashequilibrium.

•A B

A 3 1B 5 0

Again, if both choose A, switch to B allows one player to improve his payoff.

If both choose B, switch to A allows one player to improve his payoff (from 0 to 1). So, in this case,A are B are no Nash equilibria.

•A B

A 5 0B 3 1

Obviously: If both choose A, any switch to B does not improve the payoff.

The same applies for B: any switch to A does not improve the payoff. So, both, A and B, are Nashequilibria.

4.2.3 Evolutionarily stable strategy (ESS)

This concept was introduced by John Maynard Smith, with the following idea: Consider a large populationwhich consists of A players. There, a single mutant (type B) is introduced. They play a “game”, whichis given by the general payoff matrix

A BA a bB c d

as before. For the fitness functions, we also use again the standard approach

fA = axA + bxB

fB = cxA + dxB .

Question: How can a invasion of B into A be prevented, which are the necessary conditions?

Idea: Introduce an infinitesimally small quantity of B invaders, with frequency ǫ (and frequency 1 − ǫ ofA). Then, the fitness of A is greater than that of B, if the following condition is satisfied:

a(1 − ǫ) + bǫ > c(1 − ǫ) + dǫ. (4.6)

In case of ǫ → 0, the inequality leads to a > c. In case of a = c, the inequality (4.6) leads to b > d.Taken together, we get: The strategy is an ESS, in case of

46

1. a > c

2. or a = c and b > d.

This means: Selection can prevent an invasion of B into A. It is true for infinitely large populations, andinfinitesimally small quantities of the invader B.

4.2.4 More than two strategies

Up to now, we only considered cases with two strategies. But of course, nature is more complex andgames with more than two strategies may be relevant. The following notation is used: E(Si, Sj) denotesthe payoff for strategy Si versus Sj . In the same way as above, we can also generalise our definitions:

• The strategy Sk is called a strict Nash equilibrium, if

E(Sk, Sk) > E(Si, Sk) ∀i 6= k.

• The strategy Sk is called a Nash equilibrium, if

E(Sk, Sk) ≥ E(Si, Sk) ∀i.

• The strategy Sk is ESS, ifE(Sk, Sk) > E(Si, Sk)

orE(Sk, Sk) = E(Si, Sk) and E(Sk, Si) > E(Si, Sk)

is satisfied ∀i 6= k.

• The strategy Sk is “weak ESS” (i.e., stable against invasion by selection), if

E(Sk, Sk) > E(Si, Sk)

orE(Sk, Sk) = E(Si, Sk) and E(Sk, Si) ≥ E(Si, Sk)

is satisfied ∀i 6= k.

Remark: The ESS ensures that selection opposes any potential invader. Also the strict Nash equilibriumhas this property, but the Nash equilibrium doesn’t. E.g., if E(Sk, Sk) = E(Sj , Sk) and D(Sk, Sj) <D(Sj , Sj), then the selection will favour Sj and allow it to invade Sk, even though it is a Nash equilibrium.So, it may make sense to introduce the definition of a “weak ESS”.Obviously, there is a hierarchy in these definitions:

strict Nash ⇒ ESS ⇒ weak ESS ⇒ Nash .

There is still another definition, which can be useful for some problems: A strategy Sk is called unbeatable,if

E(Sk, Sk) > S(Si, Sk) and E(Sk, Si) > E(Si, Si)

is satisfied ∀i 6= k. (This concept was introduced by William Hamilton). Obviously, an unbeatablestrategy is also a strict Nash equilibrium; it is somehow “perfect”, but in reality, these strategies are rare.

4.2.5 Replicator dynamics

The next step is just a logical consequence of the preceding considerations.Goal is to consider the interaction between n different strategies. Let aji denote the payoff for the strategyi when interacting with strategy j (this forms the payoff matrix A = [aij ], and let xi denote the frequencyof the strategy i. fi =

∑nj=1 xjaij describes the expected payoff of strategy i, then we can compute the

average payoff by φ =∑n

i=1 xifi and get as a description from the frequency-dependent selection (amongn different strategies = phenotypes):

xi = xi(fi − φ), i = 1, . . . , n (4.7)

47

the famous replicator equation, one of the central topics in evolutionary game dynamics. It is for aninfinitely large population without any mutations. From the definition, we can see that the fitness fi

depends on the whole population respectively on its composition ~x = (x1, . . . , xn) . The big differenceto the equation for the constant selection dynamics, (as introduced above: xi = xi(fi − φ), whereφ =

∑ni=1 xifi ) is, that in the replicator equation the fitness is frequency-dependent.

Some properties for the replicator equation (4.7):The replicator equation is defined on the simplex

Sn = ~x :

n∑

i=1

xi = 1.

Obviously, it is invariant.A subset of the simplex, which is defined in such a way hat there at least one strategy has frequencyzero, is called a “face”. Also each face of the simplex is invariant! (it follows from the factor xi on theright hand side) This means: This model system cannot create strategies, due to the lack of mutation -it only describes pure selection!In each case, the corners of the simplex consist of fixed points of (4.7). It is possible to have further fixedpoints in the interior and in the faces of the simplex, but this depends on the payoff matrix.

4.3 Hardy-Weinberg law

There is an open question: How can a population still keep “variability” in it (which is needed for naturalselection). The idea of discrete genotypes and inheritance comes from Gregor Mendel (1866). Thecorresponding simple mathematical analysis was introduced by G.H. Hardy (a British mathematician)and generalised by W. Weinberg (a german physician).The object of interest here is an infinite (realistically: very large) population of diploid individuals, withone locus for genes / alleles a, A. [12]Assumptions: no migration, no mutation, no selection, no overlapping generations, random matingThen there are

3 genotypes AA Aa aawith relative frequencies α 2β γ

From these data, we can deduce the frequencies of the genes A and a:

p = α + β, q = β + γ. (4.8)

Assumption: The next generation is developed by random mating, without respect to the genotypes ofparents, and each mating produces the same number of descendants.

Mating possibilitiesof genotypes Frequency AA Aa aaAA × AA α2 α2 0 0AA × Aa 4αβ 2αβ 2αβ 0Aa × Aa 4β2 β2 2β2 β2

AA × aa 2αγ 0 2αγ 0Aa × aa 4βγ 0 2βγ 2βγaa × aa γ2 0 0 γ2

Frequencies of genotypes (α + β)2 2(α + β)(β + γ) (β + γ)2

in the next generation =: α =: 2β =: γ

These equations define a map of the set

S = (α, β, γ) : α, β, γ ≥ 0, α + 2β + γ = 1

into itself. This set S (which is a subset of R3) corresponds to a triangle with the barycentric coordi-

nates α, 2β, γ and is called Finetti diagram. (A nice Finetti diagram generator can be found e.g. athttp://finetti.meb.uni-bonn.de).

48

The frequencies of genotypes in the ν-th generation are denoted as αν , 2βν , γν (ν = 0 corresponds to theinitial population). So we come to the following (discrete) system of difference equations:

αν+1 = (αν + βν)2

βν+1 = (αν + βν)(βν + γν) (4.9)

γν+1 = (βν + γν)2

From (4.8) we get:α = p2, 2β = 2pq, γ = q2.

Hence, the frequencies of genotypes in the next generation depend only on the gene frequencies of theparent generation. The gene frequencies in the next generation are

p2 + pq = p and pq + q2 = q,

which are the same as in the the parent generation! Therefore, the gene frequencies are constant and thegenotype frequencies stay constant, starting from the next generation on:

αν = p2, βν = pq, γν = q2, ν = 1, 2, . . . (4.10)

Result: Each solution of (4.9), which starts inside

(α, β, γ) : α, β, γ ≥ 0, α + 2β + γ = 1,

reaches an equilibrium after one step. This state is called the Hardy-Weinberg equilibrium.

Further properties:

• From (p− q)2 = p2 − 2pq + q2 ≥ 0 we get p2 + q2 ≥ 2pq; this means that only half of the individualscan be heterocygotes.

• There is a “test”, if (α, β, γ) corresponds to a Hardy-Weinberg equilibrium. This is the case, if andonly if

αγ = β2 (4.11)

This equation can easily be deduced from (4.10). Vice versa, the iteration formulae yield

α = (α + β)2 = α2 + 2αβ + β2 = α2 + 2αβ + αγ = α(α + γ + 2β) = α,

β = (α + β)(β + γ) = αβ + β2 + βγ + αγ = αβ + βγ + 2β2 = β,

γ = (β + γ)2 = β2 + 2ηγ + γ2 = αγ + 2βγ + γ2 = γ(α + 2β + γ) = γ

and the equivalence is shown.

These considerations can be transferred to the case of n alleles a1, . . . , an. Then there are n2 genotypesa1a1, a1a2, . . . , anan (where ajak and akaj can be identified). Let

αjk = αkj ≥ 0, j, k = 1, . . . , n

be the frequencies of genotypes of the initial population, and assume

n∑

j,k=1

αjk = 1.

The (symmetric) matrix A = (αjk) describes the state of the population. By

pj =

n∑

k=1

αjk, j = 1, . . . , n

the frequencies of the gene pj are computed. Suppose again random mating, then the frequencies of thegenotypes in the next generation are computed by:

αjk = pjpk =

(∑

i

αji

)

·(

∑

l

αkl

)

, j, k = 1, . . . , n.

49

Hence, the matrix A = (αjk) is a dyad (a matrix of rank 1). The frequencies of genes in the daughtergeneration are

n∑

k=1

αjk = pj , j = 1, . . . , n.

Then the gene frequencies are constant for all generations, and the frequencies of the genotypes do notchange, beginning from the daughter generation. After one step, the population is in a Hardy-Weinbergequilibrium, which are characterised analogously to (4.10) by the relation

αjk = pjpk, j, k = 1, . . . , n

By the above introduced matrices and vectors, this Hardy Weinberg law for n alleles can be described byA = ppT . There is no formula similar to (4.11), but it is A2 = A (only in case n = 2 it follows that Ahas rank 1).

4.4 Fisher-Wright-Haldane model

We consider a population of diploid individuals with distinct generations (e.g. an insect population, withseasonal life-time). It is assumed, that the population does not have overlapping generations.Let the genotypes be identical except for one autosomal local (i.e. a locus which is not on an allosome)with n ≥ 2 different alleles a1, a2, . . . , an. Then, there are n2 possible genotypes a1a1, a1a2, . . . , anan.From a biological view, the genotypes ajak and akaj could be identified (if there is no distinction betweenpaternal and maternal genetic material desired), but the equations are more simple and symmetric, if thisidentification is not performed. In each case, we assume that the genotypes ajak and akaj appear withthe same frequency. A basic assumption of the model is the random mating, i.e. the mating happensindependently of the genotypes of the involved individuals.Since the population consists of individuals of the same generation and is originated by a random mating,it is in Hardy-Weinberg equilibrium at the beginning of a new generation. At this time, the geneticcomposition of the population is completely fixed by the frequencies p1, . . . , pn of the genes a1, . . . an.These frequencies have the properties

pj ≥ 0, j = 1, . . . , n;

n∑

j=1

pj = 1. (4.12)

Due to the Hardy-Weinberg equilibrium, the genotype frequencies at birth time are

αjk = pjpk, j, k = 1, . . . , n. (4.13)

In general, the genotypes are differently adapted to the constant environment. Thus, the proportion ofindividuals that die before reaching their reproductive phase (called differential mortality) and the numberof descendants (differential fertility) depend on the genotype. These effects can be taken together in asingle fitness or viability parameter fjk = fkj > 0 (sometimes, this is also called “Malthusian parameter”).The genotype ajak participates at the creation of the next generation with the fraction

fjkαjk/∑

r,s

frsαrs, j, k = 1, . . . , n.

Corresponding to the assumption of random mating and the relation (4.13), the genotype frequencies ofthe next generation pj are given by

pj =

∑

k fjkαjk∑

r,s frsαrs=

∑

k fjkpjpk∑

r,s frsprps, j, k = 1, . . . , n.

This is a difference equation for n variables pi. The following discussion can be simplified by the in-troduction of vector- and matrix notation. The gene frequencies pi are taken together to a vectorp = (p1, . . . , pn)T . The state is described by the vector p. With respect to (4.12), the set of possi-ble genetic states is the simplex

S = p ∈ Rn : p ≥ 0, eT p = 1.

The fitness parameters fjk are taken together to a matrix F = (fjk), which is symmetric (F = FT ) andnonnegative (F ≥ 0) (but not semidefinite in general). For reasons of simplicity of notation the vector

50

p = (pj) is assigned to the diagonal matrix P = (pjδjk). With these notations, the selection-reproductionmechanism which translates the parental gene frequencies into the filial gene frequencies p is given by themap T : S → S,

Tp =PFp

pT Fp(4.14)

These equations describe the Fisher-Wright-Haldane model, which is a basic model in classical determin-istic population genetics.Hence, to each genotype ajak a constant fitness value fjk and a frequency αjk are assigned, the latterchanges from one generation to the next. The mean fitness of population W in state p is the arithmeticmean of the fitness values of the distinct genotypes, thus linear in αjk. This term becomes a quadraticfunction if the genotype frequencies are replaced by the gene frequencies:

W (p) =n∑

j,k=1

fjkαjk =n∑

j,k=1

fjkpjpk = pT Fp. (4.15)

The mean fitness is a sum of terms which are assigned to the distinct genes, which form the differentgenotypes. The expression

Wj(p) =n∑

k=1

fjkpk, j = 1, . . . , n

is called the fitness of the gene aj . This notation yields the following useful relation:

n∑

j=1

pjWj(p) = W (p)

It says that the mean fitness of the total population is the mean of the contributions of the distinct genes,weighted with the gene frequency.The expression (4.15) defines a function W : S → (0,∞). If the mean fitness has indeed a biologicalmeaning, one would expect that W (p) increases from one generation to the next, since the populationadapts genetically to its environment. This is not obligatory and not true for all models in populationgenetics, but here, it is the case.

Proposition 4 For each p ∈ S it isW (T (p)) ≥ W (p)

Equality holds if and only if Tp = p.

This proposition is called sometimes the “fundamental theorem of natural selection”. The proof is leftout here, but can be found e.g. in [17].The stationary points of the selection model (4.14) are the solutions of the equation Tp = p or

PFp = pT Fp · p. (4.16)

This can be studied in general, but here we restrict to the case n = 2, i.e. two alleles. Since the behaviourdoes not depend on a common factor of the fjk and the case of f12 = 0 is trivial, the matrix F can beassumed to form

F =

(f 11 g

)

.

In the equations

(Tp)1 = (fp21 + p1p2)/(fp2

1 + 2p1p2 + gp22)

(Tp)2 = (p1p2 + gp22)/(fp2

1 + 2p1p2 + gp22),

p1 = x, p2 = 1 − x, (Tp)1 =: T (x) can be substituted; thus one gets an equation

T : [0, 1] → [0, 1], T (x) =(f − 1)x2 + x

(f + g − 2)x2 + 2(1 − g)x + g.

A stationary point corresponds to a fixed point of the mapping T . From T (x) = x, it follows for x 6= 0:

(f + g − 2)x2 + 2(1 − g)x + g = (f − 1)x + 1

51

respectively((f + g − 2)x − g − 1)(1 − x) = 0.

This yields the fixed point x = 1 and furthermore the solution of the equation

(f + g − 2)x − (g − 1) = 0,

if this in contained in the interval [0, 1]. Under the assumption of f, g < 1 or f, g > 1, thus there is anadditional fixed point in [0, 1], the so-called heterozygote equilibrium

p =

(1 − g

2 − f − g,

1 − f

2 − f − g

)

.

If the heterozygotes have an advantage during selection (i.e., f, g < 1), then this point is stable. If theheterozygotes are disadvantaged compared to the homozygotes (i.e., f, g > 1), then this point is unstable.

The case of a stable heterozygote equilibrium is very common in nature. The natural selection doesnot necessarily lead to an optimal genotype, but to an optimal population. Probably the case of anunstable heterozygote equilibrium is realised rarely in nature. The existence of two favoured homozygotegenotypes presumably leads to the formation of two distinct homozygote populations and finally to theformation of two species. Then, the assumption of random mating is not appropriate anymore (takenfrom [12]).

52

Chapter 5

Reaction kinetics

5.1 Enzyme kinetics: Michaelis-Menten

Literature: Murray [27], a nice short introduction can be found in Wikipedia.

The enzyme kinetics was established by Leonor Michaelis and Maud Menten in 1913.

Generally, enzymes can compensate fluctuating concentrations of substrate, i.e. they adapt their ac-tivity and thereby tune a steady state. This behaviour is the common one, but of course, there are alsoexceptions.

In opposite to the kinetics of chemical reactions, in enzyme kinetics there is the phenomenon of sat-uration. Even for very high concentrations of the substrate, the metabolic rate cannot be increasedunlimitedly, there is a maximum value vmax.

Enzymes E, in their function as biocatalytic converter, form together with their substrate S a com-plex ES, from which the reaction to the product performs. Shortly, this can be noted in the followingway:

E + Sk1

k−1

ESk2

k−2

EPk3

k−3

E + P.

k1 and k−1 are the rate constants for the association of E and S respectively the dissociation of theenzyme - substrate complex ES. k2 and k−2 are the corresponding constants for the forward reaction tothe product respectively the reverse reaction to the substrate. This reverse reaction does not occur underthe conditions of enzyme kinetics (short after the mixing of the components E and S). Furthermore, theconversion of ES to EP is measured (not the spontaneous release of P ), thus the following simplificationis justified:

E + Sk1

k−1

ESk2→ E + P.

(There is a nice idea, how to understand that kind of kinetics by a descriptive example: S are potatoes,the cook corresponds to the enzyme E - as he has to transform the potatoes into mashed potatoes, theproduct P . Obviously, the cook cannot work infinitely fast with the potatoes, only up to a limit speed;he has to deal with each potato for a certain time - there he forms a complex with the potato. And thereis no chance to get back potatoes from the mushed potatoes :-) ).k2 measures the maximal velocity of reaction under saturated substrate and is also called “turnovernumber”. The Michaelis constant, which is the concentration of substrate, where the metabolic rateassumes half of its maximum, results in

Km =k−1

k1

(in the so-called Michaelis-Menten case, if k2 ≪ k1), or more generally in

Km =k−1 + k2

k1

(called Briggs Haldane situation).

