POPULATION DYNAMICS IN RIVERS: ANALYSIS OF STEADY · PDF fileANALYSIS OF STEADY STATES OLGA...

CANADIAN APPLIED

MATHEMATICS QUARTERLY

Volume 18, Number 4, Winter 2010

POPULATION DYNAMICS IN RIVERS:

ANALYSIS OF STEADY STATES

OLGA VASILYEVA AND FRITHJOF LUTSCHER

ABSTRACT. We study a diffusion-reaction-advection equa-tion describing population dynamics of aquatic organisms sub-ject to a constant downstream drift in a bounded or semi-infinite domain. We analyze the nontrivial steady state so-lutions satisfying reflecting upstream and outflow downstreamboundary conditions (Danckwerts’ boundary conditions). Wedetermine the behavior of the domain size as a function ofthe upstream/downstream density, show stability of the steadystate, and investigate how certain qualitative aspects of thesteady-state solution depend on advection speed.

1 Introduction Streams, rivers and coastlines with longshore cur-rents are aquatic ecosystems characterized by unidirectional water move-ment. As a result, many organisms that inhabit these systems are carrieddownstream by the bias in movement. Examples include plankton, al-gae, stream insects, or larvae of benthic organisms such as sea urchins.Despite this bias, populations resist washout and manage to persist overmany generations in such advective environments. This biological phe-nomenon has been recognized and studied for more than half of a cen-tury, and is known as the “drift paradox” (see [15, 16]). The mostcommonly cited resolution for the paradox is that many stream insectshave winged adult stages, during which individuals can travel upstream[22].

A different mechanism for persistence that does not require a wingedadult stage was given by Speirs and Gurney [21]. Using a linear model,these authors showed that a sufficient amount of (unbiased) randommovement can balance the biased movement and lead to population per-sistence. A similar (nonlinear) model with biased and unbiased move-ment arises in models for microbes in the gut [2] and in the study ofphytoplankton blooms [8, 19]. Gravity causes phytoplankton to sink(biased) whereas diffusion in the water column gives rise to unbiased

Copyright c©Applied Mathematics Institute, University of Alberta.

439

440 O. VASILYEVA AND F. LUTSCHER

movement. Huisman et al. obtained a variety of numerical results forsuch a model [8], several analytical results were recently given for a sim-ilar model by Kolokol’nikov et al. [9]. The model by Speirs and Gurneywas recently extended to study more realistic situations by including abenthic compartment [14, 17], spatial heterogeneity [11] or a competingspecies [12].

Most of the results pertaining to streams and rivers cited above arebased either on linear analysis [17, 21] or on numerical simulation [12].The topic of this paper is to analytically study the underlying non-linear equation, its nontrivial steady state and its dependence on pa-rameters. More specifically, we study the (nondimensional) reaction-advection-diffusion equation

(1.1)∂u

∂t=∂2u

∂x2− q

∂u

∂x+ u(1− u),

where u(t, x) is the population density at location x at time t, and q isthe advection speed. We consider this equation together with reflectingboundary conditions upstream, i.e., ∂u

∂x= qu at x = 0, and “outflow”

boundary conditions downstream, i.e. ∂u∂x

= 0. A derivation of theseboundary conditions from random walks together with a biological in-terpretation was given in [11].

Equation (1.1) is a generalization of the well-known and well-studiedFisher equation

(1.2)∂u

∂t=∂2u

∂x2+ u(1 − u),

that describes unbiased random movement [6]. One typically consid-ers this equation on a bounded domain [0, L] with “hostile” boundaryconditions u = 0 at x = 0 and x = L. The steady-state solutionsof (1.2) satisfy a two-dimensional system of ODEs that happens to beHamiltonian (see, e.g., [10]). Using the explicitly available Hamiltonianfunction, one can show under what conditions nontrivial steady statesexist, give explicit formulas for the domain length L and establish thebifurcation structure [10].

Unfortunately, the system with advection (1.1) is not Hamiltonian,and none of the analysis mentioned above carries over to the generalcase. The goal of this paper is to establish existence, uniqueness, stabil-ity and qualitative dependence on parameters of the nontrivial steadystate of (1.1), using phase-plane methods for the steady-state equations(3.4). Some of our analysis is similar in spirit to the recent work by

POPULATION DYNAMICS IN RIVERS 441

Kolokol’nikov et al. [9], who studied a similar equation, but with zero-flux second boundary condition and a different nonlinear reaction term.

This paper is organized as follows. In Section 2, we briefly discussthe linear model and deduce the formula for the “critical domain size”Lc (the minimal length of the interval for which the trivial steady stateis unstable) in our context, i.e., with boundary conditions different fromthe ones used by Speirs and Gurney [21]. In Section 3, we introducethe nonlinear model and make some preliminary observations regard-ing existence and nonexistence of steady-state solutions. In Section 4,we analyze the behavior of the domain size L = Lµ or L = Lν as afunction of downstream density µ = u(L) or upstream density ν = u(0)respectively. We show that Lν (Lµ) is a strictly increasing function ofν (of µ). Furthermore, Lµ approaches the critical domain size Lc fromthe linear model as µ → 0 and goes to infinity as µ approaches thecarrying capacity (scaled to one). In Section 5, we show existence anduniqueness of a nontrivial steady-state solution for L > Lc for the caseof finite and infinite (L = ∞) domains. We also show that the positivesteady state solution is stable in case of finite domains. The remainingsections are devoted to the qualitative behavior of the positive steadystate solution. In Sections 6 and 7, we show that the density at thesteady state decreases pointwise for increasing advection q for finite andinfinite domains respectively. In Section 8, we investigate the conditionsunder which the steady-state profile has an inflection point and derivean approximate expression for the distance of the inflection point to theupstream boundary. In Section 9, we summarize our results, give theirbiological interpretation and present some real life examples.