53

The saturation function of a “Michaelis-Menten enzyme” is described by

v =vmax · [S]

Km + [S],

where v = ˙[P ], the graph looks obviously as follows:

[S]

v

vmax

−Km

vmax/2

Km

The following variables are introduced:

s density of S (substrate)e density of E (enzyme)c density of SE (complex substrate/enzyme)p density of P (product)

Then we get the following system of differential equations:

s = −k1se + k−1c

e = −k1se + k−1c + k2c

c = k1se − k−1c − k2c

p = k2c

Obviously, it isd

dt(e + c) = 0,

thus e + c = e0, where e0 is the initial amount of the enzyme. Thus the system reduces to

s = −k1s(e0 − c) + k−1c

= −k1se0 + (k1s + k−1)c, s(0) = s0

c = k1s(e0 − c) − k−1c − k2c

= k1se0 − (k1s + k−1 + k2)c, c(0) = 0.

Usually, it is assumed that c is essentially at equilibrium, thus dcdt ≈ 0, this leads to the description of c

in terms of s:

c(t) =e0s(t)

s(t) + Km

Substituting this into the equation for s yields

s = − k2e0s

s + Km.

This is called the pseudo- or quasi-steady state approximation.

In order to eliminate superfluous parameters, we do the following rescaling:

τ = (k1e0)t, u(τ) = s(t)/s0, v(τ) = c(t)/e0

54

and set

λ =k2

k1s0, K =

k−1 + k2

k1s0=

Km

s0, ǫ =

e0

s0.

This yields the following dimensionless equations:

u = −u + (u + K − λ)v, u(0) = 1

ǫv = u − (u + K)v, v(0) = 0.

It is assumed that the amount of the enzyme is small compared to the substrate, i.e.

ǫ =e0

s0≪ 1.

Then the system can be considered under two different time scales. The other scaling is τ = (k1s0)t.Obviously, τ ≫ τ .

Long-time behaviour Short-time behaviouru = −u + (u + K − λ)v

ǫv = u − (u + K)vu = ǫ(−u + (u + K − λ)v)v = u − (u + K)v

lim ǫ → 0

u(t) = −u + (u + K − λ)v0 = u − (u + K)v

u = 0v = u − (u + K)v

(t and τ denote different time scales).

(a) The slow manifold (under the pseudo-steady-state assumption) is

u − (u + K)v = 0 ⇔ v =u

u + K

(b) In the fast system, v tends into the following directions:Above slow manifold (i.e. v is large): v < 0, downwardsBelow slow manifold (i.e. v is small): v > 0, upwards.The slow manifold is stable!

Slow Manifold

v = u/(K+u)

v>0.

v<0.

Initialstate

v

u

(c) In order to analyse the slow dynamics, we consider u on the slow manifold v = uu+K . There, it is

u = −u + (u + K − λ)u

u + K= −u + u − λ

u + Ku = − λ

u + Ku.

5.2 Oscillations in chemical systems

Literature: Edelstein-Keshet [6]

First discovery of oscillations in chemical reactions was in 1828 in an electrochemical cell (by A.T.H.Fechner). Lotka performed a first theoretical analysis in 1910. As it was usual for a long time, thescientific community believed that chemical reactions tend to an equilibrium. One reason for that wasthe influence of thermodynamics of closed systems, neglecting that many biological or chemical systemsare open.More and more oscillating reactions were discovered, one famous is the so-called Belousov-Zhabotinskywhich we will study later in detail.

55

Lotka model

Lotka considered the following system:

A + x1k1−→ 2x1

x1 + x2k2−→ 2x2

x2k3−→ B

This system can be described by the following equations:

dx1

dt= −k1Ax1 + 2k1Ax1 − k2x1x2 = k1Ax1 − k2x1x2

dx2

dt= −k2x1x2 + 2k2x1x2 − k3x2 = k2x1x2 − k3x2 (5.1)

Assuming that substance A has constant concentration (e.g. artificially), then these equations correspondto the predator-prey model. This yields the existence of a stationary state with closed orbits around it.There is one problem with the Lotka-model: The equations show up periodic orbits which can easily bedisrupted when the dynamics are modified a little bit (called structural instability). This behaviour istypical for so-called conservative systems, where some quantity is conserved (e.g. energy). This is alsothe case for the Lotka system. Analogously to the basic predator prey model, it can be shown that thefollowing expression is an invariant of motion:

v = x1 + x2 −k3

k2lnx1 −

k1A

k2lnx2.

Each value of this quantity, here v, determines a distinct periodic solution. Hence, there are infinitelymany closed orbits, leading to structural instability (i.e. the closed orbits are not asymptotically stable).

Limit cycle oscillations

It is possible to modify the equations in such a way that they show up limit-cycle trajectories. There aresome criteria for oscillations in a chemical system:

1. In 2D the theory of Poincare-Bendixson and in n-dim Hopf-bifurcation apply as usual to a givensystem.

2. To obtain oscillations, a feedback is necessary, e.g. with activation and inhibition (special kind offeedback), e.g.

a a . . . a a

a a a . . . a

a a . . . a a

a a . . . a a

1

2

2a a . . . a a

1

1

21

2

activation +

+

−

−

inhibition

inhibition

activation

n−1 n

n3

n−1 n

a n2n−1

In biology, enzymes play a big role for the mediation of reactions, e.g. by so-called allostericmodification.Considering chemical systems which can be described by a system of two ODEs,

dx1

dt= v1(x1, x2)

dx2

dt= v2(x1, x2)

(done by Nicolis and Portnow, 1973), one of the following two cases must be satisfied in order toobtain limit-cycle oscillations.

56

2a : At least one of x1 or x2 is autocatalytic (which means that it catalyses its own production oractivates a producing substance)

2b : There is a cross catalysis (x1 activates x2 or vice versa)

Here, e.g. the Lotka system is included. The conditions are connected with the Poincare-Bendixsontheory. In higher dimension, oscillations can appear also in the case of inhibition without additionalcatalysis.

3. In closed chemical systems (without any exchange with their environment) sustained oscillationsare impossible, the system tends to a steady state.

4. Oscillations cannot occur close to a thermodynamic equilibrium (for details, see Nicolis and Port-now).

Schnakenberg model

Additional observations made by Schnakenberg (as they were cited in Edelstein-Keshet):

a: Chemical systems with two variables and less than three reactions cannot have limit cycles.

b: Limit cycles can occur only if one of the three reactions is autocatalytic

c: Limit cycles in such systems will only occur if the autocatalytic step involves the reaction of at leastthree molecules (with exactly three, the reaction is called trimolecular)

Illustrating example:

2X + Y → 3X

A → Y (5.2)

X → B

This network has the following properties:

• there is an autocatalytic step included (X produces more X)

• three reaction steps

• a trimolecular step is included

; there oscillations should show up.The corresponding equations to (5.2) read

dx

dt= x2y − x

dy

dt= a − x2y (5.3)

(x, y, a denote the concentrations of X,Y,A).There is a stationary state at (x, y) = (a, 1

a ), the general Jacobian is

J =

(2xy − 1 x2

−2xy −x2

)

,

at (x, y) it becomes

J =

(1 a2

−2 −a2

)

.

By

β = tr J = 1 − a2

γ = det J = −a2 + 2a2 = a2

the eigenvalues can easily be computed:

λ1,2 =(1 − a2) ±

√

(1 − a2)2 − 4a2

2.

57

For a2 = 1 we have λ = ±ai (purely imaginary), for a2 < 1 the eigenvalues have positive real part, thus(x, y) is an unstable focus.If a is decreased from a > 1 to a < 1, then the Hopf bifurcation yields a small-amplitude limit cycle,but its stability is uncertain (calculations are left out; the stability criterion of Marsden and McCrackenrequires a normal form, which is possible to gain, but many many calculations ...).There is a problem which can be easily understood by having a look at the phase-plane diagram:

(a,1/a)

y

xy=0. x=0

.

The isoclines are:

x = 0 : x2 − x = 0 ⇔ x = 0 or xy = 1 ⇔ y =1

x

y = 0 : a − x2y = 0 ⇔ x2y = a ⇔ y =a

x2

Trajectories (near the y-axis) can “escape” and tend to ∞, then the theorem of Poincare-Bendixsoncannot be applied.In order to avoid that problem, Schnakenberg modified the last reaction step:

X←−−−→ B,

(now reversible), then the equations read

dx

dt= x2y − x + b

dy

dt= −x2y + a.

The stationary state changes into(

a + b, a(a+b)2

)

and the Jacobian becomes

J =

(a−ba+b (a + b)2−2aa+b −(a + b)2

)

,

with

β = tr J =a − b

a + b− (a + b)2

γ = det J = −(a − b)(a + b) + 2a(a + b) = (a + b)2

Obviously, γ is always positive; when β = 0, then a bifurcation occurs, i.e. a−ba+b = (a+b)2, the eigenvalues

are

λ1,2 =±√−4γ

2= ±(a + b)i

If b is much smaller than a, then approximately

a − b

a + b≈ a

a= 1, (a + b)2 ≈ a2

; a2 ≈ 1.

At a = 1, there is the bifurcation (For a > 1, the steady state is stable; for a < 1 it is an unstable focus).The Hopf bifurcation theorem yields closed periodic orbits.The modified equations correspond to the following phase plane diagram:

58

y

x

(a+b,a/(a+b)^2)

y=0.x=0.

The nullclines satisfy following equations:

x = 0 : y =x − b

x2,

y = 0 : y =a

x2.

Thus, there exists a finite region in the first quadrant which cannot be left by a trajectory. In this case,the theorem of Poincare-Bendixson yields the existence of a stable periodic orbits in case of the stationarystate is unstable.

What is necessary for limit-cycle dynamics? Biological factors:

1. Structural stability: Small changes will not change qualitatively the dynamic behaviour of thesystem. (Counterexample: Lotka-Volterra model)

2. Open systems: certain exchange with environment should be possible, otherwise some nonrenewablecomponents are consumed and the system cannot show up undamped oscillations.

3. Feedback: Feedback or some form of autocatalysis is necessary to maintain oscillations.

4. Steady state: The system should have at least one stationary state (strictly positive for populationsor chemical concentrations). It must be unstable in order to allow oscillations.

5. Limited growth: Growth rates of all intermediates should be limited to allow bounded oscillations.

Mathematical criteria:

1. There has to be at least one stationary point which becomes unstable

2. At this stationary point, there have to be complex eigenvalues λ = a ± bi which change sign in thereal part depending on a parameter (Hopf bifurcation)

3. There has to be a bounded annular region in the xy phase space, where a flow outwards is impossible(Poincare-Bendixson)

4. There must be a region in the xy phase space where ∂F∂x + ∂G

∂y changes sign (Negative criterion of

Bendixson - necessary, but not sufficient)

5. If there are one or more S-shaped nullclines which are “well-positioned”, there show up oftenoscillations or excitability (e.g. Lienard equation; van der Pol oscillator; or also Fitzhugh-Nagumo)

5.3 Belousov-Zhabotinsky Reaction

5.3.1 Chemical background

Literature: Some nice introductions and overviews can be found in:

• http://www.ux.his.no/∼ruoff/BZ Phenomenology.html

• http://online.redwoods.cc.ca.us/instruct/darnold/deproj/Sp98/Gabe/bzreact.htm

59

• Casey R. Gray, An Analysis of the Belousov-Zhabotinskii Reaction

• Wikipedia!

The Belousov-Zhabotinsky reaction is one of the most famous examples for an oscillating chemical system.The reaction itself was detected in 1950 by Belousov, more or less accidentally. Since oscillating reactionswere thought to be impossible (to that time) he published a short article about his observation notuntil 1959. S.E. Schnoll recognised the importance and instructed A. Zhabotinsky to investigate thisphenomenon.The recipe for oscillations in a well-stirred beaker:

• 0.3 M malonic acid

• 0.1 M NaBrO3

• 2 · 10−3 M (NH4)2 Ce (NO3)6, all in 1 M sulfuric acid.

Dissolve the chemicals in the 1 M H2SO4 in the following order:

• malonic acid

• Ce (IV)

• finally bromate

After a few minutes, the oscillations start (changing the color of the solution from orange to blue andvice versa) and last for ≈ 1 hour ⇒ great lecture demonstrations!

The original Belousov-Zhabotinsky reaction (short: BZR) was considered in detail by R.M. Noyes, R.J.Fields and E. Koros at the University of Oregon (early 1970s), creating a system of 18 reactions and 21distinct chemical species (and a correspondingly large ODE system).

5.3.2 The ODE system

We will consider a simplified version of the Fields-Koros-Noyes model, the so-called Oregonator (coveringthe essential behaviour of the BZR). It contains the following reaction steps:

BrO−3 + Br− → HBrO2 + HOBr Rate = k1[BrO−

3 ][Br−]HBrO2 + Br− → 2HOBr Rate = k2[HBrO2][Br−]

BrO−3 + HBrO2 → 2HBrO2 + 2Ce4+ Rate = k3[BrO−

3 ][HBrO2]2HBrO2 → BrO−

3 + HOBr Rate = k4[HBrO2]2

B + Ce4+ → 12fBr− Rate = kc[Z][Ce4+]

B: all present oxidisable organic speciesf: stoichiometric factor (encapsulates the organic chemistry involved)All rate constants ks depend upon temperature and have to be determined experimentally (the valuesare known but left out here). Notations:

A : BrO−3

B : all oxidisable organic speciesP : HOBrX : HBrO2

Y : Br−

Z : Ce4+

The reactants A and B are assumed to be constants, thus we get the following rate equations:

X = k1AY − k2XY + k3AX − 2k4X2

Y = −k1AY − k2XY +1

2fkcBZ

Z = 2k3AX − kcBZ

60

These equations can be simplified by transforming X,Y,Z (which describe concentrations) into dimen-sionless variables:

x =2k4X

k3A

y =k4Y

k3A

z =kck4BZ

(k3A)2

τ = kcBt

Then, the dimensionless version of the ODE system reads

dx

dτ=

qy − xy + x(1 − x)

εdy

dτ=

−qy − xy + fz

δ(5.4)

dz

dτ= x − z,

where

ε =kcB

k3A, δ =

2kck4B

k2k3A, q =

2k1k4B

k2k3A.

The dimensionless system can be interpreted to describe relative concentrations of the species.Even the simplified system (5.4) can be reduced further, by using the fact, that δ is quite small comparedto ε (δ = 4 · 10−4, ε = 4 · 10−2). Thus, in the equation

δdy

dτ= −qy − xy + fz

the left hand side can be regarded to be near zero, yielding

y ≈ fz

q + x.

So we come to the following 2D system:

εdx

dτ= x(1 − x) +

f(q − x)

q + xz =: g(x, z), (5.5)

dz

dτ= x − z =: h(x, z). (5.6)

Also ε in (5.5) is a small parameter; thus we consider here a so-called stiff system of ODEs. It is expectedto find so-called “relaxation oscillator” behaviour for the orbits in the phase plane.ε is a small parameter, so the right-hand side of equation (5.5) might be close to zero and x “tries” tomaintain a suitable value (with respect to z). If such a value can not be achieved, then there will berapid jumps in x (typical behaviour of relaxation oscillators). Let us consider first the nullclines of thesystem:

z = x(1 − x)x + q

(x − q)f, (g(x, z) = 0) (5.7)

z = x, (h(x, z) = 0). (5.8)

We consider now the x-z phase plane. There are two equilibria, called E1 (the origin) and E2 (the positiveequilibrium). Above the curve of h(x, z) = 0, there is z > x and thus dz

dτ < 0 (see Eq. (5.6)), and vice

versa dzdτ > 0 below the curve h(x, z) = 0. In the same way, we get dx

dτ < 0 above and dxdτ > 0 below the

curve. This yields the following figure:

h(x,z)=0

g(x,z)=0

E2

E1

Oscillator

z

x

A

B C

D

61

The stoichiometric factor is chosen to be f = 2/3 (there are some reasons for that, which are not discussedhere; from experimental data f is between 1

2 and 1).

By some computations, the coordinates of the point A, B, C and D can be determined:

Point x z y

B (1 +√

2)q (1+√

2)2qf 1 + 1√

2

D 12 − q 1+4q

4f12 + 2q

A (1 + 8q)q 1+4q4f

18q

C 1 − 6q (1+√

2)2qf (1 +

√2)2q

Let E2 = (x∗, z∗) be the positive equilibrium. E2 will be unstable (important for allowing the system tooscillate), if

(1 +√

2)q = xB < x∗ < xD =1

2− q. (5.9)

Using x∗ = z∗ reformulates (5.7) into

(1 − x∗)(q + x∗) + f(q − x∗) = 0. (5.10)

Since this equation has a positive root (for any positive q and f), there exists a positive equilibriumpoint (x∗, z∗). Now we can determine those values of f which can induce oscillations, using (5.9) and thecoordinates of B and D: If x∗ reaches xB , then f = 1 +

√2 − (3 + 2

√2)q; on the other hand x∗ = xD

corresponds to f = 1+2q2−8q . Thus, oscillations show up if

1 + 2q

2 − 8q< f < 1 +

√2 − (3 + 2

√2)q.

Using typical concentrations (q = 8 · 10−4) leads to 0.5024 < f < 2.41.

Some more background about what happens at the equilibrium point: If the trace of the correspondingJacobian matrix at position (x∗, z∗) is zero, then a Hopf bifurcation will show up there. The positiveequilibrium point can be computed by just taking the positive root of (5.10), this yields

x∗ =1

2

(

1 − f − q +√

f2 + (1 + q)2 + f(6q − 2))

, z∗ = x∗. (5.11)

Computation of the Jacobian matrix:

∂g

∂x=

1

ε

[

1(1 − x) + x(−1) +fz(−1) · (q + x) − f(q − x)z

(q + x)2

]

=1

ε

[

1 − 2x +−fzq − fzx − fzq + fzx

(q + x)2

]

=1

ε

[

1 − 2x − 2fzq

(q + x)2

]

∂g

∂z=

1

ε· f(q − x)

q + x

∂h

∂x= 1

∂h

∂z= −1,

thus

J(x, z) =

(1ε (1 − 2x − 2fzq

(q+x)2 ) 1ε

f(q−x)q+x

1 −1

)

.

Then the trace reads

tr J =1

ε

(

1 − 2x − 2fzq

(q + x)2

)

− 1.

Thus, in the equilibrium point, there will exhibit a Hopf bifurcation, if

0 =1

ε

(

1 − 2x∗ − 2fx∗q

(q + x)2

)

− 1 ⇔ ε = 1 − 2x∗ − 2fx∗q

(q + x∗)2. (5.12)

62

If q is given, equation (5.12) can be used to determine ε as a function of f , see the following figure(showing the locus of Hopf bifurcation points):

ε

f

Oscillatory region

The oscillatory region is exactly the same as determined before.As a result, it can be remarked that under realistic experimental conditions there is a nonempty set ofparameter values that shows up oscillatory behaviour!

5.3.3 Spatial pattern formation

Literature: [7]

If the homogeneous solution (with the chemical ingredients as mentioned above) is put into a petridish, such that the fluid is just a thin layer, spatial patterns will evolve (some are starting from theboundary, or by using a small metal wire, additional “sources” can be initialised):

(the experiment was performed at Helmholtz Centre Munich, “Glasernes Labor”)

Several phenomena can happen:

• Target pattern: In these time-dependent structures one can find circular waves which propagateradially from a centre

• Spiral waves: A spiral pattern rotates around a centre (a typical period is around one minute)

From a mathematical point of view, the analysis of spiral waves is not so easy. Opposite to travellingwaves, it is not possible to reduce the equations to ODEs - so many things were examined just numerically(and especially for complicated kinetics of the Belousov-Zhabotinsky reaction).

Another possibility is to perform nice simulations with Cellular Automata models (e.g. with the so-called Greenberg-Hastings automaton). But we will not do that in this lecture.

5.4 Metabolic pathways

Metabolic pathways consist of a series of chemical reactions which occur in a cell. The “players” arechemicals, which are catalysed by enzymes etc. . Often, further substances are needed, the so-calledcofactors. Due to the large number of different chemicals which are involved in these systems, thecorresponding pathways may be quite elaborate. (And of course, there are many different metabolicpathways involved in the “life” of a cell - which is then called the “metabolic network”). So, for themodelling, it is essential to restrict on some smaller “sub-systems”, which can be somehow treatedseparately. Still then, it may be difficult to determine the reaction parameters etc. Some major metabolicpathways can be seen in the following “survey”, taken from Wikipedia:

63

Such pathways are very important in order to maintain the homeostasis within an organism (homeostasismeans: the systems maintains a stable, constant condition). There are three possible “results”:

• to store something in the cell

• to use a metabolic product immediately

• to initiate other metabolic pathways (this is called a “flux generating step”)

The principle of putting up the corresponding mathematical system can be done similarly to othernetworks, including mass action law, enzyme kinetics etc.Since the systems can become very large, we consider here only a very small system as example, theglycolysis.

5.4.1 Glycolysis model

Literature: [1, 35, 6]

A very “typical”, ubiquitous biochemical reaction (as part of metabolic systems) is the so-called gly-colysis. The structure looks as follows:

Glucose GGP FGP

ATP ADP

PFKFDP products

Table of notations:

GGP glucose-6-phosphateFGP fructose-6-phosphateFDP fructose-1,6-diphosphateATP adenosine triphosphateADP adenosine diphosphatePFK phosphofructokinase

The interesting part of the pathway concerns the following: The PFK catalysis the process FGP → FDP(thus PFK acts as an enzyme, notation: E), by that process also ATP is dephosphorylated to ADP. ThePFK itself can occur in an active and an inactive form (called “allosteric enzyme”, i.e. it can changeits form - a conformational change how the enzyme is folded exactly); the binding of n ADP molecules(denoted by S2 to PFK activates it. This “activation” can be formulated as the reaction

nS2 + Ek3

k−3

ESn2

64

The “complex” ESn2 (the activated form of PFK) can then bind with an ATP molecule (denoted by S1)

and dephosphorylates it into ADP. This reaction is of Michaelis-Menten type, i.e.

S1 + ESn2

k1

k−1

S1ESn2

k2→ ESn2 + S2

Furthermore, we assume that S1 is just added constantly (rate a) and S2 is removed (rate b). Let X1

denote the complex ESn2 and X2 denote the complex S1ESn

2 . Now we can try to apply the law of massaction and get:

S1 = a − k1S1X1 + k−1X2

S2 = −bS2 − k3Sn2 E + k−3X1 + k2X2

E = −k3Sn2 E + k−3X1

X1 = −k1S1X1 + k−1X2 + k2X2 + k3Sn2 E − k−3X1

X2 = k1S1X1 − k−1X2 − k2X2.

The total enzyme mass is conserved: E + X1 + X2 = 0 ⇔ E + X1 + X2 = E0.For X2, the quasi-steady state assumption can be used, i.e. X2 = 0 ;:

X2 =k1S1X1

k−1 + k2.

Also the activation is a fast process and can thus be assumed to be in quasi-steady state, i.e.

E = 0 ; E =k−3X1

k3Sn2

.

Hence we get:

X1 = E0 − X2 − E = E0 −k1S1

k−1 + k2X1 −

k−3

k3Sn2

X1

⇔ X1

(

1 +k1S1

k−1 + k2+

k−3

k3Sn2

)

= E0

⇔ X1 = E0/

(

1 +k1S1

k−1 + k2+

k−3

k3Sn2

)

Using these equations to replace the variables X1, X2 and E in the upper 5D system yields the reduced2D system:

S1 = a − k1S1E0/

(

1 +k1S1

k−1 + k2+

k−3

k3Sn2

)

+k−1k1S1

k−1 + k2E0/

(

1 +k1S1

k−1 + k2+

k−3

k3Sn2

)

S2 = −bS2 − k3Sn2

k−3

k3Sn2

E0/

(

1 +k1S1

k−1 + k2+

k−3

k3Sn2

)

+k−3E0/

(

1 +k1S1

k−1 + k2+

k−3

k3Sn2

)

+ k2k1S1

k−1 + k2E0/

(

1 +k1S1

k−1 + k2+

k−3

k3Sn2

)

.

Let Km := k−1+k2

k1and Ke := k−3

k3, then these equations can be rewritten to

S1 = a − k2

KmS1E0/

(

1 +k1S1

k−1 + k2+

k−3

k3Sn2

)

S2 = −bS2 +k2

KmS1E0/

(

1 +k1S1

k−1 + k2+

k−3

k3Sn2

)

.

It can be shown that this system shows up oscillatory solutions, but the computations are quite lengthy,so we omit it here.As an example, a numerical solution (for arbitrarily chosen parameter values a = 1, k2 = 2,Km =10,Ke = 5, E0 = 30, n = 3, b = 1):

65

S2

S1

5

5

4

3

4

2

1

30

210

S1

t

5

10

4

3

8

2

1

60

420

S2

t

5

10

4

3

8

2

1

60

420

(phaseplane S1 S2 and the time courses of S1 and S2)

5.5 Gene regulatory networks

Literature: Ellner & Guckenheimer [7]

In this section, we want to deal with the modelling of gene regulatory networks. Some basic knowl-edge from biology:

• Two main steps are involved in gene expression:

1. Transcription of DNA: mRNA is produced with complementary nucleotide sequences

2. Translation: The sequence of nucleotides (“organised” in triplets, in parts of the DNA called“coding sequences”) is translated to proteins with the corresponding amino acid sequence.

• Proteins can influence the regulation of gene expression by binding to the DNA at special places,and by that can block (repressors) or enhance (activators) the “expression” of particular genes. Bythat mechanisms, feedback loops can be constructed in the gene regulatory networks.

• Proteins can also influence each other, form complexes, act as enzymes etc.

So, the modelling itself can be done in a similar way as for chemical reactions. For detailed models,concentrations of mRNA and the proteins can be taken as variables. Remark, that these networks maybe very complex and only few rates etc. may be known. So it often makes sense, to reduce the modelto the most essential parts of the network; e.g. by picking out small parts of such regulation systems. Ifthe model output should be compared with experimental data (e.g. for parameter fitting), there can beanother problem: Often, it is not possible to observe all “players” in the network experimentally.

5.5.1 Clock

Literature: Elowitz & Leibler [8]

Idea: Construct an oscillatory network by using three transcriptional repressors which are inserted intothe bacterium Escherichia coli (which is often used for experiments and thus well-known) by a so-calledplasmid. The structure, how the repressors influence each other, can be seen in the following schematicdiagram:

TetR

cl LacI

Thus, LacI inhibits the transcription of the gene, which codes for TetR; TetR inhibits the transcriptionof the gene, which codes for cl; and cl inhibits the transcription of the gene, which codes for LacI. Thus,we consider here an example for an negative feedback loop.In the following we use index 1 for LacI, index 2 for TetR, index 3 for cl as notation. The followingassumptions were made for the basic model:

• constant decay probability for each mRNA molecule (with same value for all types)

66

• the mRNA synthesis rate depends on the concentration of the corresponding repressor, i.e. adecreasing function of it (also the same for all types)

• constant decay probability for each protein molecule (same for all types)

• The synthesis rate for the repressors depends linearly on the corresponding mRNA concentration

• The synthesis of mRNA and proteins does not depend on other variables

Thus, we get the following model equations, already in their nondimensional form (for i = 1, 2, 3):

mi = −mi +α

1 + pnj

+ α0

pi = −β(pi − mi),

where α0 denotes the transcription rate in presence of a high concentration of the repressor, α thetranscription rate in absence of the repressor, β describes the ratio of the decay rate of the proteincompared to the decay rate of the mRNA, n can be interpreted as a “cooperativity” coefficient for thedependency of the repression on the protein concentration (sometimes also called “Hill coefficient”).How can we find out, if a certain parameter constellation will lead to oscillatory solutions, or tend to anequilibrium, or if other types of long-time behaviour could appear? ; mathematical analysis is necessary!First, we compute the stationary states of the given system, i.e.

mi = −mi +α

1 + pnj

+ α0 = 0

pi = −β(pi − mi) = 0.

We can show: There exists exactly one equilibrium of the system, with

mi = pi = p, i = 1, 2, 3

(i.e. all variables assume the same value). (; the proof is left for the exercises)In the next step, we want to check, under which condition this equilibrium is stable. For that purpose,it is useful to linearise the system, as usual. Let (for i = 1, 2, 3, the corresponding j is “i − 1”)

fi(m1, p1,m2, p2,m3, p3) := −mi +α

1 + pnj

+ α0

gi(m1, p1,m2, p2,m3, p3) := −β(pi − mi)

The partial derivatives read

∂fi

∂mi= −1,

∂fi

∂pj= −

α · n · pn−1j

(1 + pnj )2

, all others = 0

∂gi

∂mi= β,

∂gi

∂pi= −β, all others = 0

So, the complete Jacobian matrix reads

J =

−1 0 0 0 0 − αnpn−1

3

(1+pn3)2

β −β 0 0 0 0

0 − αnpn−1

1

(1+pn1)2 −1 0 0 0

0 0 β −β 0 0

0 0 0 − αnpn−1

2

(1+pn2)2 −1 0

0 0 0 0 β −β

.

(the entries are taken in the order of m1, p1, m2, p2, m3, p3). In the stationary point, we have p1 = p2 =

p3 = p, inserting that into the Jacobian matrix and using the short notation X := − αnpn−1

(1+pn)2 yields

J =

−1 0 0 0 0 Xβ −β 0 0 0 00 X −1 0 0 00 0 β −β 0 00 0 0 X −1 00 0 0 0 β −β

.

67

Using Maple (or being patient and computing it “by hand”) yields for the eigenvalues:

λ1,2 = −1

2− 1

2β ± 1

2

√

(1 − β)2 − 4βX

λ3,4 = −1

2− 1

2β ± 1

2

√

(1 − β)2 − 2βX + 2IβX√

3

λ5,6 = −1

2− 1

2β ± 1

2

√

(1 − β)2 − 2βX − 2IβX√

3

For stability, we need all real parts of the eigenvalues to be negative. By some computations, the followingcondition can be determined:

3X2

4 + 2X<

(β + 1)2

β.

This means: if the given parameters satisfy this condition, then the equilibrium is stable. Let Y = 3X2

4+2X .We can consider the region of stability dependent on the parameter β (the degradation rate of theproteins):

6420

Stable

Unstable

y

14

12

10

8

6

4

2

0

beta

108

Stability ´ region

Obviously, the minimum is reached for β = 1. Due to the rescaling, the mRNA underly degradation withrate 1; so, it corresponds to the situation of similar degradation rates for proteins and mRNAs.Since it is necessary, to be in the unstable situation, to allow the system to oscillate, one could say,that the repressilator has the greatest propensity for oscillations in case of similar degradation rates forproteins and mRNAs. Vice versa, if Y is very large, it allows for a broad range of β values to oscillate.These oscillations can be used to produce something like a clock mechanism.

5.5.2 Toggle-Switch

Literature: Gardner et al. [9]

In this example, also the bacterium E. coli was used for the experimental part of the study. The approachwas quite similar, so the modifications were done by an insertion of plasmids, carrying the informationabout the gene regulatory network. Opposite to the repressilator system considered before, here a networkincluding just two repressors which can inhibit the synthesis of a mRNA.

u v

Typical Toggle−switch

68

Goal of the approach: Find a system with bistable behaviour, i.e. which can exhibit two stable equilibria.Also here, the network structure was translated into a system of ODEs, but in a simpler way thanbefore: Only one variable was used per repressor, i.e. no extra equation for the mRNA, just the proteinconcentrations are considered. The “justification” for that proceeding is again a kind of time scaleargument (similar to the derivation of the Michaelis-Menten equations): If β is large, that pi will changefast until pi ≈ mi is reached (quasi-steady state). So, mi can be replaced by pi, leading to a reducedsystem. Thus, the basic model here reads:

u = −u +αu

1 + vβ

v = −v +αv

1 + uγ.

Several assumptions were made: There is no production left, if the repressor reaches high levels (i.e.,compared to above: α0 = 0).For simplifying the analysis of the model, we assume at the moment that αu = αv = a and β = γ = b(we only consider the case b > 1 here, which corresponds to a “true cooperativity in inhibition” (e.g. bythe binding of multimers to the coresponding position on the DNA). Thus, the system reads

u = −u +a

1 + vb=: f(u, v)

v = −v +a

1 + ub=: g(u, v)

By a simple observation we find immediately, that this system does not admit periodic solutions. Forthat, we compute

div(f, g) = −1 − 1 < 0,

thus, the negative criterion of Bendixson and Dulac can be applied and excludes periodic orbits.We also can show the boundedness of the solutions: Let Ω = (u, v) ∈ R

2+ | 0 < u + v < 2a. Then we

get for (u, v) ∈ R2+ \ Ω:

d

dt(u + v) =

a

1 + vb+

a

1 + ub− (u + v)

≤ 2a − (u + v)

< 0.

Obviously, positivity is preserved by the system. So, it follows that Ω is positively invariant (solutionscannot leave Ω) and absorbing.Now we are in a situation, where we can apply the theorem of Poincare-Bendixson: As periodic orbits areexcluded and the solutions are bounded, the only remaining possibility for an ω-limit set is a stationarypoint. Thus, each solution will tend to a stationary point.In the next step, we look for the stationary states. Again, we start by looking for symmetric equilibriumsolutions, of the form (u, u), i.e. f(u) = 0 where f(u) = −u + a

1+ub . There is just one unique solution,

since f ′(u) = −1 − baub−1

(1+ub)2< 0 ; f is decreasing in u, f(0) = a > 0 and f(a) = − a1+b

1+ab < 0.

Next, we look for asymmetric equilibrium solutions:

0 = u = −u +a

1 + vb

0 = v = −v +a

1 + ub

⇔ u =a

1 + vb

v =a

1 + ub,

i.e., the condition can be reformulated as

u = g(v), v = g(u) where g(u) =a

1 + ub.

Let us consider two examples (both with b = 2), left picture: a = 1, right picture: a = 3:

69

v

u

4

4

3

2

3

1

0210

u-Nullcline

v-Nullcline

v

u

4

4

3

2

3

1

0210

u-Nullcline

v-Nullcline

Obviously, the number of intersection points depends on the actual parameter configuration. We cansee: The bifurcation in this case happens, when the isoclines are tangent to each other (in the symmetricequilibrium).

v

u

4

4

3

2

3

1

0210

u-Nullcline

v-Nullcline

So, we can take this idea for a mathematical analysis. Due to the symmetry, it is necessary to haveg′(u) = −1. The derivative reads

g′(u) = − abub−1

(1 + ub)2

In the symmetric equilibrium, we have u = a1+ub , so we can use that, hence we get

−1 = g′(u) = − abub−1

(1 + ub)2= − abub

(1 + ub)21 + ub

a= − bub

(1 + ub)

⇒ b =1 + ub

ub= u−b + 1

⇒ u−b = (b − 1)

⇒ u = (b − 1)−1b ,

which corresponds to the value of u at the bifurcation point. Due to g(u) = u we can compute thecorresponding value for the parameter a:

a = u(1 + ub) = (b − 1)−1b (1 + (b − 1)−1)

= (b − 1)−1b

(b

b − 1

)

= b(b − 1)−1b−1.

So, the dependency of a on b looks as follows:

70

4

b

3

2

10

1

08642

a

5

This example shows up a pitchfork bifurcation, which is quite rare in a biomathematical context; theunderlying reason is the symmetry of the problem here. (Remark, that even small disturbances ofthe symmetry can split up the pitchfork bifurcation into a transcritical bifurcation and a saddle-nodebifurcation.How can this dependency be interpreted biologically?

• If the protein synthesis rate is large, then the two stable stationary points will be positioned farfrom each other (and far from the symmetric equilibrium point). The larger the distance betweenthe two stable states is, the more “stable” is the switch behaviour - for “noise”, it is hard to causea switch to the other state.