One final comment is on order before we embark on the analysis.Some of the results presented here could be obtained by PDE methods,e.g., monotone iteration, maybe even in a more elegant way. The chal-lenge that we set ourselves here was to remain within the theory of ODEsand phase-plane analysis and see how many of the results for the Hamil-tonian system (q = 0) can still be obtained without the Hamiltonianstructure available.

2 Model and its linearization Let u(t, x) be the population den-sity at distance x from the upstream boundary at time t. We considerthe single population model in an advective environment, described bythe reaction-diffusion-advection equation:

(2.1)∂u

∂t= D

∂2u

∂x2−Q

∂u

∂x+ ru

(

1 − u

K

)

.


The first term on the right in (2.1) corresponds to diffusive movementof individuals (due to self-propelling and/or water turbulence) with dif-fusion coefficient D. The second term represents movement of the or-ganisms that is caused by drift (Q is the effective speed of the current).The third term reflects the assumption that the population grows logis-tically, with intrinsic growth rate r and environmental carrying capacityK. We assume that all parameters are positive. We consider somebiological examples in the last section of this paper.

In addition, we consider so-called “Danckwert’s boundary conditions”[2]

(2.2)

Qu(t, 0)−D∂u

∂x(t, 0) = 0,

∂u

∂x(t, L) = 0.

These boundary conditions are well-established in the context of so-called Plug Flow Tubular Reactors; see [1, p. 569] and models for nu-trient transport in the gut [2]. They have also been derived from anindividual random walk in [11].

The reflecting upstream boundary condition tells us that the individ-uals cannot cross the upstream boundary (x = 0) and move beyond thetop of the stream. The downstream condition indicates that net out-fluxfrom the domain is due to advection only and not to diffusion. This canbe seen in a variety of ways. For example, if one considers the flux J ofindividuals, as a combination of advective and diffusive fluxes,

J = Jdiff + Jadv = −D∂u

∂x+Qu,

then this flux reduces to the advective flux at the boundary. Therandom-walk interpretation in [11] gives the same. Alternatively, if oneconsiders the movement equation only, i.e., (2.1) with r = 0, and inte-grates over the domain, then one obtains

d

dt

∫ L

0

u(t, x) dx = −Qu(L).

Hence, the change is due to individuals leaving the domain by advection.Biologically, this situation may most closely describe a river flowing intononadvective freshwater habitat, such as a lake. Flow takes individuals


into the lake, but since conditions in the lake are not hostile, individualscan also diffuse back and forth so that the net diffusive flux is zero.

Speirs and Gurney [21] studied the reaction-diffusion-advection equa-tion with a linear growth term:

∂u

∂t= D

∂2u

∂x2−Q

∂u

∂x+ ru.

Instead of the “outflow” boundary condition ∂u∂x

(t, L) = 0, the authorsconsidered “hostile” downstream boundary condition u(t, L) = 0, i.e.,organisms are being removed from the system as soon as they reach theleft border of domain.

In our case, we get the Speirs-Gurney equation (with outflow down-stream boundary condition) if we linearize (2.1) at zero:

(2.3)

∂u

∂t= D

∂2u

∂x2−Q

∂u

∂x+ ru,

Qu(t, 0) −D∂u

∂x(t, 0) = 0,

∂u

∂x(t, L) = 0.

Proposition 2.1. The general solution of (2.3) is given by

u(t, x) = Σ∞k=1e

λkt

(

AkeQ

2Dx cos

√

4D(r − λk) −Q2

2Dx

+BkeQ2D

x sin

√

4D(r − λk) −Q2

2Dx

)

,

where λk are eigenvalues of the operator

D∂2f

∂x2−Q

∂f

∂x+ rf

with boundary conditions

Qf(0) −Df ′(0) = 0, f ′(L) = 0.

Proof. Standard separation of variables technique.


Classical Sturm-Liouville theory states that the eigenvalues λk aboveform an infinite decreasing sequence λ1 > λ2 > . . . [4]. If λk are allnegative, then the population goes extinct in the long term. On the otherhand, if for some k we have λk > 0, the population exhibits unboundedgrowth. Setting λ1 = 0 and applying boundary conditions to the generalsolution of (2.3) gives us the threshold value for the domain size, referredto as critical domain size Lc:

(2.4) Lc(Q) =

arctan

(

Q√

4rD−Q2

2rD−Q2

)

θ, 0 < Q ≤

√2rD,

π + arctan

(

Q√

4rD−Q2

2rD−Q2

)

θ,

√2rD < Q < 2

√rD ,

where

θ =

√

4rD −Q2

2D.

Note that for Q =√

2rD, both formulas give us Lc(√

2rD) = π/2θ,and thus Lc depends continuously on Q.

Remark 2.2. When advection reaches its critical value Qc = 2√Dr,

the critical domain size becomes infinite and the entire population iswashed downstream, i.e. persistence is not possible. For Q < Qc andL < Lc, the population goes extinct as well. For Q < Qc and L > Lc,the population in the linearized model experiences unlimited growth.

3 Nonlinear system, steady state solutions, and connectionwith the Fisher equation We now consider the nonlinear model.We will make our first observations regarding the existence and nonex-istence of a nontrivial steady-state solution as we vary the advectionspeed. We start by nondimensionalizing (2.1) and (2.2). We rescale thepopulation density by the carrying capacity, time and space by charac-teristic time and length:

u =u

K, t = rt, x =

√

r

Dx, q =

Q√Dr

.


We omit the tildes for convenience, so that (2.1, 2.2) become

(3.1)

∂u

∂t=∂2u

∂x2− q

∂u

∂x+ u (1 − u) ,

qu(t, 0) − ∂u(t, 0)

∂x= 0,

∂u(t, L)

∂x= 0.

We investigate the properties of nonzero steady state solutions of(3.1), i.e., solutions that do not depend on time. Such a steady statesolution satisfies

(3.2) u′′ − qu′ + u(1 − u) = 0


(3.3)

{

u′ = qu, x = 0,

u′ = 0, x = L.

Equation (3.2) is equivalent to the following system of two differentialequations:

(3.4)

{

u′ = v,

v′ = qv − u(1 − u),


(3.5)

{

v = qu, x = 0,

v = 0, x = L.

Hence, we are looking for orbits of (3.4) connecting the straight linesv = qu and v = 0. Let us refer to such solutions as “connecting orbits.”

Next, we use the classical result of Fisher [6] and point out the sim-ilarity of equation (3.1) with the Fisher equation written in travellingwave coordinates.

Remark 3.1. (a) For Fisher’s equation (1.2) on the real line, thereexists a special solution in form of a monotone, positive travelling waveu(t, x) = φ(x+ct) (moving from the right to the left) iff c ≥ 2 (c ≥ 2

√Dr


in the dimensional case) [6, 10] . In travelling wave coordinates, Fisher’sequation takes the form

(3.6) cφ′ = φ′′ + φ(1 − φ)

with boundary conditions φ(∞) = 1 and φ(−∞) = 0. More specifically,a travelling wave solution of Fisher’s equation corresponds to a (unique)heteroclinic connection, located in the first quadrant of the uv-plane andconnecting two fixed points: (0, 0) and (1, 0) of (3.4) obtained from (3.2).Since equation (3.2) is the same as (3.6), with c = q, we can use Fisher’sresults in our setting. Note that one can consider a solution of theform u(t, x) = φ(x − ct) with the boundary conditions φ(−∞) = 1 andφ(∞) = 0 (travelling wave moving from the left to the right). Howeverthe corresponding heteroclinic orbit is located in the fourth quadrant,and this is not applicable in our discussions.

(b) Note that, if q ≥ 2 the fixed point (0, 0) of the system (3.4) isan unstable node and we have “node-saddle” heteroclinic connection,the “traveling wave.” For q < 2, point (0, 0) is an unstable spiral andwe observe “focus-saddle” heteroclinic connection from (0, 0) to (1, 0),approaching the fixed point (1, 0) from the first quadrant. There is nononnegative travelling wave in this case.

First we show that when the advection speed is greater than thethreshold value q∗ = 2 (Q∗ = 2

√Dr in the dimensional case), then the

population will not be able to persist.

Lemma 3.2. There are no nontrivial solutions of (3.4, 3.5) for q ≥ q∗.

Proof. By Remark 3.1(a), we have a heteroclinic connection

u = u1(x), v = v1(x),

between the origin and the fixed point (1, 0), located entirely in thefirst quadrant, approaching (0, 0) and (1, 0) as x → −∞ and x → ∞,respectively.

Note that the slope of the vector field defined by (3.4) at any pointin the uv-plane is given by

v′

u′=qv − u(1 − u)

v= q − u(1 − u)

v< q.

Thus for any 0 < u < 1 and v > 0, the slope of any solution of (3.4)(including the heteroclinic orbit) is less than q. Therefore, for u > 0,the line v = qu will always stay above the curve u = u1(x), v = v1(x).


Suppose there exists a solution of (3.4) that starts at v = qu for x = 0and reaches v = 0 when x = L for some L > 0. In this case the solution(connecting orbit) must intersect v = 0 when u ∈ [0, 1]. Indeed, if u > 1,then the slope of the vector field on {v = 0} is positive, and we will notbe able to reach v = 0. Thus, since the connecting orbit reaches thesegment [0, 1] of {v = 0}, it must intersect the heteroclinic orbit or passthrough the fixed point. Neither one can happen, because two solutioncurves cannot intersect (by uniqueness), and the fixed point (1, 0) cannotbe reached for a finite L.

From here on, we assume that 0 ≤ q < q∗ = 2. We show that whenadvection is less than critical, there are nontrivial steady states for somedomain size L.

Lemma 3.3. For any 0 ≤ q < q∗, there exists L > 0 for which (3.4,3.5) has a nontrivial solution.

Proof. The linearization of system (3.4) at (0, 0) is given by

(3.7)

{

u′ = v,

v′ = qv − u.

The Jacobian of this system has the two complex roots λ1,2 =q±

√q2−4

2 .Therefore the origin of the linear system is an unstable focus. ByGrobman-Hartman Theorem (see [18]), the origin for the nonlinear sys-tem is an unstable focus as well. Thus, there exist solutions for appro-priately chosen L. Namely, in a small neighborhood of the origin, thetrajectories of system (3.4) spiral away from the origin and cross bothlines corresponding to the boundary conditions. Consequently, there ex-ist solutions of the nonlinear system (3.4) that start on the line {v = qu}and end on the line {v = 0} (second boundary condition).