• A large “cooperativity parameter” b allows the system to bifurcate already for small synthesis ratesa.

• Taking all together: large a and b cause a robust switch (but vice versa, it can be complicated touse the switch)

In the next step, we want to study the stability behaviour of the symmetric equilibrium, denoted now by(u, v) = (u, u). The Jacobian matrix reads in general:

J =

(

−1 − abvb−1

(1+vb)2

− abub−1

(1+ub)2−1

)

and in the symmetric equilibrium

J =

(

−1 − bub

(1+ub)

− bub

(1+ub)−1

)

(computation similar to above). Obviously, there is one eigenvalue

λ1 = −1 − bub

1 + ub

(the corresponding eigenvector is (1, 1)). Considering the sign of the determinant, we can estimate thesign of the second eigenvalue. The determinant reads:

det(J) = 1 −(

bub

1 + ub

)2

.

We find:

• If g′(u) = − bub

1+ub > −1, then det(J) > 0 and thus λ2 < 0.

• If g′(u) < −1, then det(J) < 0 and thus λ2 > 0.

This means, that the change of stability happens in g′(u) = −1, which again corresponds to the al-ready determined bifurcation point. So, dependent on the parameter a, the system behaves qualitativelydifferent (left picture: one stable stat. point, right picture: two stable stat. points)

71

u

v v

u

5.5.3 Positive Feedback loop / Modelling Quorum sensing

Literature: [38, 26]

There is another, also very famous possibility to get bistability - to have a positive feedback loop inthe regulation system. We consider that behaviour in detail in the example of a model for bacterial“Quorum sensing”.

Biological background: Some bacterial species have the ability to “communicate” via certain signalmolecules. By such a mechanism, they can e.g. coordinate processes, which are efficient only in situ-ations, where complete colonies (or at least cell clusters) can act in the same way. The first example,where such a behaviour was examined, was the bacterium Vibrio fischeri. These bacteria normally livein the open sea, but some of them also in the light organs of squids. There, they produce light, but only,if their density is quite high (producing light in a low density would be waste of energy, since nobodycan see it). The advantage for the squid is that this light organs allows them not to appear as a “blackshadow” - compared to the night sky, with moon, stars etc. - for predators, which swim deeper than thesquids.

The bacteria do not only produce the signal molecule, but have also a “sensing mechanism”. So, ifthe concentration of the signal molecules exceeds a certain threshold, then different genes are activated.Many of such bacteria include an upregulation of the signal molecule production, leading to a positivefeedback loop. We will study below, what happens in such a case.

Let us consider the underlying regulation process (which is typical, also for other bacterial species)in greater detail:

72

AHL

Diffusion

AHL

DiffusionCytoplasm

LuxI AHL/LuxR Polymer

LuxR−

LuxR

External Medium

AHL=3OC6HSL

Luminescence

AHL

lux−Box DNA

AHL

• LuxI is the enzyme, which produces the signal molecule AHL (with an approximately constantrate).

• AHL can diffuse out and into the cell; furthermore, it can form a complex with the so-called receptormolecule LuxR (in the cell; or more precisely: in the cytoplasm of the cell).

• The LuxR-AHL complex forms higher clusters, so-called “oligomers” or “polymers” - a kind of“chains” formed of identical LuxR-AHL complexes

• These polymers can bind to the DNA (more exactly: tot the so-called lux box, which is the promoterregion of the lux operon and act there as “transcription factor”. This means: The transcriptionof several genes is activated by that binding. In our example the important processes are: Theproduction of LuxI is increased (additional to a kind of “background production”, which alwaystakes place), and the bioluminescence is regulated (via several steps, a light-producing enzyme,called “Luciferase” is formed).

This obviously leads to a positive feedback loop!

Basic model

First, we consider a small submodel, concerning the binding of the polymer to the DNA: The lux box mayassume two states: “bound” or “unbound”. Assumption: The probability of binding to the lux box isproportional to the concentration of the polymer (constant rate κ+). The probability of the correspondingdissociation is assumed to occur with the constant rate κ−. So, the equation for the dynamics off theprobability of a lux box to be bound reads:

d

dtP = κ+yn(1 − P ) − κP ,

where yn denotes the concentration of the polymer. The quasi-steady state assumption yields then:

P =κ+yn

κ+yn + κ−.

Since we assumed that the increase of the LuxI production is proportional to that probability, the so-called“induced production rate” can be described as follows:

Induced production rate = βyn

1 + (β/κ)yn.

β denotes the increased production rate per polymer (when only a few polymer molecules are present)and κ denotes the “asymptotic efficiency” in case of a high polymer density in the system.In our modelling approach, we neglect the production / degradation of mRNA completely, which can bejustified as above by the assumption that these processes happen quite fast and can be assumed to bemore or less in equilibrium.The notations for all variables and parameters in the complete system can be found in the following table:

73

Variable NameMass of AHL outside of the cell xe

Mass of AHL within the cytoplasm xc

Mass of AHL-producing enzyme (LuxI) lMass of receptor molecule (LuxR) rMass of the complex (LuxR—AHL) y1

Mass of the i-mer consisting of two (LuxR—AHL) molecules yi

Parameter NameProduction rate of AHL by LuxI α1

Degradation rate AHL in the cytosol γc

Diffusion rate out of the cell of AHL d1

Diffusion rate into the cell of AHL d2

Rate of AHL binding to LuxR complex π+1

Rate of AHL/LuxR complex dissociation (one AHL bound) π−1

Rate of AHL/LuxR dimer association (binding of two AHL/LuxR complexes) π+2

Rate of AHL/LuxR dimer dissociation π−2

Rate of AHL/LuxR i-mer association (binding of AHL/LuxR to AHL/LuxR (i-1)-mer) π+i

Rate of AHL/LuxR i-mer dissociation π−i

Degradation rate of AHL outside of the cell γe

Background production rate of LuxI α3

Degradation rate of LuxI γ3

Slope of increase of LuxI-production (low dimer concentration) β1

Asymptotics of increase of LuxI-production (high dimer concentration) κ1

Background production of LuxR α4

Degradation rate of LuxR γ4

The pathway is described by the following ODE system:

xc = α1l − (γc + d1)xc + d2xe − π+1 rxc + π−

1 y1

xe = d1xc − d2xe − γexe

l = α3 − γ3l + β1yn

1 + (β1/κ1)yn

r = α4 − γ4r − π+1 rxc + π−

1 y1

y1 = π+1 rxc − π−

1 y1 − 2π+2 y2

1 + 2π−2 y2 −

n∑

j=3

π+j y1yj−1 +

n∑

j=3

π−j yj

y2 = π+2 y2

1 − π−2 y2 + π−

3 y3 − π+3 y1y2

yi = π+i y1yi−1 − π−

i yi + π−i+1yi+1 − π+

i+1y1yi for 2 < i < n

yn = π+n y1yn−1 − π−

n yn.

As usual, we are interested in the stationary points of the system. There, the following conditions arerequired:

0 = π+i yiyi−1 − π−

i yi for i > 2

0 = π+2 y2

1 − π−2 y2

0 = π+1 rxc − π−

1 y1.

Due to r = 0 we get r = α4/γ4. This can be inserted into the upper equations and yields:

yi =

(i∏

m=2

π+m

π−m

) (π+

1

π−1

)i (α4

γ4

)i

xic for 1 ≤ i ≤ n.

Using the equation l = 0 yields:

l =α3

γ3+

β1

γ3

xnc

(∏n

m=2π−

m

π+m

) (π−

1

π+

1

)n (γ4

α4

)n

+ xnc β1/κ1

.

74

For a stationary state, also the intracellular and the extracellular AHL concentration are balanced,

xe =d1

d2 + γe

xc.

From xc = 0 we get:

α1l = (γc + d1)xc − d2xe = (γc + d1)xc −d1d2

d2 + γe

xc

=

(

γc(d2 + γe) + d1d2 + d1γe − d1d2

d2 + γe

)

xc

=

(

γc(d2 + γe) + d1γe

d2 + γe

)

xc,

thus

xc = α1

(

d2 + γe

γc(d2 + γe) + d1γe

)

·

α3

γ3+

β1

γ3

xnc

(∏n

m=2π−

m

π+m

)(π−

1

π+

1

)n (γ4

α4

)n

+ xnc β1/κ1

.

Lumping parameters together, the underlying structure of this equation is of the form

xc = α + βxn

c

xnthresh + xn

c

.

Obviously, the left hand side is just a line, whereas the right hand side is a s-shaped curve. There maybe one or three intersection points, i.e. up to three equilibria. We will consider that later more detailed.

In order to simplify the model (i.e. to reduce the number of ODEs), we can assume different time-scales. In the following, we assume that all dynamics is fast, except for that of AHL (xc and xe). Thecorresponding fast equations get a small ε (on the left hand side - indicating the fast dynamics):

xc = α1l − (γc + d1)xc + d2xe − π+1 rxc + π−

1 y1

xe = d1xc − d2xe − γexe

εl = α3 − γ3l + β1yn

1 + (β1/κ1)yn

εr = α4 − γ4r − π+1 rxc + π−

1 y1

εy1 = π+1 rxc − π−

1 y1 − 2π+2 y2

1 + 2π−2 y2 −

n∑

j=3

π+j y1yj−1 +

n∑

j=3

π−j yj

εy2 = π+2 y2

1 − π−2 y2 + π−

3 y3 − π+3 y1y2

εyi = π+i y1yi−1 − π−

i yi + π−i+1yi+1 − π+

i+1y1yi for 2 < i < n

εyn = π+n y1yn−1 − π−

n yn.

Now we let ε → 0, in the limit we obtain a function for l which only depends on xc (same computationas above):

l =α3

γ3+

β1

γ3

xnc

(∏n

m=2π−

m

π+m

) (π−

1

π+

1

)n (γ4

α4

)n

+ xnc β1/κ1

.

Inserting this equation for l into the equation for xc yields:

xc = α1

α3

γ3+

β1

γ3

xnc

(∏n

m=2π−

m

π+m

) (π−

1

π+

1

)n (γ4

α4

)n

+ xnc β1/κ1

− (γc + d1)xc + d2xe

xe = d1xc − d2xe − γexe.

Also here, we can lump parameters together and get as a simple model:

xc = f(xc) − d1xc + d2xe

xe = d1xc − d2xe − γexe,

75

where

f(xc) := α +βxn

c

xnthresh + xn

c

− γcxc.

In some sense, the external AHL concentration can be controlled (one can add it artificially, or maybewash it out chemostat-like), so we can take the variable xe as a kind of bifurcation parameter. Lookingnow for the stationary states of the system, it depends on the value of xe, how many stationary pointswill be available. We find the following structure for the bifurcation diagram:

stable resting state

extracell. AHL concentration

state

unstable

stable activated state

intr

acel

l. A

HL

conc

.

There, two saddle-node bifurcations appear: for a certain xe two additional stationary points appear,i.e. a stable branch and an unstable branch. For a certain xe > xe, the unstable branch meets thelow stable branch and they “disappear”. This behaviour can be interpreted biologically in the followingway: The system starts with a low bacterial population density, i.e. also with a low AHL concentration.In the time course, the bacterial population grows (with it the still low-level AHL concentration, butonly slightly), until the extracellular AHL-concentration reaches xe. Then, there is no possibility to stayon/near the lower stable branch, the system has to jump to the upper stable branch, corresponding tothe “ON” state. (Further population growth doesn’t change much anymore, just a slight further increaseof the already high AHL concentration). Being on the upper stable branch and reducing the extracellularAHL concentration (e.g. artificially), slightly reduces the intracellular AHL, until xe reaches xe, then thesystem is forced to jump down to the “OFF” state). This is a nice example of so-called “bistability”.(The unstable branch in between does not play a role for the biology, just separates the area of attractionof the two stable branches in the bistable interval).Apart from the switching behaviour, this kind of hysteresis may also help the system near the threshold tostabilise its decision against perturbations; i.e. the system doesn’t have to switch on and off permanently,but can stay for some time in the chosen state.

Parameter fitting

From experiments, the time course of the bacterial population growth is known (the logistic growth iswell-suited to describe it).Problem: For a long time, it was difficult to measure the (extracellular) AHL concentration directly.Instead of the AHL concentration, in many experiments the luminescence was measured (remember:the bacteria V. fischeri produce light, the production of Luciferase should be cotranscribed with thetranscription of luxI). So, we have to use such an “indirect” measurement for fitting our AHL model tothe experimental data. There are two possibilities how to deal with that problem:

Assumption:

Luminescence ~ AHL

(see Schaefer et al., 2000)

production of luciferase

into the model system

Introduce also

(Here we assume that luminescence and AHL are proportional; for the proportionality constant, there isan estimate known from literature). So we can just fit our model to the data for the luminescence andget e.g. the following result:

0 5 10 15 20

−4

−2

02

4

ZEIT

log(

Lum

ines

zenz

/OD

)

76

• The increase of the AHL production between the OFF and the ON state is a factor of ≈ 1600 (fitsnice to experimental data where an increase of luminescence up to a factor of 10000 was observed)

• The average degree of polymerisation of the LuxR-AHL complex is around 40 molecules. Thisseems to be a little bit high, but considering the fitting procedure respectively the dependency ofthe solution curve on n in greater details, there is not much difference between e.g. n = 5 (muchmore realistic) and n = 40. So, the result of the fitting is maybe “too exact” for the data.

Spatial model

Up to now, we only considered homogeneous concentrations. But in reality, the bacteria are very non-homogeneously distributed in space (e.g. they often grow in small clusters e.g.), so there are a lot ofsituations, where the spatial arrangement should also be considered. For that, we want to introduce aspatial model. Let ue denote the extracellular AHL concentration and uc the intracellular.

• The bacteria are assumed to be balls (with radius R)

• There are N cells in the system, cell i is centred around position xi.

• Inside the cells: The concentrations in there are distributed more or less homogeneously, so we donot deal with spatial structure there, just use the ODE dynamic as before.

• Extracellular space: Due to the “geometrical definition” of the positions of the bacteria, it is definedto be

Ω = x ∈ R3 | |x − xi| > R, 1 ≤ i ≤ N.

The AHL just diffuses here freely, and it is still degraded with rate γ.

• The exchange of AHL between intracellular and extracellular space has to go via the cell wall. Theinflow can be described by integration; the outflow is a boundary condition at ∂Ω (of Robin type,the flow depends on the concentration).

Taken together, we get:

d

dtui

C = f(uic) +

∫

|x−xi|=R

(d1ue(x, t) − d2uic) do

∂

∂tue(x, t) = D∆ue − γue

(

−D∂

∂νue(x, t)||x−xi|=R

)

= d1ue(x, t) − d2uic.

In principle, one can use this model for simulations (i.e. use numerical approaches for solving the system).Here, two typical situations with 11 bacteria and an impermeable surface:

Bacteria on flat surface, in biofilm with reduced diffusibility:

77

Bacteria in a pocket with small aperture:

From these pictures, we can observe also the influence of the spatial distribution. So, obviously, thebacteria do not just sense their number or density (what “pure” quorum sensing would mean), but alsothe free diffusible space around (which is called diffusion sensing) and something like their “clustering”.Taken together, this is called “efficiency sening” - with the idea, that the bacteria use the AHL moleculesas “cheap test molecules” for checking how efficient it could be to produce more costly molecules, whichare often involved in this kind of regulation system.

Coming back to Mathematics, there is just a problem for realistic situations: The number of bacte-ria in a natural system may be quite high, leading to numerical problems (fine discretisation necessarynear bacteria). Thus, other approaches could be useful. One possibility is a kind of approximation and isbased on the assumption that bacteria are quite small compared to the “outer space”. In the approxima-tion, the radius R of the (ball-shaped) bacteria is considered in the limit R → 0 (remark, that a correctscaling of influx and efflux is necessary). For the stationary case, with some further requirements, it ispossible to find corresponding approximative equations (just algebraic equations) - much easier to solvenumerically.

A first consideration deals with the special case of just one bacterium:

Lemma 3 Consider for Ω = |x| > R ⊂ R3 the equation

0 = D∆u − γu, u||x|=R = u0.

The unique solution in C2 which exhibits radial symmetry and vanishes for |x| → ∞ reads

U(r) = u0R

re−

√γ/D (r−R).

Proof: Since we look for a classical solution with radial symmetry, the equation is rewritten in radialcoordinates,

d2

dr2u(r) +

2

r

d

dru(r) − γ

Du(r) = 0, u(R) = u0.

The general solution of this system reads

u(r) =1

r

[

C1 sinh(√

γ/Dr) + C2 cosh(√

γ/Dr)]

.

Since sinh(r) as well as cosh(r) tend to infinity for r large, we have necessarily C1 = −C2, leading to

u(r) =C

re−

√γ/D r.

The constant C is determined by the initial condition.

78

2

Remark: In the sense of generalised functions, we find

(

D∆ − γ

)1

|x| e−√

γ/D |x| = 4πδ0(x), (5.13)

i.e. U(r) is the singularity solution of the diffusion operator combined with degradation.

Remark also, that in equilibrium, we have a defined relation between AHL concentration within thecell and AHL concentration just outside of the cell wall that is independent of the dynamics within thecell. Since ν is the outer normal of |x| > R at the boundary |x| = R, we find

−D∂

∂νu(x) = D

d

dru(R) = −u0

[D

R+

√

γD

]

and hence

−u0

[D

R+

√

γD

]

= d1u0 − d2uc

⇒ u0 =d2

d1 + D/R +√

γ Duc.

We thus obtain one single equation for the stationary states of the cell,

0 = f(uc) + 4π R2

[

d1d2

d1 + D/R +√

γ Duc − d2uc

]

= f(uc) − 4π R2 d2D/R +

√γD

d1 + D/R +√

γDuc.

The term 4π R2 d2 (D/R+√

γD)/(d1+D/R+√

γD)uc obviously formulates the net outflow of signallingsubstance.

The correct scaling for the influx and the efflux has to be done in the following way:

Efflux: The efflux of signalling substance is proportional to the cell surface (O(R2)). Thus, the constantd2 has to be rescaled by 1

R2 to become “independent” of the influence of the chosen cell radius.