4 More on steady state: domain size as the function of up-stream/downstream density In this section, we analyze the rela-tionship between the domain size L and the upstream/downstream den-sity in the case of a positive steady state of our model. Essentially, weshow that higher density corresponds to larger domains.

Let (u1(x), v1(x)) be a solution of

(4.1)

{

u′ = v,

v′ = qv − u(1 − u),


satisfying (u1(−∞), v1(−∞)) = (0, 0) and (u1(∞), v1(∞)) = (1, 0). Sucha solution exists (and its orbit is unique) by Remark 3.1(b). Since sucha curve (the heteroclinic connection) will necessarily intersect the linev = qu, we may assume that v1(0) = qu1(0), and u1(x), v1(x) > 0for x > 0 (i.e., the “last” intersection of heteroclinic connection with{v = qu} happens when x = 0). Then such a solution is unique.

Let νmax = u1(0). For any 0 < ν < νmax let (uν(x), vν (x)) be the(unique) solution of (4.1) satisfying (uν(0), vν(0)) = (ν, qν). (see Fig-ure 4.1 for illustration).

Considering the region bounded by the line v = qu, the positive u-axis, and the heteroclinic connection, we see that the curve (uν(x), vν (x))will eventually cross the u-axis between u = 0 and u = 1. For any0 < ν < νmax, let Lν > 0 be such that vν(Lν) = 0. Let µν = uν(Lν).Then 0 < µν < 1 (again, see Figure 4.1). Note that (uν(x), vν (x)) is acontinuous function of x and ν (e.g., see [18, p. 78]), and hence both Lν

(as the solution of vν(x) = 0) and µν = uν(Lν) are continuous functionsof ν ∈ (0, νmax).

FIGURE 4.1: Connecting orbit for a finite domain and heteroclinic orbitin the uv-plane.

In this and later sections of the paper, some proofs are more con-veniently formulated using upstream density ν whereas others becomeeasier using downstream density µ. The following lemma connects thetwo parameters.


Lemma 4.1. The mapping ν 7→ µν is a continuous strictly increasingfunction from (0, νmax) onto (0, 1). In particular, limν→0 µν = 0 andlimν→νmax

µν = 1.

Proof. Continuity is observed above, and the fact that µν is strictlyincreasing with respect to ν follows from the observation that solutioncurves of (4.1) do not intersect. Note for any 0 < µ < 1 there existsa solution curve of (4.1) passing through (µ, 0). It will necessarily passthrough a point (ν, qν) for some 0 < ν < νmax. Hence, µ = µν , and themapping ν 7→ µν is onto.

For 0 < µ < 1 let Lµ = Lν where µ = µν . So, Lµ is a continuousfunction of µ ∈ (0, 1). Now, we look at the behavior of Lµ as µ→ 0. Ourgoal is to prove that limµ→0 Lµ = Lc, where Lc is the critical domainsize for the linear system, given by (2.4).

It is slightly more convenient to consider the change of variables x 7→−L + x, i.e., we consider the boundary conditions v(0) = 0, v(−Lµ) =qu(−Lµ). The solution of the nonlinear system (3.4) is given by thevariation of constants formula as

[

u(x)v(x)

]

= eAx

[

µ0

]

+

∫ x

0

eA(x−s)

[

0u2(s)

]

ds, A =

[

0 1−1 q

]

.

We denote by [u, v]T the solution of the linearized system (3.7) withu = 1, v = 0. Then v(−Lc) = qu(−Lc). The solution with u = µ, v = 0is given by µ[u, v]T

If u(0) = µ, then u(x) < µ for all x < 0. Hence, we can bound thedistance between the solution of the nonlinear problem and the linearproblem, starting at (µ, 0) for x = −Lµ from above by

‖[u(x) − µu(x), v(x) − µv(x)]T ‖ ≤ µ2

∥

∥

∥

∥

∫ x

0

eA(x−s)ds

∥

∥

∥

∥

.

Therefore, there is a constant C > 0, for which

|µv(−Lµ) − qµu(−Lµ)| ≤ µ2C.

Since the solution of the linear problem satisfies the condition v(−Lc) =qu(−Lc), we have proved the following theorem.

Theorem 4.2. Lµ → Lc as µ→ 0 (equivalently, Lν → Lc as ν → 0).


FIGURE 4.2: The graph of domain size Lν vs. upstream density ν, forq = 1 (obtained numerically).

Remark 4.3. Figure 4.2 shows the graph of Lν vs. ν for q = 1,obtained numerically. Note that by (2.4) for q = 1, Lc(1) = 2π

3√

3≈ 1.2,

which agrees with the graph. Note also that Lν appears to increase withν, and goes to infinity as ν approaches a threshold value νmax ≈ 0.212.

Next, we give an analytical proof that Lν is an increasing functionof ν (as suggested by the numerics), following the idea of the proof ofLemma 2.1 in [3].

Proposition 4.4. If ν1 < ν2 < νmax, then Lν1 < Lν2 .

Proof. Let u(x) be the steady state solution of (1.1). Then

−(e−qxux)x = −e−qx(−qux + uxx) = e−qxu(1 − u).

Thus the following equalities take place:

−(e−qxuν1

x )xuν2 = e−qxuν1(1 − uν1)uν2 ,

−(e−qxuν2

x )xuν1 = e−qxuν2(1 − uν2)uν1 .

Taking difference between the above expressions and then integrating itbetween 0 and any α ∈ (0,min(Lν1 , Lν2)] we obtain


e−qx [uν2

x (x)uν1(x) − uν1

x (x)uν2(x)]∣

∣

∣

α

0

=

∫ α

0

e−qxuν1(x)uν2 (x)(uν2(x) − uν1(x)) dx.

Using the boundary conditions at x = 0 we get

(4.2) e−qα [uν2

x (α)uν1(α) − uν1

x (α)uν2(α)]

=

∫ α

0


Next we want to show that for all x ∈ [0,min(Lν1 , Lν2)] uν1(x) <uν2(x). Note that since uν1(0) = ν1 < ν2 = uν2(0) this is true for x = 0.

Suppose the statement is not true, then there exists 0 < β ≤ min(Lν1 ,Lν2) such that uν1(x) < uν2(x) for x ∈ [0, β), but uν1(β) = uν2(β). Thentaking α = β in (4.2) and using

uν1(β) = uν2(β),

we get

(4.3) e−qβuν1(β) [uν2

x (β) − uν1

x (β)]

=

∫ β

0


Note that the right hand side of (4.3) is positive, and therefore uν2x (β) >

uν1x (β). On the other hand, for z(x) = uν2(x)−uν1(x) we have z(x) > 0

for x ∈ [0, β) and z(β) = 0, which implies z′(β) = uν2x (β)−uν1

x (β) ≤ 0, acontradiction. Thus we have proved that for any x ∈ [0,min(Lν1 , Lν2)]uν1(x) < uν2(x).

Now, suppose Lν2 ≤ Lν1 , so min(Lν1 , Lν2) = Lν2 . Then taking (4.2)with α = Lν2 and using the boundary condition uν2

x (Lν2) = 0, we get

e−qLν2

[−uν1

x (Lν2)uν2(Lν2)]

=

∫ Lν2

0

e−qxuν1(x)uν2(x)(uν2 (x) − uν1(x)) dx > 0.

In the above equality, the right hand side is positive since uν2(x) >uν1(x) on [0, Lν2 ], while the left hand side is negative, a contradiction.Thus Lν1 < Lν2 .


Finally, we look at the behavior of Lν as ν → νmax, or, equivalently,the behavior of Lµ as µ → 1. In the following theorem we confirm thenumerical observations made in Remark 4.3.

Theorem 4.5. Lµ → ∞ as µ → 1 (equivalently, Lν → ∞ as ν →νmax).

Proof. We use the standard result on continuous dependence on initialdata to prove this theorem; see, e.g., [18, Theorem 1, Section 2.3]. Pickany 0 < X < ∞ and ε > 0. Since (1, 0) is a steady state, we can pickδ > 0 small enough so that the solution with initial data (µ, 0) and|µ− 1| < δ remains within ε of (1, 0) up to ”time” X. Hence, as µ → 1,it will take the solution arbitrarily long to leave an ε-neighborhood of(1, 0). In particular, Lµ → ∞.

5 Existence, uniqueness and stability of the steady stateWe use the results about Lν to show existence and uniqueness of thesolution of (3.4, 3.5) for any L > Lc.

Theorem 5.1. For any L > Lc (3.4, 3.5) has a unique positive solution.Equivalently, for any L > Lc (3.1) has a unique positive steady state.

Proof. Case 1: finite domain.We know that Lν (as a function of ν) is continuous and increasing on(0, νmax). It has finite limit Lc at 0 and goes to infinity as ν → νmax.Clearly, for any L > Lc, there is exactly one ν ∈ (0, νmax) such thatLν = L. By the definition of Lν , this means that there exists (u(x), v(x))satisfying (3.4, 3.5), such that (u(0), v(0)) = (ν, qν). Since ν 6= 0, thissolution is positive. Moreover, such a solution is unique (as a solutionof an initial value problem).

Case 2: infinite domain.We now turn to the case of infinite domain. Suppose 0 ≤ q < 2. Weknow that the solution (u1(x), v1(x)) of (4.1) with u1(0) = νmax satisfieslimx→∞(u1(x), v1(x)) = (1, 0). This implies existence of a steady statesolution. Uniqueness follows from the fact that there is a unique solutionof (4.1) satisfying (u(0), v(0)) = (νmax, qνmax), and if the solution doesnot pass through this point, it either reaches u-axis in finite “time” L,or does not approach to it at all.

Hence, we have


Theorem 5.2. For any 0 ≤ q < 2 there exists a unique solution (u(x), v(x))of (4.1) satisfying v(0) = qu(0) and v(∞) = 0.

Our next goal is to prove stability of the positive steady state solutionof (3.1) (when it exists). We linearize around the steady state, and makethe ansatz u(t, x) = u(x)+φ(x)e−λt. Substituting into (3.1) and keepingthe leading order terms, gives the following eigenvalue problem:

(5.1)

−λφ(x) = φ′′(x) − qφ′(x) + φ(x)(1 − 2u(x)),

φ′(0) = qφ(0),

φ′(L) = 0.

To eliminate the advection term, we consider ψ(x) = e−qx2 φ(x). Then

(5.1) becomes

(5.2)

ψ′′(x) +

(

1 − q2

4− 2u(x) + λ

)

ψ(x) = 0,

ψ′(0) − q

2ψ(0) = 0,

ψ′(L) +q

2ψ(L) = 0.