Influx: In the limit R → 0, we expect point sources, which menas that the concentraion behaves likethe solution of D∆u + Aδxi

(x) close to the center xi of cell i. This pole of order one thus scales byO( 1

R ). But also the influx is proportional to the cell surfaces (which scales by O(R2)). So, takentogether the influx scales by O( 1

R )O(R2) = O(R) and the constant d1 has to be scaled by 1R .

Remark: In the cells, we consider total masses uic(t), but in the extracellular space a concentration is

considered, ue(x, t); that is the reason behind for the different scaling of efflux and influx.

Now, we can write down the rescaled system:

0 = f(uic) +

∫

|x−xi|=R

(d1

Rue(x, t) − d2

R2ui

c ) do

0 = D∆ue − γue(

−D∂

∂νue|∂Ωi

)

=d1

Rue −

d2

R2ui

c.

Let JR ⊂ L2(Ω)×Rn be the set of solutions of the rescaled system for a given scaling R of the cell radius.

Starting with the idea of the solution for a single bacterium, a kind of superposition principle and alot of computations, one can derive the following ansatz for an approximative solution for the signalling

79

substance outside of the cells:

0 = f(uic) +

4π d2

D + d1

− (D − d1

√γ

DR)ui

c + Rd1D

(D + d1)

∑

j 6=i

ujc

‖xi − xj‖exp

(

−√

γ

D‖xi − xj‖

)

u(x) =

N∑

i=1

Ai + R Bi

‖x − xi‖exp

(

−√

γ

D‖x − xi‖

)

(5.14)

Ai = exp(

√γ

DR)

d2

D + d1 + DR√

γD

uic

Bi = −d2d1

(

exp(√

γDR)

D + d1 + DR√

γD

)2∑

j 6=i

ujc

‖xi − xj‖exp

(

−√

γ

D‖xi − xj‖

)

.

Let JR ⊂ L2(Ω) × Rn denote the set of solutions of this equation for given radius R.

The following theorem shows that the rescaled equations and the approximative equations describeessentially the same model. Therefore we need to define the distance between the sets JR and JR. Anelement U ∈ JR is a vector, consisting of the description of the AHL concentration u(x) in the mediumand the AHL concentration in the N cells ui

c. Alike, an element U ∈ JR is a vector (u(x), uic)i=1,..,N .

We define as the distance between U and U the distance of the projection to uic res. ui

c, since the densitywithin the cells determine the exterior field,

‖U − U‖ := max|uic − ui

c| | i = 1, .., N.

The distance between the sets JR and JR is defined in a symmetric way,

dist(JR, JR) := sup inf‖U − U ′‖ | U ′ ∈ JR, inf‖U ′ − U‖ |U ′ ∈ JR | U ∈ JR, U ∈ JR.

Theorem 2 Consider the stationary, scaled equations (5.14) resp. the approximative stationary equa-tions (5.14). Assume f ∈ C2 is a bounded function and require that the set of nonnegative roots of

f(x) − 4πd1d2

D + d1x = 0

is finite and

f ′(x) 6= 4πd1d2

D + d1

for any root (i.e. the roots are hyperbolic). Then limR→0 J(R) = limR→0 J(R) and there is an R0 > 0,s.t.

distJ(R), J(R) ≤ CR3/2 for 0 ≤ R < R0.

Remark: We find JR = JR for R = 0. Furthermore, there is R0 > 0 s.t. JR 6= and JR 6= for0 ≤ R ≤ R0 and – if all elements in JR|R=0 are hyperbolic, we find

dist(JR, JR) = O(R2) for 0 ≤ R ≤ R0.

80

Chapter 6

Modelling of diseases / Medicalapplications

6.1 Vector-borne diseases / Dynamics of Malaria spread

Literature: Murthy [29]

Malaria is a tropical infectious parasitic disease, with annually 100-200 million victims. The controlof Malaria is difficult and of limited success; improved knowledge about population dynamics and epi-demiology is essential. Here, mathematical models can be very useful, for understanding the underlyingmechanisms and interactions, and for (quantitatively) selecting optimal control strategies.

Malaria is caused by parasites (Plasmodium ); these parasites are transmitted to humans by femalemosquitoes (Anopheles), the mosquitoes become infected by biting infected humans (; two hosts for theparasite).

For the life cycle of Plasmodium spp. (one of the relevant Plasmodium species) see http://www.biosci.ohio-state.edu/∼parasite/lifecycles/plasmodium lifecycle.html for a nice overview.

Shortly, a mosquito injects sporozoites into a human

→ Sporozoites enter liver cells, reproduce asexually, enter blood stream

→ Parasites enter red blood cells

→ Parasites reproduce asexually in red blood cells, which burst releasing merozoites

→ some merozoites develop into male/female gametocytes (in red blood cells)

→ Mosquitoes ingest gametocytes (while feeding on blood)

→ Gametocytes fuse in vector’s gut, sporozoites are produced which migrate to the vector’s salivaryglands

→ restart

System characterisation:

1. Variables and Interactions

• Human population, e.g. division into infected and uninfected (differ event. between immunesand susceptibles), or more refined

• Mosquito population, e.g. division into infected (infective or incubating) and uninfected

• Parasite: four different species, in simple characterisation treat them as one

• Climatic and geographical factors (essential: water pools, temperature ; Malaria only inregions between 60N and 40S latitudes, altitude less than 3000 metres), region-dependentrelevant parameters

81

• Mosquito/Human interaction: Only adequate contact is relevant, certain stage in Plasmod-ium life cycle necessary ; basically probabilistic; high certainty leads to an epidemic, smallprobability to a slow spread. Superinfection possible (multiple adequate contacts, resulting inmore than on brood of parasites present in one human host)!

• Immunity: can be caused by previous exposures, vaccination, immunisation, genetic factors ...

2. Deterministic ↔ Stochastic Characterisationdepending e.g. on the population sizes, large sizes allow a deterministic approximation

3. Time Scales, list of the most important ones:

• ≈ 30 minutes: sporozoites enter the liver after their deposition into blood

• ≈ 10-30 days incubation period

• ≈ 1-3 days cycles of symptom occurrence

• ≈ 10-22 days extrinsic incubation period, depending on parasite species and temperature

• ≈ 14 days life cycle of mosquito

• ≈ 70-80 years life cycle of human

• ≈ ≥ 0 duration of immunity

4. Discrete ↔ Continuous Time CharacterisationUsing discrete time, the discretisation interval should be chosen appropriately to the necessary timescales and the desired degree of detail. In most cases, continuous time is chosen.

Now we will shortly introduce some different mathematical models which describe the spread of Malaria.

Model I: Ross, 1916:

Deterministic process, using a single variable x(t) (the proportion of Malaria-affected human population); 1 − x(t) is the unaffected proportion [34].Let r be the recovery rate, h the proportion of unaffected people receiving infectious bites per unit time(both are assumed to be constant in time), then the model reads

x(t) = h(1 − x(t)) − rx(t)

= h − (r + h)x(t).

Let x(0) = x0 be the proportion of infected population at time t = 0. Then the solution can be easilycomputed (e.g. by variation of constants):

x(t) =h

r + h−

(h

r + h− x0

)

e−(r+h)t

Asymptotically, the solution tends to

limt→∞

x(t) =h

r + h

Let p(t) be the population of detected infected people at time t. An additional parameter k, 0 < k ≤ 1,can be introduced, describing the detection probability of infection. Considering a cohort of newbornchildren (x0 = 0) leads to

p(t) = kx(t) =kh

h + r

(

1 − e−(r+h)t)

.

It is possible to adapt the model to concrete data - for detailed data and graphics see [29] page 246. Thefitting procedure yields the parameter values

h = 0.0089 ± 0.00059

r = 0.00050 ± 0.00148

k = 0.637 ± 0.078

82

Model II, Macdonald, 1950

This model should improve model I (small values for r) - it allows the treatment of superinfection, wherea human host can carry multiple broods of parasite (but no explicit precise notion is contained) [19]. Themodel reads:

x(t) =

h − rx(t) for h ≤ rh(1 − x(t)) for h > r

Interpretation: For h < r, all individuals exhibit new infections at rate h. For h > r, an infectedindividual would never recover.Let x0 be the initial condition. The solution can be computed as

x(t) =

hr (1 − e−rt) for h ≤ r1 − e−ht for h > r

with the asymptotic limit value

limt→∞

x(t) =

hr for h ≤ r1 h > r

Now, the estimates for the parameters are more realistic, r ≈ 0.005 (i.e. an improvement compared tomodel I).

Model III, Dietz, 1970

Following the deterministic Macdonald-model, here the total population is divided into groups j (accord-ing to the number of broods that are present in an individual) [2].Let fj(t), 0 ≤ j < ∞, represent the proportion of total population with j broods, x(t) the proportionaffected by malaria, thus

x(t) =∞∑

j=1

fj(t)

and for the unaffected proportion follows directly

1 − x(t) = f0(t) = 1 −∞∑

j=1

fj(t).

Let r be the recovery rate for an individual brood and h be the rate of new infections. So the followingmodel approach is introduced:

f0(t) = −hf0(t) + rf1(t) (6.1)

fj(t) = hfj−1(t) − (h + rj)fj(t) + r(j + 1)fj+1(t), j ≥ 1, (6.2)

with the initial conditions

fj(0) = f0j , 0 ≤ j < ∞, with f0

j ≥ 0 and

∞∑

j=0

f0j = 1.

Again, a cohort of newborns is considered:

f00 = 1; f0

j = 0 for 1 ≤ j < ∞.

Introduce the following function:

F = F (ω, t) =

∞∑

j=0

ωjfj(t)

Taking equation (6.2), multiplying it by ωj and summing over all j yields

∞∑

j=0

ωj fj(t) = h ·∞∑

j=0

ωjfj−1(t) −∞∑

j=0

(h + rj)ωjfj(t) + r∞∑

j=0

(j + 1)ωjfj+1(t)

= h ·∞∑

j=0

ωj(fj−1(t) − fj(t)) + r · (∞∑

j=0

(−j · ωjfj(t) + (j + 1)ωjfj+1(t))

= h ·∞∑

j=0

(ωj+1fj(t) − ωjfj(t)) + r ·∞∑

j=0

(ω · (−jωj−1fj(t)) + (j + 1)ωjfj+1(t))

83

thus∂F

∂t= h(ω − 1)F − r(ω − 1)

∂F

∂ω(6.3)

and from the initial conditions it follows directly

F (ω, 0) = 1.

The result of this operation (6.3) is a first-order linear partial differential equation, but simple enough tobe solvable, we get

F (ω, t) = exp(−h

r(ω − 1)(−1 − e−rt))

Using the definition of x(t) we come to

x(t) = 1 − f0(t) = 1 − F (0, t)

= 1 − exp(−h

r(1 − e−rt)).

For t → ∞, the solution approximates

limt→∞

x(t) = 1 − exp(−h

r).

Taking again p(t) be the proportion of detected infected people, then p(t) = x(t) in case of a perfectdetection, otherwise p(t) = kx(t) with 0 < k ≤ 1.

Problems of the considered models up to now:

• over-simplified

• mosquito population and interaction with it neglected

• deterministic models, but interaction and spread are highly stochastic

• parameters r and h taken from cohorts of newborns, should be based more generally

Additional definitions:

m mosquito density per humana average number of humans bitten per mosquito per days proportion of mosquitoes with sporozoitesb proportion of actually infectious mosquitoes (with sporozoites)n extrinsic incubation periodp survival probability for a given mosquito over a given dayγ mosquito death rate

Generally, this yields for the rate of new infections

h = masb

Assumption: An infected mosquito stays infected during its life; the infection happens by an adequatecontact with an infected person. Due to the extrinsic incubation period, it is not possible to have infectedmosquitoes with an age < n.Further assumption: Adequate contacts between a mosquito of age t > n and infected humans aregoverned by a Poisson process with parameter xa(t − n), then

1 − exp(−xa(t − n))

describes the probability of at least one adequate contact and the proportion of infected mosquitoes ofage [t, t + dt] is

(1 − exp(−xa(t − n))) · γe−γtdt

84

The overall proportion of infected mosquitoes (corresponding to s) we get by integrating:

s =

∫ ∞

n

(

1 − e−xa(t−n))

· γe−γt dt

=

∫ ∞

n

γe−γt − γexan−(xa+γ)t dt

=[−e−γt

]∞n

+

[γ

xa + γexan−(xa+γ)t

]∞

n

= e−γn − γ

xa + γe−γn

= axe−γn/(ax + γ).

The survival probability of a mosquito for one day is

p = e−γ ⇔ − ln p = γ

(where γ is the mosquito death rate), yielding

s = axpn/(ax − ln p)

andh = (ma2bxpn)/(ax − ln p)

Obviously, h depends on several basic parameters, including the interaction between humans and mosquitoes.

Model IV, Dietz, Molineaux and Thomas, 1974

This is also a deterministic, but discrete and more detailed model, developed for a field project in Nigeria[3]. The following variables:

x1(t) non-immune / negativex2(t) non-immune / incubatingy1(t) non-immune / positive infectiousy2(t) non-immune / positive - non infectiousx3(t) immune / negativex4(t) immune / incubatingy3(t) immune / positive - non-infectious

(“Negative” means no parasites in the blood, “incubating” means: parasites only in the liver). Thetransition rates are:

h(t) from negative to incubating (in both cases)Q(t) from incubating to infectious (non-immunes) resp. from incubating to non-infectious (immunes)α1 from infectious to non-infectious (non-immunes)α2 from non-immune non-infectious to immune non-infectiousR1 from non-infectious positive to negative in non-immuneR2 from non-infectious positive to negative in immune

and constants:

n incubation periodη extrinsic incubation periodδ new birth and death rate for for all groups

Remark: We assume constant population size = 1.The (discrete) changes are given by the following equations:

∆x1 = δ + R1y2 − (h + δ)x1

∆x2 = hx1 − Qx1(t − n) − δx2

∆x3 = R2y3 − (h + δ)x3

∆x4 = hx3 − Qx3(t − n) − δx4

∆y1 = Qz1(t − n) − (α1 − δ)y1

∆y2 = α1y1 − (α2 + R1 + δ)y2

∆y3 = α2y2 − Qx3(t − n) − (R2 + δ)y3

85

(some variables refer to the incubation period with a time lag of n). ∆ denotes the “difference operator”,i.e. it means for example ∆x1 = x1(t + 1) − x1(t).Let qi be the probability of detecting a parasite, then the proportion of positive detected population reads

p(t) =

3∑

i=1

qiyi(t)

and the true proportion of positives is

x(t) =

3∑

i=1

yi(t).

The model can be fitted to experimental data, this yields values for the parameters. A comparison ofexperimental data and model output can be found in [29], Figure 14.3.

Model V, Bekessy, Molineaux and Storey, 1976

In this model for the change between infected and uninfected there is a uncertain change mechanismincluded. The following definitions are used:

N total population (fixed)X(t) number of infected individuals at time t

So, the number of susceptibles at time t reads N − X(t).Assumption: The probability of one new infection in the time interval [t, t+∆t] is given by h(N−X(t))∆tand the probability of one recovery is rX(t)∆t (again h is the rate of new infections and r the recoveryrate).Use the following definitions:

pj = probX(t) = j, 0 ≤ j ≤ N

P (z, t) =

N∑

j=0

pj(t)zj

pj satisfy the following set of differential equations:

dp0(t)

dt= −hNp0(t) + rp1(t)

dpj(t)

dt= −(h(N − j) − rj)pj(t) + h(N − j + 1)pj−1(t) + r(j + 1)pj+1(t) for 1 ≤ j ≤ N − 1

dpN (t)

dt= −rNpN (t) + hpN−1(t)

and P (z, t) satisfies the PDE

∂P

∂t= (1 − z)(r + hz)

∂P

∂z+ γh(z − 1)P.

For t = 0, the number of infected individuals is assumed to be x(0) = a, then we get

pj(0) =

1 if j = a0 otherwise

andP (z, 0) = za

This initial condition allows to get the analytical solution

P (z, t) = (r + h)−N ((r + hz) + r(z − 1)e−(r+h)t)a · ((r + hr) − h(z − 1)e−(r+h)t)N−a.

86

6.2 Diabetes

Literature: [39]

In this section, we consider the regulation of blood glucose by insulin. Roughly speaking: If the bloodglucose level is increased permanently, this disease is called diabetes. Being more precise, “Diabetesmellitus” denotes a group of metabolic diseases of the metabolism, with the main diagnostic findings ofa hyperglycemia. There are different possibilities for the underlying reason: Either, there is a lack ofinsulin (a pancreatic hormone, controlling blood glucose), or a kind of insulin resistance .There are mainly two distinct forms of the disease:

Type I: The pancreatic cells which produce insulin (β-cells in the so-called islets of Langerhans ) aredestroyed (e.g. by a mislead autoimmune process)

Type II: The secretion of insulin or the response to insulin is disturbed (especially under conditionswhen the blood glucose level is increased)

Here, we have a look on the regulation system of blood glucose in general.

6.2.1 Model for glucose and insulin dynamics, including β-cell mass dynamics

Literature: Topp [37]

Up to now, most mathematical models for glucose regulation only contained the dynamics of glucoseand insulin. But additionally, also the available mass of β-cells may play a role, even though it works ona slower time scale. Such an approach is considered in the model of Topp et al. .It aims to describe relatively slow glucose kinetics, i.e. situations with a time-scale of days to years (e.g.fasting blood glucose levels). For that purpose, a single-compartment model is sufficient.

Glucose dynamics

Let G denote the concentration of glucose in the blood (dimension: mg dl−1). Time t is measured indays. The general approach for the mass balance equation of glucose reads

dG

dt= Production − Uptake.