This problem has the same eigenvalues as (5.1). They form an in-creasing sequence λ1 < λ2 < . . . [4]. To prove stability we need to showλ1 > 0. Suppose ψ1(x) is the eigenfunction corresponding to the domi-nant eigenvalue λ1. From classical Sturm-Liouville theory it follows thatψ1(x) is of one sign in [0, L], so we may assume that ψ1(x) > 0 for anyx ∈ (0, L).

Let w(x) = e−qx

2 u(x). Substituting into (3.2) we get

(5.3)

w′′(x) +

(

1 − q2

4)w(x) − e

q

2x(w(x)

)2

= 0,

w′(0) − q

2w(0) = 0,

w′(L) +q

2w(L) = 0.

Multiplying the equations in (5.3) by ψ1(x), and in (5.2) by w(x),integrating between 0 and L, and taking the difference of the two ex-pressions, we get


∫ L

0

ψ′′1 (x)w(x) dx −

∫ L

0

w′′(x)ψ1(x) dx + λ1

∫ L

0

ψ1(x)w(x) dx

− 2

∫ L

0

u(x)ψ1(x)w(x) dx +

∫ L

0

eq2x(w(x))2ψ1(x) dx = 0.

Note that

∫ L

0

ψ′′1 (x)w(x) dx−

∫ L

0

ψ1(x)w′′(x) dx

=

(

ψ′1(x)w(x)|L0 −

∫ L

0

ψ′1(x)w

′(x) dx

)

−(

ψ1(x)w′(x)|L0 −

∫ L

0

ψ′1(x)w

′(x) dx

)

= − q2ψ1(L)w(L) − ψ′

1(0)w(0)

−(

− q

2w(L)ψ1(L) − w′(0)ψ1(0)

)

= 0.

Also note that eq

2x(w(x))2 = u(x)w(x). Thus we have

λ1

∫ L

0

ψ1(x)w(x) dx −∫ L

0

u(x)w(x)ψ1(x) dx = 0,

or

λ1 =

∫ L

0u(x)w(x)ψ1(x) dx

∫ L

0ψ1(x)w(x) dx

> 0.

With this result on eigenvalues, we are ready to prove stability of thesteady state.

Theorem 5.3. The positive steady state solution u = u∗(x) of (3.1)with 0 < L <∞ is stable.

Proof. The operator A = ∂2/∂x2, defines an analytic semigroup onL1(0, L) with domain

(5.4) D = {U ∈ W2,1(0, L)|Ux(0)− q1U(0) = 0, Ux(L) + q1U(L) = 0},

(q1 = q/2) whose spectrum equals the point spectrum; see [5, Sec-tion 4b), Chap. VI]. The semigroup generated by A + (1 − q2

1 − 2u∗)I


is also analytic so that the spectral mapping theorem applies ([5, 3.12,Chap. IV]). By the preceding eigenvalue calculation and Theorem 11.20in [20], the steady state u∗ is linearly stable, and by Theorem 11.22 in[20] it is stable.

6 Dependence of the steady state on advection speed for infi-nite domains In this section, we investigate how changes in advectionaffect the steady state profile in the case of infinite domain.

Consider the diffusion-advection-reaction equation with logistic growthterm, reflecting boundary condition upstream and no-flux condition at∞.

(6.1)

∂u

∂t=∂2u

∂x2− q

∂u

∂x+ u (1 − u) ,

qu(t, 0) − ∂u(t, 0)

∂x= 0, lim

x→∞

∂u(t, x)

∂x= 0.

We are interested in the steady state solutions, i.e., we set ut = 0 andu = u(x). Thus we consider the equation u′′ − qu′ + u(1 − u) = 0, orwritten as a first order system

(6.2)

{

u′ = v,

v′ = qv − u(1 − u).

The above system has two fixed points: (0, 0) and (1, 0). It is known(see Remark 3.1b) that for q < q∗(= 2) the origin is an unstable spiraland (1, 0) is a saddle point. The heteroclinic orbit that connects thesetwo fixed points also intersects the line corresponding to boundary con-dition v = qu, in the first quadrant of the uv-space. More specifically,there exists an orbit (uq, vq) such that

(6.3) vq(0) = quq(0)

and

(6.4) limx→∞

(uq(x), vq(x)) = (1, 0).

We have changed notations to stress the fact that we study depen-dence of steady states on advection speeds, assuming that µ = 1. Thuswe are interested in the behavior of (uq(x), vq(x)) with respect to the


advection speed q. We may view this orbit as the graph of v = vq(u)(since u′(x) = v(x) > 0 in the first quadrant). Note that the curvev = vq(u) is the stable manifold of the fixed point (1, 0). Therefore, atthis point, the curve is tangent to an eigenvector of the Jacobian of (6.2)at (1, 0) corresponding to the negative eigenvalue. Thus we can find theslope of v = vq(u) at u = 1 by analyzing that Jacobian. Namely, wehave

J(u, v) =

(

0 1−1 + 2u q

)

and

J(1, 0) =

(

0 11 q

)

.

The eigenvalues of J(1, 0) are

λ1 =q −

√

q2 + 4

2< 0 and λ2 =

q +√

q2 + 4

2> 0.

An eigenvector corresponding to λ1 is given by v1 = (1,q−

√q2+4

2 ). Thus

the slope of v = vq(u) at (1, 0) is m(q) =q−

√q2+4

2 .In the following, let 2 > q1 > q2.

Lemma 6.1. m(q1) > m(q2).

Proof. Note that m′(q) = 12

(

1 − q√q2+4

)

> 0. Therefore, m(q) is an

increasing function of q and m(q1) > m(q2).

Lemma 6.2. There exists 0 < u∗ < 1 such that vq1(u) < vq2

(u) for allu ∈ (u∗, 1).

Proof. Let w(u) = vq1(u) − vq2

(u). The statement now follows fromw(1) = 0 and w′(1) = v′q1

(1) − v′q2(1) = m(q1) −m(q2) > 0.

Lemma 6.3. vq1(u) < vq2

(u) for all max(uq1(0), uq2

(0)) ≤ u < 1(common domain of vq1

(u) and vq2(u)).

Proof. We know that vq1(u) < vq2

(u) for all u∗ < u < 1. If this is nottrue for all

max(uq1(0), uq2

(0)) ≤ u < 1,


there exists 0 < u < 1 such that vq1(u) = vq2

(u) = v. Then

(vq2)u(u) = lim

u→u+

vq2(u) − vq2

(u)

u− u= lim

u→u+

vq2(u) − vq1

(u)

u− u

≥ limu→u+

vq1(u) − vq1

(u)

u− u= (vq1

)u(u).

(6.5)

On the other hand,

(6.6) (vq2)u(u) = q2 −

u(1 − u)

v< q1 −

u(1 − u)

v= (vq1

)u(u),

a contradiction.

Lemma 6.4. uq1(0) < uq2

(0).

Proof. Note that the slope of the line v = q2u is q2. The slope of thesolution v = vq2

(u) is less than q2:

dv

du= q2 −

u(1 − u)

v< q2.

Therefore, vq2(u) ≤ q2u for any u ∈ [max(uq1

(0), uq2(0)), 1). Thus, by

the above lemma, we have

(6.7) vq1(u) < vq2

(u) ≤ q2u < q1u,

so vq1(u) < q1u for all u ∈ [max(uq1

(0), uq2(0)), 1). Since vq1

(uq1(0)) =

q1uq1(0), we conclude that uq1

(0) < max(uq1(0), uq2

(0)) = uq2(0).

We are now ready to prove the main result of this section: the steady-state density decreases pointwise with increasing advection.

Theorem 6.5. uq1(x) < uq2

(x) for any x ≥ 0.

Proof. By the above lemma, this is true for x = 0. If this is not truefor some x > 0, then there exists x > 0 such that uq1

(x) = uq2(x). We

may assume that x is the smallest such. Let u = uq1(x) = uq2

(x). First,note that by Lemma 6.3,

(uq1)x(x) = vq1

(uq1(x)) = vq1

(u) < vq2(u)(6.8)

= vq2(uq2

(x)) = (uq2)x(x).


On the other hand, by the choice of x, for any 0 < x < x we haveuq1

(x) < uq2(x), so

(6.9) uq1(x) − uq1

(x) = uq1(x) − u < uq2

(x) − u = uq2(x) − uq2

(x).

Since x− x < 0, we get

(6.10)uq1

(x) − uq1(x)

x− x>uq2

(x) − uq2(x)

x− x.

Taking a limit as x→ x−, we get

(6.11) (uq1)x(x) ≥ (uq2

)x(x).

This is a contradiction.

7 Dependence of the steady state on advection speed forfinite domains Now we will consider the case of a domain of finitelength and investigate the dependence of the steady state solution of(3.1) on the advection speed. The situation here is somewhat morecomplicated than in the previous section for an infinite domain, wherethe “endpoint” (u(∞) = 1) was the same for all possible solutions.

Setting ut = 0, we get the first order system:

(7.1)

u′ = v,

v′ = qv − u(1 − u),

v(0) = qu(0),

v(L) = 0.

We want to show that given 0 < q2 < q1 < q∗ = 2, a population insidea habitat of length L with advection speed q1 will have lower densitythan a population in the same habitat with advection speed q2, at anypoint of the domain [0, L].

Our goal is to prove

Theorem 7.1. If 0 < q2 < q1 < 2, uq1(x) and uq2

(x) are the steadystate solutions of (3.1) with q = q1 and q = q2 respectively, then

(7.2) uq1(x) < uq2

(x), x ∈ [0, L].


Note that in the first quadrant (u, v > 0) any trajectory of thesystem (7.1) can be viewed as a graph of a function v = v(u) (sinceu′ = v 6= 0). Moreover, such curves are solution curves of the ODE

dv/du = q − u(1−u)v

. In particular, no two such curves intersect.First, we notice the effect of increasing the advection on the phase

portrait of ODE dv/du = q − u(1−u)v

.

Lemma 7.2. Given q1 > q2, at any point (u∗, v∗) (in the first quad-

rant), the slope of the direction field of dv(u)du

|u=u∗ = q1 − u∗(1−u∗)v∗

is

greater than that of the equation dvdu

|u=u∗ = q2 − u∗(1−u∗)v∗

.

Proof. Follows easily from assumption q1 > q2.

Lemma 7.3. Suppose q1 > q2. Let v = vq1(u) and v = vq2

(u) be thesolutions of

(7.3)dv

du= q1 −

u(1 − u)

v

and

(7.4)dv

du= q2 −

u(1− u)

v,

respectively, both passing through a point (u∗, v∗) with u∗, v∗ > 0. Thenvq1

(u) < vq2(u) for u < u∗ and vq2

(u) < vq1(u) for u∗ < u (on the

common domain of vq1and vq2

; see Figure 7.1).