To obtain the proper units, the rates of glucose production and uptake have to be normalised by thevolume of the glucose distribution, i.e. to be of dimension mg dl−1 d−1.From different experiments (e.g. using the so-called glucose clamp technique), it is known:

• Glucose production decreases linearly with respect to glucose levels (at constant insulin levels)

• Glucose uptake increases linearly with respect to glucose levels (at constant insulin levels)

This leads to the following approach:

Production = P0 − (EG0P + SIP I)G

Uptake = U0 + (EG0U + SIUI)G,

where P0 and U0 are rates of glucose production and uptake at zero glucose (dimension mg dl−1 d−1),EG0P and EG0U are called the glucose effectiveness for production and uptake at insulin level 0 (dimensiond−1), SIP and SIU denote the insulin sensitivity for production and uptake (dimension µU−1ml d−1),and I gives the blood insulin concentration (dimension µU ml−1). Taking all these informations together,we end up with

dG

dt= R0 − (EG0 + SII)G, (6.4)

where we use the net rate of production at zero glucose R0 = P0 − U0, the total glucose effectiveness atzero insulin EG0 = EG0P + EG0U , and the total insulin sensitivity SI = SIP + SIU .

87

Insulin dynamics

Also for the insulin, we use a single-compartment equation, under the assumption that we consider mainlythe long-time evolution of fasting insulin levels (with a slow time-scale). The general approach reads

dI

dt= Secretion − Clearance.

To obtain the proper units, the rates of secretion and clearance have to be normalised by the volume ofthe insulin distribution, i.e. to be of dimension µU ml−1 d−1. From different experiments/publications itis known:

• Near the steady state, the rate of insulin clearance is proportional to the blood insulin levels

• The rate of insulin secretion (from the pancreatic tissue) can be described appropriately by asigmoidal function of the glucose concentration.

This leads to the following approach:

Clearance = kI

Secretion = βσG2/(α + G2),

where β denotes the mass of pancreatic β cells (dimension: mg). σ describes the maximum insulinsecretion by an β cell (it is assumed, that all β cells have the same σ). The Hill function G2/(α + G2) isobviously a sigmoidal function, with values between 0 and 1, and reaches its half maximum at G = α1/2.Taking all information together, we get

dI

dt= βσG2/(α + G2) − kI. (6.5)

β-Cell dynamics

Last, but not least, also the β-cell dynamics can be described by a single-compartment equation, withthe general approach

dβ

dt= Formation − Loss.

There are three possibilities for the formation of new β-cells: Replication of existing β-cells, neogenesisfrom stem cells, and transdifferentiation of other cells. Since the latter two seem to play a minor role, weneglect them for our modelling approach.Cell loss can be caused by regulated cell death (apoptosis), unregulated cell death (necrosis), or trans-differentiation to other cell types. Transdifferentiation is neglected also for the cell loss.From experimental observations, it is known:

• The percentage of β cells which takes part in replication depends nonlinearly on the glucose levelin the medium. In principle, the replication rates increase when the glucose level increases, just forvery high glucose levels (called extreme hyperglycemia), the replication can be reduced.

• β cell death depends nonlinearly on glucose, with increasing glucose levels, it first decreases, laterincreases again

This leads to the following approach:

Replication = (r1rG − r2rG2)β

Death = (d0 − r1aG + r2aG2)β

(The death equation is just a simple approach, can be done maybe in a more suitable way). Here, d0

denotes the death rate at glucose level zero (dimension: d−1); the further constants r1a resp. r2a havethe dimensions mg−1dl d−1 resp. mg−2dl2d−1. The rate constants r1r and r2r have the dimensionsmg−1dl d−1 and mg−2dl2d−1. Taken together, this yields

dβ

dt= (−d0 + r1G − r2G

2)β, (6.6)

where r1 = r1r + r1a and r2 = r2r + r2a. So, the equations (6.4), (6.5) and (6.6) form the basic βIGmodel.

88

Analysis of the system

There are different timescales contained in our system:

• glucose and insulin levels can change within minutes ; fast subsystem

• β-cell mass changes within days ; slow subsystem

First, we consider the fast subsystem:Obeying the different time-scales can be done by

G = R0 − (EG0 + SII)G

I = βσG2/(α + G2) − kI

β = ε(−d0 + r1G − r2G2)β

(which means: the change in β happens only very slowly, due to ε ≪ 1): Let ε → 0, thus, the fastdynamics is described by

G = R0 − (EG0 + SII)G

I = βσG2/(α + G2) − kI

(β = 0)

The condition for a stationary state is, as usual,

G = 0

I = 0

⇔G = R0

EG0+SII

I = βσG2

k(α+G2)

Let f(I) = R0

EG0+SII describe the G = 0 isocline. Also the I = 0 isocline can be described in terms of Iby the following transformation:

I =βσG2

k(α + G2)⇔ Ik(α + G2) = βσG2

⇔ IkG2 − βσG2 + Ikα = 0

⇔ G2 = − Ikα

Ik − βσ

⇔ G =

√

Ikα

−Ik + βσ=: g(I).

Obviously, f(I) is strictly monotone decreasing, whereas g(I) is strictly monotone increasing, i.e. max.one intersection point is possible. Due to f(0) = R0

EG0> 0, g(0) = 0 and limI→ βσ

kg(I) → +∞, there

exists exactly one stationary solution (G∗, I∗), where 0 < G∗ < R0

EG0, 0 < I∗ < βσ

k . The correspondingJacobian matrix reads:

J =

(

−(EG0 + SII) −SII2βσG·(α+G2)−βσG2·2G

(α+G2)2 −k

)

=

( −(EG0 + SII) −SIG2βσαG

(α+G2)2 −k

)

.

Then:

trJ = −(EG0 + SII) − k < 0

detJ = k(EG0 + SII) +2βσαSIG

2

(α + G2)2> 0

; the stationary solution is stable. The sign of the discriminant will decide if it is a stable node or astable spiral. For realistical parameter values, the discriminant is negative, i.e. the stationary solutionis a stable spiral and the glucose and the insulin underlies a damped oscillation when approximating thestationary point.We use the following parameter values:

89

Parameter Value (and units)SI 0.72 mlµU−1d−1

EG0 1.44 d−1

R0 864mg dl−1d−1

σ 43.2 µU ml−1 d−1

α 20000 mg2dl−2

k 432 d−1

d0 0.06 d−1

r1 0.84 · 10−3 mg−1 dl d−1

r2 0.24 · 10−5 mg−2 dl2 d−1

An example for a time-course (blue line: Glucose, green dashed line: Insulin); after a while, both tendto their stationary state:

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

20

40

60

80

100

120

140

160

180

200

time

Time course of Glucose and Insulin

The position of the stationary point depends on the available β-cell mass: The G = 0 isocline does notdepend on β. But g(I) (i.e. the I = 0 isocline) depends on β, the higher β, the lower is g(I). This means:Higher values for β shift the position of the stationary point more downwards ⇔ more to the right. Theinfluence on the position of the isoclines can be seen in the following figure:

Insulin

Glucose

353025

400

15 20

300

0

100

10

600

0

200

5

500

The red line is the G = 0 isocline; the others are I = 0 isoclines for different values for β (cyan: β = 150,blue: β = 300, black: β = 450).

Next step: We consider the slow subsystem, which means: The fast subsystem is assumed to be insteady state - and this steady state is just slowly shifted by the available β-cell mass. The equation forthe slow subsystem reads:

β = (−d0 + r1G − r2G2)β

Obviously, this 1D slow system shows up three stationary solutions:

• β = 0 called “pathological state”

90

• r2G2 + r1G − d0 = 0 ⇔ G1,2 =

−r1±√

r21−4r2d0

−2r2For the given parameter values, these stationary

states correspond to

G = 100mg

dlcalled “physiological state”

and G = 250mg

dlwhich is unstable

Splitting up the β-cell replication and the death rate, dependent on the glucose level G, we can distinguishdifferent zones of behaviour:

Zone I = hypoglycemia : The death rate is greater than the replication rate ; decreasing β-cell mass; increasing glucose level ; physiological state

Zone II = mild hyperglycemia : The replication rate is greater than the death rate ; increasing β-cellmass ; decreasing glucose level ; physiological state

Zone III = extreme hyperglycemia : The death rate is greater than the replication rate ; increasingglucose level (positive feedback loop) ; pathological state, with zero β-cell mass

Obviously, starting in Zone I or II ends up in the physiological state, whereas Zone II always leads toa dysregulation. The unstable stationary state of the slow subsystem divides the two different basins ofattraction of the other two stationary states.In the next step, we take together what we have learned about the subsystems into a whole βIG model.By the standard analysis, using the given parameter values we find three stationary states in (β, I,G):

• (300, 10, 100) is a stable spiral, the physiological state

• (37, 2.8, 250) (saddle point)

• (0, 0, 600) is a stable node

In the same way as in the Michaelis-Menten case, the solutions will quickly tend to the slow manifold(along trajectories which are more or less perpendicular to the β-cell mass axis), later move along thatslow manifold (according to the dynamics of the β-cells), until they reach one of the stationary points.We can consider e.g. the projection onto the β − G plane, it looks as follows:

300 600 900 1200

β−cell mass (mg)

0

100

Glu

cose

(m

g/dl

)

600

500

400

300

200

saddle

pathological

physiological

Again, it depends on the starting position, where the solution will tend to (as it can be seen in the picture,similarly to the observations in the slow subsystem only).Even though this model seems to be quite simple, it agrees quite nicely with the experimental observations.

How can a regularly working regulation system end up in hyperglycemia?Of course, the choice of parameters can influence the behaviour of the system; e.g. the parameters of theβ-cell mass equation may influence the number of stationary states, whereas all parameters may influencethe positions of the stationary states. There are very different possibilities which may occur in a sickperson: Defects in glucose dynamics, defects in insulin dynamics or defects in β-cell mass dynamics.

91

6.3 HIV

Literature: [24, 28]

Some basics about the so-called human immunodeficiency virus, HIV: It is a retrovirus and leads tothe “acquired immune deficiency syndrome”, AIDS. There are several properties of the virus which makeit (up to now) impossible to cure this disease. The virus attacks the so-call CD4+ T-cells, also calledT-helper cells, which are essential for the human immune system. To be infected with the virus is notequivalent with having AIDS - the latter one is in some sense the “end-stage” of the disease, it may take5-10 years to reach this stage, even though the viral replication dynamics is very high (but the immunesystem counteracts). So, there is a very long incubation period.

6.3.1 Anderson’s First Model

This is really a very simple model, only suitable as a first approach or as a first idea: One considers apopulation which consists only of people which are infected with HIV and is interested in the time coursewhich leads from the HIV infection to AIDS.Let M(t) denote the size of the “only infected” part of the population, N(t) the part of the populationwhich developed already AIDS. As initial conditions, we start with M(0) = M0 and N(0) = 0, the totalpopulation size is kept constant, i.e.

M(t) + N(t) = M0.

The approach can be nondimensionalised by dividing throughout by M0, i.e. then

x(t) =M(t)

M0, y(t) =

N(t)

M0, x(t) + y(t) = 1.

A simple conversion rate, from infection to AIDS is assumed (and denoted by ν(t)), so the correspondingODEs read

x = −ν(t)x, x(0) = 1 (6.7)

y = ν(t)x, y(0) = 0 (6.8)

Remark: In this model approach, all infected people will get AIDS (this is not necessarily true in allcases).It may make sense to assume that the rate ν increases in time, due to the gradual break-down of theimmune system; a simplest approach for that may be

ν(t) = at, a > 0, a constant .

Then, the solution of the system (6.7, 6.8) is

x(t) = e−at2/2,

y(t) = 1 − e−at2/2;

obviously, x → 00 and y → 1 for t → ∞.Even though, this model is so simple, one can compare it to real-world data (from people who got thevirus via a blood transfusion); the best fit value is then around a = 0.2341/year.

6.3.2 Anderson’s improved HIV model

Also in this model, no cellular process, but only the epidemic as such is considered. The total population(here assumed to consist of a homosexual population), N(t) is splitted up into several “compartments”:

• X(t): number of susceptibles at time t

• Y (t): number of infectious at time t

• A(t): number of AIDS patients at time t

• Z(t): number of the seropositives at time t, who are not infectious

Maybe, the easiest way to find the desired equations, is to look at a rough flow diagram of the disease:

92

Susceptible X

Infectious Y

Seropositive Znon−infectious

Natural deathAIDS A

Natural death

Disease−induced death

Natural death

Natural death

B

d

p

c

µ

λ

µ

(1−p)ν

µ

µ

ν

So, the basic model equations can be formulated as follows:

X = B − µX − λcX

Y = λcX − (ν + µ)Y

A = pνY − (d + µ)A

Z = (1 − p)νY − µZ

N(t) = X(t) + Y (t) + Z(t) + A(t).

Most parameters are clear in there meaning; λ denotes the infection probability from a randomly chosenpartner (λ = βY/N , β is the transmission probability), c the number of sexual partners, ν the conversionrate from infection to AIDS.Obviously, the total population N(t) is not constant in time, as

N = B − µN − Ad.

Similar to other epidemic models, it makes sense to consider the basic reproductive rate R0 ≈ βc/ν. If itis > 1 (that means, the number of secondary infections, caused from a primary infection, is greater thanone), then an AIDS epidemic will spread out.As usual, we try to determine the stationary points, i.e.

0 = B − µX − λcX

0 = λcX − (ν + µ)Y

0 = pνY − (d + µ)A

0 = (1 − p)νY − µZ

In reversed order, we get from these equations:

Z =(1 − p)ν

µY

A =pν

d + µY

Y = 0 or X =ν + µ

βcN

Y =B − µX

βc

N

Xor X =

BN

µN + βcY.

In the first case (Y = 0), this leads to Z = 0, A = 0, X = Bµ , N = X, i.e. this corresponds to a system

without HIV.In the second case (Y 6= 0), we start with an (at the moment arbitrary) N∗ (its size will be determined

93

later). Then we get step by step:

X∗ =ν + µ

βcN∗

; Y ∗ =B − µν+µ

βc N∗

βc

1ν+µβc

=B

ν + µ− µ

βcN∗

; A∗ =pν

d + µY ∗ =

pν

d + µ

(B

ν + µ− µ

βcN∗

)

; Z∗ =(1 − p)ν

µY ∗ =

(1 − p)ν

µ

(B

ν + µ− µ

βcN∗

)

.

Using that, we can compute the “resulting” N∗:

N∗ = X∗ + Y ∗ + A∗ + Z∗

=

(ν + µ

βc− µ

βc− pν

d + µ

µ

βc− (1 − p)ν

µ

µ

βc

)

N∗

+B

ν + µ+

pν

d + µ

B

ν + µ+

(1 − p)ν

µ

B

ν + µ

=ν

βc

(

1 − pµ + d + µ − dp − pµ

d + µ

)

N∗ +B

ν + µ

((d + µ)µ + pνµ + (1 − p)ν(d + µ)

(d + µ)µ

)

=ν

βc

(dp

d + µ

)

N∗ +B

ν + µ

((d + µ)µ + ν(d + µ) − pνd

(d + µ)µ

)

,

from that it follows:(

1 − ν

βc

(dp

d + µ

))

N∗ =B

ν + µ

((d + µ)(ν + µ) − pνd

(d + µ)µ

)

and

N∗ =B

ν + µ

((d + µ)(ν + µ) − pνd

(d + µ)µ

)/ (βc(d + µ) − νdp

βc(d + µ)

)

=Bβc

(ν + µ)µ

((d + µ)(ν + µ) − pνd

βc(d + µ) − pνd

)

.

Remark: Not all parameters allow for an “endemic equilibrium”, i.e. that the coordinates of the secondstationary point are biologically meaningful!As usual, one can apply now linearisation and by that get, that the “endemic equilibrium”, in case ofbiological relevance, corresponds to a stable spiral; all solutions tend to it (proofs are left out here).

6.3.3 Modelling Combination Drug Therapy

Literature: [27]

Some more basics about the mechanisms of a HIV infection: The HI virus enters the CD4+ T-cells(there is a high affinity of a protein on the surface of the virus towards the so-called CD4 protein on theCD4+ T-cell). The HI virus itself has a very simple structure and cannot reproduce itself, so it relies onreproduction by the host cell. The virus bears a copy of the RNA (as precursor of DNA); the RNA istranslated into a DNA copy inside the T-cell. When the T-cell reproduces, it also produces a copy of thecirus. In principle, there are two possibilities for the virus to “leave” the host cell:

• slowly, the host cell survives

• fast, the host cell collapses

(Here, we consider only a simpler case for the modelling, where only the virus production after lysis, i.e.cell death, is taken to be relevant)For the model, four components are taken:

• uninfected T-cells, denoted by T

94

• infected T-cells, denoted by T ∗ (being more precise: “productively infected T-cells”, since not allinfected T-cells really produce the virus)

• infectious viruses, denoted by VI

• noninfectious viruses, denoted by VNI

The underlying ideas for the model are a little bit similar to the approach for the Hepatitis C modelwhich we treated in lecture part I.There is a source of new, uninfected T-cells (they are produced in the thymus), this term is called s.In order to have a non-zero stationary state in absence of infection, an additional term for the T-cellproduction, limiting their number. Additionally, there is a clearance term (rate dT ), and of course, byinfection uninfected cells switch to the infected ones (this term depends on the available infectious viruses;rate k).For the infected T-cells, there is the positive term according to the newly infected cells (it can be reducedby drugs with an efficacy called nrt (0 ≤ nrt ≤ 1), i.e. for nrt = 1, it is completely effective, no productionof infected T-cells possible anymore); also here: natural death is possible, rate δ. As an additionalcomponent of the multiple drug treatment, a protease inhibitor is used, let np be its effectiveness - thismeans: a part np of the new produced viruses will be noninfectious (np would correspond to completelyeffective drug). A factor N gives information about how many viruses will be set free by cell death.Both, infectious and noninfectious viruses are assumed to have the same natural death rate, c. So we canformulate the model:

T = s + pT

(

1 − T

Tmax

)

− dT T − kVIT

T ∗ = (1 − nrt)kVIT − δT ∗

VI = (1 − np)NδT ∗ − cVI

˙VNI = npNδT ∗ − cVNI .

It is easy to see that there should be two stationary states for the system: (Ts1, 0, 0, 0) - without anyinfected cells and viruses - and (Ts2, T

∗, VI , VNI) - the “infected steady state”. One can compute:

Ts1 =Tmax

2p

[

(p − dT ) +

√

(p − dT )2 +4sp

Tmax

]

. (6.9)

Question: What happens after a small perturbation by introduction of HIV? This is analysed by theusual linearisation, the eigenvalues are found to be:

λ1 = p

(

1 − 2Ts1

Tmax

)

− dT

λ2,3 = −c + δ

2± 1

2

√

(c + δ)2 − 4cδ + 4δNkTs1(1 − nc)

λ4 = −c,

where nc = 1 − (1 − nrt)(1 − np), it is called the “effectiveness of the combination treatment”.Obviously, all eigenvalues are real. For stability, we have to check their sign: λ3 and λ4 are negative, i.e.would allow for stability. λ1 is negative, if

p

(

1 − 2Ts1

Tmax

)

− dT < 0

⇔ 1 − 2Ts1

Tmax<

dT

p

⇔ 2Ts1

Tmax> 1 − dT

p

⇔ Ts1 >Tmax

2

(

1 − dT

p

)

.

This condition is obviously satisfied, due to (6.9).λ2 is negative, if

c + δ >√

(c + δ)2 − 4cδ + 4δNkTs1(1 − nc),

95

i.e., the uninfected stationary state is stable if

−4cδ + 4δNkTs1(1 − nc) < 0

⇔ NkTs1(1 − nc) < c

⇔ 1 − nc <c

NkTs1

⇔ nc > 1 − c

NkTs1.

Biological interpretation: The virus can be (more or less; in reality not completely) eliminated in case ofthe drugs are strong enough.Let us consider the infected stationary state, with the coordinates

Ts2 =c

Nknc

T ∗ =cVI

δN(1 − np)

VI =s

kTs2+

1

k

[

p

(

1 − Ts2

Tmax

)

− dT

]

¯VNI =npVI

1 − np.

It has to be checked, if the infected stationary state is biologically relevant. The only problem couldappear in VI . Simple computations show: For VI = 0, the uninfected and the infected stationary statecross each other; 0 < Ts2 < Ts1 is equivalent to VI > 0. The biological interpretation here is clear: Thereis a lower level of T-cells in the infected stationary state than in the uninfected.As usual, it is possible to consider now the Jacobian matrix, to find out something concerning the stability.Details are left out here (boring to do all these computations ... but it is possible to compute everythingexplicitely), but the standard approach (with applying the Routh-Hurwitz criterion) yields: The infectedstationary state is stable, if

c < NkTs1(1 − nc),

which is also the condition for being positioned inthe biologically relevant area.Taken together, this yields: There is a transcritical bifurcation (exchange of stability)!

Special case for the T-cell dynamics:A very simple approach is T = const = T0. But an antiretroviral therapy can, to some extend, lead to arecovery of T-cells; from some patient data, one could assume a linear or exponential function of time.Observation / Assumption: VI(t) becomes very small soon after start of therapy (e.g. for np = 1:VI(t) = V0e

−ct), so there is some reason to neglect the term −kVIT . Furthermore, for a simple approachs is assumed to be the major mechanism how T-cells are replaced, thus

T = s − dT T,

or even simplerT (t) = T0 + at

(a constant) is taken as T-cell dynamics.Assumption: nrt = 0 (only protease inhibitor drugs are used), then the complete system reads:

T ∗ = kVI(T0 + at) − δT ∗

VI = (1 − np)NδT ∗ − cVI

˙VNI = npNδT ∗ − cVNI .

Remark: This is a nonautonomous system! (We do not know many tools how to analyse it). Fortunately,it is very easy to transform it into an autonomous system, just by replacing T (t) = T0 + at by thecorresponding ODE, T = a, with initial condition T (0) = T0. Even that simple model can be fitted topatient data quite well (and by that, parameter values can be estimated).

96

6.3.4 Delay model for HIV infection with drug therapy

Biological problem: When a virus enters a cell, this cell will only start after a certain time delay withthe viral production. For keeping the model simple, it is assumed that the uninfected T-cells stay on aconstant level, i.e. T = T0.In an example, a fixed time delay τ was assumed; although, it would be more realistic to take a distributeddelay function. So, the model equations read:

T ∗ = kT0VI(t − τ) − δT ∗

VI = (1 − np)NδT ∗ − cVI

˙VNI = npNδT ∗ − cVNI .

It can be interpreted as follows:

• Average life span of a virus: 1/δ

• Average life span of a cell from the time point of infection until it dies: τ + 1/δ

6.4 Tumour growth

Literature:

Here, we do not consider a special type of a tumour, but some basic principles which arise more orless for all “solid tumours”.The growth of a tumour consists of different stages:

• The cells in a very small tumour receive nutrients just by simple diffusion and can grow exponentially(no lack of nutrient).

• When the tumour becomes larger, the nutrient concentration at the centre of the tumour willdecrease (the nutrients are consumed up on their “way” to the centre).

• If the nutrient level is below a certain level, the cells die ; a central necrotic core will develop, thegrowth rate decreases more and more.

• In the next stage, the “angiogenesis” is initialised, leading to the dangerous vascular stage: Thetumour induces the building of blood vessels - which are used for the supply with nutrients, but alsofor the invasion of other tissues in the body, and the forming of metastases (; “explosive growth”).

From a mathematical point of view, one has to consider a diffusion process for the nutrient supply. Theinteresting point is that the tumour boundary is moving, due to the growth of the tumour. So, alsothe position of the boundary has to be described in the mathematical model. We use the followingassumptions for the model:

1. There is a diffusive equilibrium state between the cell colony and the surrounding medium. Thetumour consists of three different layers:

• In the centre: core of necrotic debris

• Next layer: quiescent, non-proliferating cells (this layer is thin)

• Out layer: proliferating cells

proliferating cells

Quiescence

Necrosis

Surrounding medium

97

2. The nutrient level is denoted by σ(x, y, z, t).

• If σ ≥ σ1 (a constant, critical value), then the cells proliferate.

• If σ ≤ σ2 (another constant, critical value), then the cells die.

• σ2 < σ < σ1 corresponds to the quiescent region.

Let σo denote the value of σ at the outer surface of the tumour. From experimental observations,it is known that

h =

ν√

σo − σ1 for σo > σ1,0 for σo < σ1,

where h is the thickness of the proliferating cell layer, ν is a constant.

3. Let dA be an element of the tumour surface area. New cell volume is produced at the rate βh dA(β constant) by the live cell volume dV = h dA. In the same way, this volume consumes nutrientsat the rate γh dA, γ constant.

4. In case of σ2 < σ < σ1, the proliferating cells become quiescent. The rate of gain of quiescent massper unit volume is constant.

5. There is a loss of necrotic mass per unit volume; it is assumed to have a constant rate.

6. The tumour is kept “compact” by a surface tension force T , which is assumed to be proportionalto the mean curvature K of the boundary.

7. The cell colony of the tumour is considered as a “incompressible fluid”, so the birth and death ofcells cause internal pressure differentials, and these lead again to some motion of the cells, i.e.

q = −∇P,

where q(x, y, z, t) is a vector describing the velocity of the cells, and P (x, y, z, t) is a magnitudewhich is proportional to the internal pressure.

Further notation: LetΓ(x, y, z, t) = 0

represent the outer surface andΓN (x, y, z, t) = 0

the outer surface of the necrotic core.From the conservation of mass, we find (for n denoting the normal vector on dA pointing out):

(q+ · n − q− · n)dA = βh dA ⇔ q+ · n = q− · n + βh on Γ = 0.

(the flow out of dV via dA (small h) has to be compensated by the production).In the same way, the nutrient diffusion into dV has to be compensated by the nutrient consumption indV , yielding

kn · ∇σ = γh on Γ = 0.

(Remark: No diffusive transport from the interior, due to σ ≤ σ1 )Let λ = βν and µ = γν, i.e.

βh = βν√

(σ − σ1) = λ√

(σ − σ1)

γh = γν√

(σ − σ1) = µ√

(σ − σ1).

λ and µ can be assumed to be of order one in case of a large proliferation rate of new cells.Pressure P and tangential velocity components are assumed to be continuous across the surfaces. Sincethe surface tension T was assumed to be proportional to the mean curvature K and is equalized b thepressure of the surface of the tumour, we get:

P = αK on Γ = 0.

(α is the mentioned proportionality constant).Let the vector r denote a typical point on the outer surface of the vector. The motion of Γ = 0 can bedescribed then by

dr

dt= q+,

98

for a known Γ(x, y, z, t) = 0. Further, conservation of mass can be written in the following way,

∇ · q = −S,

if S(x, y, z, t) denotes the cell loss rate at a point inside the tumour, and q denotes the cell velocity. Thereare two possibilities for the cell loss:

• Apoptosis (programmed cell death), constant rate S1, in the proliferating and the quiescent region.

• Necrosis, constant rate S2, only in the core

We can formulate that mathematically by using the Heaviside step function H:

S(x, y, z, t) = S1H(|r| − |rN |) + S2H(|rN | − |r|).

(rN is a point on ΓN = 0, i.e. on the surface of the necrotic core).The nutrient concentration σ is assumed to be in the diffusive equilibrium, so the corresponding equationreads

∆σ = 0.

For the moment, we assume that there is a constant supply of nutrient (and the surrounding medium islarge compared to the size of the tumour), i.e.

σ → σ∞ for |r| → ∞.

Taking together q = −∇P and ∇ · q = −S yields ∆P = S inside the tumour.

Now the complete system of equations:

∆P = S inside Γ = 0 (6.10)

∆σ = 0 (6.11)

The complete boundary conditions on Γ = 0 read:

P = αK (6.12)

q+ · n ( = q− · n + βh) = −n · ∇P + λ√

(σ − σ1) (6.13)

q+ × n ( = q− × n) = −∇P × n (6.14)

kn · ∇σ = µ√

(σ − σ1) (6.15)

(in the following µ corresponds to the original µ/k).The boundary surface is governed by

dr

dt= q+, r(0) = a.

P and σ and their normal derivatives are assumed to be continuous across the surface ΓN = 0. Further-more, σ is defined to assume a certain value σ2 on ΓN = 0 (or vice versa).This is a moving boundary problem! It is difficult to solve in general, but with some good luck, specialgeometric configurations allow for an exact solution. Therefore, we consider the spherically symmetricsituation in the next step.

6.4.1 Spherical tumour

Initially, we start with a spherical tumour of radius a and assume that it grows, but keeps its sphericalshape. Thus, the outer surface of proliferating cells can be described by

r = R(t), R(0) = a,

the radius of the tumour at time t.Now we can express the model equations in spherical polar coordinates:

1

r2

∂

∂r

(

r2 ∂P

∂r

)

= S, inside Γ = 0, i.e. r ≤ R(t)

1

r2

∂

∂r

(

r2 ∂σ

∂r

)

= 0.

99

For finding an explicit solution, we have to split up into the different regions of the tumour (proliferatingcell region RN < r ≤ R, necrotic core 0 < r ≤ RN ).

Case RN < r ≤ R:Here, the equations read:

1

r2

∂

∂r

(

r2 ∂P

∂r

)

= S1, RN < r ≤ R(t)

1

r2

∂

∂r

(

r2 ∂σ

∂r

)

= 0,

and the solutions can be determined to be

P (r) = S1r2

6+

A1

r+ B1

σ(r) =C1

r+ D1

(A1, B1, C1, D1 constants; they allow to adapt the solution to the given boundary conditions).The mean curvature of a sphere of radius R is K = 1

R , so from this boundary condition we obtain

P (R) = S1R2

6+

A1

R+ B1 = αK =

α

R;

analogously, from

limR→∞

σ(R) = limR→∞

(C1

R+ D1

)

= σ∞,

we findD1 = σ∞.

Case 0 < r ≤ Rn:Here, the equations read

1

r2

∂

∂r

(

r2 ∂P

∂r

)

= S2, 0 < r ≤ RN

1

r2

∂

∂r

(

r2 ∂σ

∂r

)

= 0,

and the solutions are

P (r) = S2r2

6+

A2

r+ B2

σ(r) =C2

r+ D2.

Further requirement: P and σ should exist in r = 0, thus we need A2 = 0 and C2 = 0.

Let us consider, what happens on the boundary r = RN : We want to have

limr→R−

n

P (r) = limr→R+

n

P (r),

thus

S2R2

N

6+ B2 = S1

R2N

6+

A1

RN+ B1,

and fromlim

r→R−

n

P ′(r) = limr→R+

n

P ′(r)

we demand

S2RN

3= S1

RN

3− A1

R2N

.

Due to σ = σ2 in r = RN we find:

C1

RN+ D1 = σ2 ⇔ C1 = RN (σ2 − σ∞) and D2 = σ2.

100

This is sufficient information now to determine A1, B1, B2 for a given boundary radius RN (explicitcomputation is left out, but not difficult):

A1 = (S1 − S2)R3

N

3

B1 =α

R− S1

R2

6− (S1 − S2)

R3N

3R

B2 =α

R− S1

R2

6+ (S1 − S2)(3R − 2RN )

R2N

6R,

so the pressure and nutrient distributions can be formulated as follows:Case RN < r < R:

P (r) =α

R− S1

6(R2 − r2) + (S1 − S2)

R3N

3

(1

r− 1

R

)

(6.16)

σ(r) =σ2 − σ∞

rRN + σ∞ (6.17)

Case 0 < r ≤ RN :

P (r) =α

R+ S2

r2

6− S1

R2

6+ (S1 − S2)

R2N

6R(3R − 2RN )

σ(r) = σ2.

In the next step, the time-dependency of the tumour boundary, r = R(t), has to be described. We knowthere:

q+ · n = −n · ∇P + λ√

(σ − σ1) and n · ∇σ = µ√

(σ − σ1),

thus

q+ · n = −n · ∇P +λ

µn · ∇σ,

which reads in the radial coordinates

q = −∂P

∂r+

λ

µ

∂σ

∂ron r = R(t).

As the boundary surface was defined by drdt = q+, we obtain:

dr

dt= −∂P

∂r+

λ

µ

∂σ

∂r. (6.18)

Now, the outer boundary R(t) can be described by inserting (6.16, 6.17) into (6.18):

dR

dt= −S1

R

3+ (S1 − S2)

R3N

3R2− λ

µ(σ2 − σ∞)

RN

R2

RN can be expressed in terms of R (again by inserting etc.):

RN =−µ2R3 + µR2

√

µ2R2 + 4(σ∞ − σ1)

2(σ∞ − σ2),

so we can end up with an ODE for R, which can be “solved” at least numerically (and then also allowsto compute RN explicitely).

101

Chapter 7

Cellular Automata

This chapter is only a very rough and short introduction in cellular automata. Further information canbe found for example in [14, 10, 40]. Additionally, there are a lot of nice websites, dealing with cellularautomata, their history, applications, and providing a lot of simulations a.s.o. Only a few examples:

• http://www.rennard.org/alife/english/acintrogb01.html

• http://mathworld.wolfram.com/CellularAutomaton.html

• http://www.collidoscope.com/modernca/

• http://www.fourmilab.ch/cellab/

• http://www.jweimar.de/jcasim/

• http://student.vub.ac.be/∼nkaraogl/drcell/drcell.htm

7.1 Basics of Cellular Automata

The idea of combining “cells” and “automata” dates from the famous scientist John von Neumann (endof the 1940s) - other sources mention Stanislaw Ulam.

Main idea of this kind of modelling: Space is arranged in so-called “cells” (1D, 2D or even higherdimensions) which have only a few states (originally they had only two states: ON or OFF). For theinterpretation of these cells, there are several possibilities, dependent on the concrete situation whichhas to be modelled: Locations, where some individuals live; the individuals themselves; biological cells... The exchange of information between the cells happens only in a certain neighbourhood, according tofixed rules.Apart from this principle of locality, cellular automata deal with discrete time, discrete space and alsodiscrete dependent variables.In general, a cellular automaton is described by:

• a grid G

• a set of elementary states E

• a neighbourhood U

• rules for changes of the state function z

The formal definition of a cellular automaton reads:

Definition 1 (Cellular automaton) A Cellular Automaton is a tuple A = (G,E,U, f) of a grid G ofcells, a set of elementary states E, a set defining the neighbourhood U and a local rule f .

We start considering the 1D case by a very simple example:

102

Grid G : Z StudentsNeighbourhood U(i) of cell i: Neighbourhood: left neighbourU(i) = i-1 , i , i+1 the person itself, right neighbour

(elementary) states: elementary states:E= 0, 1, . . . , N 0 − N gummi bearsstate function: state function respectively corresponding table

z : G → E, xi 7→ z(xi)Student# of his/her gummi bears

Rules: Rules:

zt+1i =

zti − 2 if zi − zi−1 ≥ 1 and zi − zi+1 ≥ 1

zti − 1 if (zi − zi−1 ≥ 1 and zi − zi+1 = 0)

or (zi − zi−1 = 0 and zi − zi+1 ≥ 1)zti + 1 if (zi−1 − zi ≥ 1 and zi − zi+1 = 0)

or (zi+1 − zi ≥ 1 and zi − zi−1 = 0)zti + 2 if zi−1 − zi ≥ 1 and zi+1 − zi ≥ 1

zti − 1 if zi−1 = zi = zi+1

zti else

If there is a gummi bear difference to the neighbour(he/she has less than you), pass one to him/her,otherwise eat one up!

7.1.1 Boundary conditions

We consider the following “problem”: If there is a finite grid, there are cells near the boundary (boundarycells), which have less neighbours than those in the interior of the area. How to handle these cells?There are several possibilities:

• “Dirichlet” boundary: the values for the boundary cells are prescribed.

• open boundary: the rules are applied on the available neighbouring cells.

• periodic boundary: the right boundary is connected with the left boundary ; torus (look at /remember the paper model)

• symmetric boundary: missing neighbouring cells at the boundary are replaced by “mirrored” ones.

7.1.2 Dynamics

During our gummi bear experiment, probably there remained one question open: In which order thegummi bear exchange or meal should be performed?There are mainly two approaches for the dynamics of cellular automata:

Synchronous dynamics: The rules are applied simultaneously in all cells, in a certain cycle (“syn-chronously”)

Asynchronous dynamics: The rules are applied consecutively, one cell after another cell. This means:at each time step, the rules are executed only at one place.There are again a lot of possibilities to choose the exact order:

• in fixed order (always the same), e.g. one line after the other

• in random order, but cells can only be recalled after a complete run (“with memory”)

• in random order without any “memory”

Additionally, stochastic models can easily be included in the cellular automata approach. But herewe consider only deterministic ones.

7.1.3 2D case

We consider now a grid in dimension 2, e.g. a quadratic grid Z × Z (in real world, grid G is finite)

103

Other geometries for the grid are of course also possible (for further examples, have a look at [10] p.20).Typical neighbourhoods:

von Neumann Moore extended Moore

More formally:U(x) = U(x1, x2) = x + y : y ∈ U(0)

is defined by the neighbourhood of 0:von Neumann: U(0) = (0,−1), (−1, 0), (0, 0), (0, 1), (1, 0)Moore: U(0) = x : |xj | ≤ 1 The states are the same as in 1D, the

state function is denoted asz : G → E, xij 7→ z(xij)

Short notation: ztij is the state in (i, j) at time t.

7.1.4 Wolfram Classification

Here, we consider a special class of cellular automata, i.e. those with A = (Z, 0, 1, xi−1, xi, xi+1, f),which are 1D cellular automata, only considering their nearest neighbours, and only have 0/1 states.Obviously, such a neighbourhood can assume 23 = 8 different configurations of states. Since the localfunction can map each configuration to a 0 or 1, there are 28 = 256 possible different local function, i.e.256 different automata. Their dynamics show up quite different patterns (in a corresponding graphic,consisting of white or black fields for the 0 or1 states). A kind of classification of these automata warintroduced by Wolfram:

Wolfram class I: Each initial configuration tends to a fixed point, thus, the pattern becomes constant.

Wolfram class II: Simple stationary or periodic structures show up and small changes in the initialconfiguration only have influence on a finite number of cells.

Wolfram class III: Here, so-called chaotic patterns show up, that means that changes that are done inthe initial configuration influence a number of cells which is linearly growing in time.

Wolfram class IV: The “worst” but maybe most exciting case, in which complex localised patterns showup, which are correlated over a long distance. Nothing can be predicted about the influence ofchanges in the initial condition.

There is a quite simple system how to identify each of the 256 local functions (which was introducedby Wolfram). The eight possible configurations of a neighbourhood are ordered like three digit binarynumbers from 0 to 7: U0 = 000, U1 = 001, U2 = 010, . . ., U7 = 111. For a given local rule f, the nextstate x with neighbourhood Ui is denoted by ci = f(Ui). Then, the number of this local rule is defined

by∑7

i=0(ci2i).

We consider a few simple examples:

104

Rule 254: A simple infection automaton is considered: A cell becomes infected just if one of its directneighbours is infected. This is formulated by a local function as follows:

f(z|U(x)) =

1 if s ≥ 10 otherwise

(In this notation, z is the state of the grid, z(x) the state of the cell x and z|U(x) the configurationof the neighbourhood of cell x). Here, s is the number of infectious neighbours of the cell x = xi,thus s = xi−1 + xi + xi+1. In the following table, it is shown, how the number of this rule (254) iscomputed:

Neighbourhood Ui 000 001 010 011 100 101 110 111Next state of x c0 c1 c2 c3 c4 c5 c6 c7

2i 1 2 4 8 16 32 64 128Example 0 1 1 1 1 1 1 1ci2

i 0 2 4 8 16 32 64 128

Rule number∑7

i=0(ci2i) = 254

The dynamics of this rule can be easily determined: As far as one cell starts with state 1 (i.e.,at least one cell is infected), a state is reached where all cells are in state 1, i.e. all cells becomeinfected. The only exception is the case of all cells are starting with state 0 (non-infected), then allcells stay in this state. Since in each case a stationary pattern is reached, this automaton belongsto Wolfram class I.

Rule 50: This rule can also be interpreted to be a simple infection model. A cell becomes infected if oneof the neighbouring cells is already infected. After one time step, the infectious cells recover. Thelocal function looks as follows:

f(z|U(x)) =

1 if z(x) = 0 and s ≥ 10 otherwise

The cellular automaton is started with a randomly chosen initial configuration and shows then apattern with cells that periodically change their state from 0 to 1 and vice versa. Hence, rule 50belongs to Wolfram class II.

Rule 90: It is defined as follows:

000, 010, 101, 111 7→ 0

001, 011, 100, 110 7→ 1.

Then there show up a chaotic patterns; rule 90 is an example of Wolfram class III.

Remark: There is no 1D nearest neighbour cellular automaton which belongs to Wolfram class IV.

Now we consider some small famous examples for cellular automata.

7.2 Greenberg-Hastings Automaton

This cellular automaton was introduced in 1978 by Greenberg and Hastings. Originally, it was introducedas a model for excitable media. There are different possibilities for an interpretation, e.g. cardiac muscle,cultures of Dictyostelium discoideum of forest fires. We consider it here as a epidemic model, that means,the individuals have different immunological states: susceptible, infectious or immune. Each cell of theautomaton represents an individual or the territory where an individual lives in. The infection is passedby a direct contact between infectious and susceptibles.The model is set up as follows:

• n × m grid

• von Neumann neighbourhood or Moore neighbourhood

• Periodic boundary conditions

105

• Elementary states:

E = 0︸︷︷︸

susceptible

, 1, . . . , a︸ ︷︷ ︸

infectious

, a + 1, . . . , a + g︸ ︷︷ ︸

immune

= 0, . . . , e − 1

• The local function reads

f(z|U(x)) =

1 if z(x) = 0 and s ≥ 1z(x) + 1 if 0 < z(x) < e − 10 otherwise

, (7.1)

where s describes the number of infectious neighbours. A newly infected cell stays infectious for atime steps, afterwards it stays immune for g time steps, until it becomes susceptible again.

Here, we restrict ourselves to cases with a, g ≥ 1 where a + g ≥ 3 and 1 ≤ a ≤ e/2.If the initial configuration has finite support, there are two possibilities: The epidemic “dies out” (whichmeans, that in any finite regions of the grid all cells have state 0 in finite time) or there is at least onecell which repeatedly becomes infected (i.e. the pattern persists). There exists a condition to predict thebehaviour, in order to understand that, we need to define the distance between states in E.

Definition 2 (Distance)

a)d(m,n) = min|m − n|, e − |m − n|

is called the distance between two states m,n ∈ E of cells.

b) Every state k ∈ E is identified with the point exp2πk/(e) on the unit circle in the complex plane. Thenthe signed distance between the two states m,n ∈ E can defined as

σ(m,n) =

d(m,n) if the arc mn is oriented in a mathematically positive sense−d(m,n) else.

Speaking less formal, the states are positioned on a circle and the distance is the shorter arc betweentwo states, therefore also d(m,n) = d(n,m). The local rule (7.1) implies, that per time step a cell canonly move for one state. This leads to 0 ≤ d(z(x)t, z(x)t+1) ≤ 1. Since the signed distance includesinformation about the orientation of the arc, we have σ(m,n) = −σ(n,m).

Definition 3 (Cycle) A cycle is an ordered n-tuple (x1, . . . , xn, xn+1) such that x1, . . . , xn are distin-guished, xn+1 = x1 and xi+1 is neighbour of xi, i = 1, . . . , n.

That means that a cycle is an ordered set of cells with two successive cells being neighbours and identifiedlast and first cell of the cycle.

Definition 4

a) A cycle C is called continuous at time t, if d(z(xi)t, z(xi+1)

t) ≤ a for 1 ≤ i ≤ n.

b) The so-called winding number Wt(C) of a continuous cycle at time t is defined as

Wt(C) =1

e

n−1∑

i=1

σ(z(xi)t, z(xi+1)

t).

Now we have all necessary definitions to understand the theorem of Greenberg et ab. predicting the fateof initial configurations:

Theorem 3 (Greenberg et ab.) Consider a Greenberg-Hastings automaton with a, g ≥ 1 where a+g ≥3 and 1 ≤ a ≤ e/2. Then a configuration with a finite support is persistent if and only if there exists atime t′ ≥ 0 where a continuous cycle C with Wt′(C) 6= 0 can be found.

106

7.3 Game of life

The so-called “Game of life” was created by John Conway in 1968, and soon became famous and well-known, in the scientific area as well as in a kind of “scientific game community”.In an abstract way, it can be described to show artificial structures that can reproduce themselves, oreven as a very simple system that can be used to described complex phenomena.Roughly, it is described formally as follows:

• n × m grid

• Moore neighbourhood U(i, j)

• arbitrary boundary conditions

• elementary states E = 0, 1:

0 “dead cell”1 “living cell”

• Rules:

zt+1ij =

1, if∑

(k,l)∈U(i,j) ztkl = 3

1, if∑

(k,l)∈U(i,j) ztkl = 4 and zt

ij = 1

0, else

In words, these rules can be described in the the following simple way:

1. One inactive cell surrounded by three active cells becomes active (”it’s born”) ;

2. One active cell surrounded by 2 or 3 active cells remains active ;

3. In any other case, the cell ”dies” or remains inactive.

The “biological” interpretation says that a cell will only be born or be able to survive if there are conve-nient environmental conditions which are given by a certain population density. They cannot survive ina too isolated population (less than two neighbours alive) and not in a overpopulation (more than threeneighbours alive).It can be shown (but not here) that a network of cells can be built which imitates logical circuits andpossibilities of data storage and transfer. In some sense, the game of life can emulate a computer. Thisbehaviour is called “computational universal”.

There is a rich collection of initial conditions leading to an interesting behaviour of the system, thebest to explore them is to try it and play with it!

Some more interesting properties of the game of life (which can be tested by “playing with it”):

• It is not possible to predict the evolution of most of the initial configurations. Very different patternscan show up if the initial configuration is modified only a little bit.

• There are examples of initial configurations with finite support (e.g. “glider” or “glider gun”) whichproduce an unbounded pattern.

• It is possible to get patterns with (sometimes very large; up to 30, e.g. “queen bee”) periods

• Thus, the game of life obviously belongs to Wolfram class IV.

A nice freeware program for the game of life can be tested and also downloaded athttp://www.bitstorm.org/gameoflife .

107

7.4 Generalised Cellular Automata and related models

There are several possibilities to generalise the concept of cellular automata models. We do not considerthem in detail here, it is only a short overview. The first collection is still within the classical definitionof cellular automata:

• Cellular automata with stochastic rules: For example, in the Greenberg-Hastings automaton, itcould make sense to describe an infection probability. Thus, the rules can contain also stochastic(random) components.

• Grid modifications: Cellular automata are not necessarily restricted to square structured grids.The grid structure somehow propagates in the patterns that show up, and it might make senseto use more or less round pattern in order to mimic an isotropic spread as good as possible. Onepossibility is a hexagonal grid (with more symmetries than squared grids). Another possibility isto introduce more structure into the automaton and use e.g. random grids

• Dynamic modification: The already mentioned asynchronous dynamics is another possibility tomodify the standard CAs.

Another possibility is to leave the classical CAs and user differently defined classes of models. Here wegive a very short introduction in the so-called Dimer automata. They are often used e.g. to describe themovement of individuals, because it is difficult to formulate that in classical CA rules. This is done asfollows:Consider a 1D square grid with 0 and 1 as elementary states, and a nearest neighbourhood. The rulesread:

1. Choose randomly a cell x (uniform choice)

2. Choose randomly a cell y out of U(x) (the neighbourhood)

3. A new state is assigned to x and y.

Of course, this kind of modelling can be extended to higher dimensions and arbitrary neighbourhoods(even on stochastic grids).

We want to use this kind of modelling to describe a particle movement. In order to do that, we considertwo adjacent cells; their states are switched in one time step (i.e. somehow we consider bonds instead ofstates). This switching ensures conservation of mass. The states of such a pair (first cell x, second celly) are denoted by xy, independent from their relative position in space. x and y can assume the statesas in the CAs. Then a possible rule for modelling movement can be described as follows:

00 7→ 00, 01 7→ 10, 10 7→ 01, 11 7→ 11.

Then a single cell walks randomly over the grid. If several cells start in state 1, then they walk aroundindependently from each other.

Another example: Simple example for alignment. We consider a 2D square grid G ∈ Z2, with a Moore

neighbourhood. The elementary states are E = 0, 1, 2, 3, 4 where 0 describes the empty space and 1,2, 3, 4 describe particles with orientation up, left, down or right, respectively. The rules are defined asfollows:

1. Choose randomly a cell x

2. Choose randomly a neighbour y of cell x

3. If y is occupied (i.e. 6= 0), oriented towards y and y has state 0 (i.e. it is empty), then the particlemoves from x to y.

4. If x and y are occupied (i.e. 6= 0), then x takes the orientation of y.

Of course, this is not a very refined model, just a basic one.

108

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110

Index

β-cell, 87β-cell dynamics, 88

Activator-inhibitor dynamics, 37AIDS, 92Allee effect, 13Anderson’s first HIV model, 92Anderson’s improved HIV model, 92Angiogenesis, 97Arc, 106Area of attraction, 76Asynchronous dynamics, 103autocatalytic, 57

Basic reproductive ratio, 40Basin of attraction, 91Belousov-Zhabotinsky, 60Bifurcation point, 71Biofilm, 29Bioluminescence, 73Bistability, 69, 76Bistable behaviour, 13Black death, 33Boundary cells, 103Boundary conditions, 26Boundary surface, 99Briggs-Haldane situation, 53Brownian motion, 22Butterflies, 13

Cells, 102Cellular Automata, 102Characteristic curves, 26Clock, 66Clusters, 77Combination Drug Therapy, 94Compartments, 92Competitive interaction, 3Conservation of mass, 98Coopertivity parameter, 71Correlated random walk, Random walk, 24Cross catalysis, 57Cycle, 106

Dampled oscillation, 89Delay differential equation (DDE), 16Diabetes, 87Differential fertility, 50Differential mortality, 50Diffusion equation, 24Diploid individuals, 50

Dirichlet condition, 26Distance between states, 106

Ecological problems, 3Effective growth rate, 40Efficiency sensing, 78Elementary states, 103endemic equilibrium, 94Epidemic model, 31Epidemic wave, 33Evolutionarily stable strategy, 46Evolutionary dynamics, 40Evolutionary game theory, 43Extinction rate, 12

Facilitated Diffusion, 34Fick’s law, 27Finetti diagram, 48Fisher-Wright-Haldane model, 51Fitness, 41, 43, 50Fitness parameters, 50Fitting, 77Food chain, 5Food webs, 5Fourier approach, 36Fundamental theorem of natural selection, 51

Game of life, 107Gene regulatory networks, 66Genetic composition, 50Genotype frequency, 49Gierer-Meinhardt model, 38Glucose dynamics, 87Glycolysis, 64Greenberg-Hastings Automaton, 105Grid, 103

Hardy-Weinberg equilibrium, 49Hardy-Weinberg law, 48Heterocygotes, 49Heterozygote equilibrium, 52Hill coefficient, 67History function, 17HIV, 92Homeostasis, 64Hopf bifurcation, 58Hysteresis, 76

Immigration rate, 12Incompressible fluid, 98Insulin, 87

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Insulin dynamics, 88Integrodifferential equation, 17Invader, 47invasion, 30

Langerhans islets, 87Leading edge, 32Levin’s Basic model, 9Life expectancy, 32Limit-cycle oscillations, 56Logit model, 13Lotka model, 56Lotka-Volterra, 18

Malaria, 81Malaria-affected population, 82Malthusian parameter, 50May and Leonard, 3May’s Predator-prey model, 20Metabolic pathways, 63Metapopulations, 9Metastases, 97Michaelis constant, 53Michaelis-Menten, 6minimal wave speed, 31Moments, 19Moore neighbourhood, 104Morphogenesis, 36Mosquito swarm, 27Moving boundary problem, 99Muscle fibre, 34Mushroom structure, 29Mutation, 43

Nash equilibrium, 46Necrotic core, 97Neighbourhood, 102Neumann condition, 26Nonautonomous system, 96Nondimensionalisation, 92Nonhomogeneous distribution, 77Normal distribution, 24

Oregonator, 60Oscillating reaction, 60Oscillatory network, 66Oscillatory solutions, 65

Pancreatic cells, 87Pattern formation, 37Payoff, 45Perturbation, 36Phytoplankton, 6Pitchfork bifurcation, 71Porous media equation, 26Positive feedback loop, 72Predator-prey model, 20Pressure, 98Proliferation rate, 98

Quasi-steady state, 35

Quasi-steady state approximation, 54Quorum sensing, 72

Random mating, 48, 50Random walk, 22Re-occupation, 9reaction-diffusion equation, 30Receptor molecule, 73Reduced diffusibility, 77Relaxation oscillator, 61Replicator equation, 48Repressor, 67Reproduction, 40Reproduction success, 43Rescaling, 80Rescue effect, 12Resistance, 87Robin boundary condition, 77Robust switch, 71Routh-Hurwitz criteria, 16Rule, 102

Saddle-Node bifurcation, 76Scaling, 79Schnakenberg model, 57Selection, 41Selection dynamics, 42Sigmoidal function, 88Single-compartment model, 87Singularity solution, 79Slow manifold, 55Spherical polar coordinates, 99Spherical tumour, 99Spiral waves, 63Stability criterion, 58Stable branches, 76State function, 102, 104Stiff system, 61Stoichiometric factor, 60Structural instability, 56Substrate, 6Superinfection, 82, 83Surface tension force, 98Survival of the first, 42Survival of all, 42Survival of the fittest, 42Susceptibles, 32Suvival of the fitter, 41Switch behaviour, 71Symmetric equilibrium solutions, 69Synchronous dynamics, 103

Telegraphers equation, 26Time delay function, 19Time delay, Delay models, 16Time-scale, 88Toggle switch, 68Transcription, 66Translation, 66travelling wave, 30

112

Trimolecular reaction, 57Tumour, 97Turing instability, 38Turing pattern, 36Turnover number, 53Two-player game, 45

unbeatable strategy, 47Unstable branch, 76

Vascular stabe, 97Volterra model, Predator-prey model, 18von Neumann neighbourhood, 104

Washout, 6wave speed, 30Winding number, 106Wolfram classification, 104

Zooplankton, 6

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