Proof. Let w(u) = vq1(u)−vq2

(u). Thus w(u∗) = vq1(u∗)−vq2

(u∗) = 0

and dwdu

|u=u∗ = ddu

(vq1(u)− vq2

(u))|u=u∗ =dvq1

du|u∗ − dvq2

du|u∗ = q1 − q2 >

0. This means that w(u) has at most one zero and we have w(u) < 0for u < u∗ and w(u) > 0 for u > u∗, as needed.

Let (uq1(x), vq1

(x)) be the solution of (7.1) with q = q1. Let v =

vq1(u) be the corresponding solution of dv

du= q1 − u(1−u)

v, defined on the

interval (uq1(0), uq1

(L)). Let (u(x), v(x)) be the solution of (6.2) withq = q2 passing through the point (uq1

(L), 0) such that v(0) = q2u(0),and let v = v(u) be the equation of this curve as a solution of dv

du=

q2 − u(1−u)v

, defined on the interval (u(0), uq1(L)) = (a, b).

Lemma 7.4. For any u ∈ (a, b), v(u) > vq1(u).


FIGURE 7.1: Intersection of orbits corresponding to different advectionspeeds.

Proof. Take any µ ∈ (a, b). Let v = vµ(u) be the solution of dv/du =

q2 − u(1−u)v

passing through the point (µ, vq1(µ)). By Lemma 7.3, with

(u∗, v∗) = (µ, vq1(µ)), vµ(u) < vq1

(u) for any u ∈ (µ, c), where c < b isthe point where the curve v = vµ(u) crosses the u-axis (see Figure 7.2).Since v = v(u) and v = vµ(u) cannot intersect, we have v(µ) > vµ(µ) =vq1

(µ), as needed.

Let L′ > 0 be such that v(L′) = 0. Let us prove that in order toreach a certain downstream density (in our case, b), a population thatis subject to a higher advection needs a larger habitat. In other words,

Lemma 7.5. L > L′.

Proof.

L =

∫ b

uq1(0)

du

vq1(u)

>

∫ b

a

du

vq1(u)

>

∫ b

a

du

v(u)= L′.


FIGURE 7.2: Relationship between orbits.

For any 0 < µ < 1, let Lµ > 0 be such that for the solution of

(7.5)

u′ = v,

v′ = q2v − u(1 − u),

v(0) = q2u(0),

v(Lµ) = 0.

We have u(Lµ) = µ. In other words, Lµ is the size of the habitatcorresponding to the downstream density µ in the case of the smalleradvection q2.

As proved earlier, Lµ → ∞ as µ → 1, and as we know, Lµ is increasingwith respect to µ. Thus if (uq2

(x), vq2(x)) is the solution of (6.2) with

advection q = q2, then L > L′ implies uq2(L) > u(L′) = uq1

(L).We are now ready to prove our theorem.

Proof of Theorem 7.1. We consider two cases.Case 1: uq1

(L) < uq2(0) (the ranges of uq1

and uq2do not overlap).

In this case, for any x ∈ [0, L] we have

uq1(x) ≤ uq1

(L) < uq2(0) ≤ uq2

(x),

as needed.


FIGURE 7.3: The case of overlapping domains.

Case 2: uq1(L) ≥ uq2

(0) (there is an overlap, see Figure 7.3).

Note first that since uq2(L) > u(L′), the curve v = vq2

(u) is locatedabove the curve v = v(u) on the common domain [uq2

(0), uq1(L)]. Thus

by Lemma 7.4 , for any u ∈ [uq2(0), uq1

(L)],

vq1(u) < vq2

(u).

Note that vq2(uq2

(0)) = q2(uq2(0)) and

(7.6)dvq2

du|u=uq2

(0) = q2 −uq2

(0)(1 − uq2(0))

vq2(uq2

(0))< q2,

and therefore, since v = vq2(u) is concave down, we have vq2

(u) < q2ufor u > uq2

(0). Now, for u ∈ [uq2(0), uq1

(L)] we have vq1(u) < vq2

(u) ≤q2u < q1u.

So, vq1(u) < q1u for all u ∈ [uq2

(0), uq1(L)].

Since vq1(uq1

(0)) = q1uq1(0), we conclude that uq1

(0) 6∈ [uq2(0), uq1

(L)],i.e., uq1

(0) < uq2(0).

We want to show that for any x ∈ [0, L] uq1(x) < uq2

(x). Supposethis is not the case, and consider the smallest x > 0 such that uq1

(x) =uq2

(x). Let u = uq1(x) = uq2

(x). Note that u ∈ [uq2(0), uq1

(L)].


Now, we have

duq1

dx

∣

∣

∣

∣

x=x

= vq1(uq1

(x)) = vq1(u) < vq2

(u)

= vq2(uq2

(x)) =duq2

dx

∣

∣

∣

∣

x=x

.

(7.7)

On the other hand, by the choice of x, for any 0 < x < x, we haveuq1

(x) < uq2(x), so

(7.8) uq1(x) − uq1

(x) = uq1(x) − u < uq2

(x) − u = uq2(x) − uq2

(x).

Since x− x < 0, we get

(7.9)uq1

(x) − uq1(x)

x− x>uq2

(x) − uq2(x)

x− x.

Taking the limit as x→ x−, we get

(7.10)duq1

dx

∣

∣

∣

∣

x=x

≥ duq2

dx

∣

∣

∣

∣

x=x

,

a contradiction. �

8 Qualitative aspects of the steady state solution Althoughwe do not have an explicit formula for the positive steady state solutionu = u(x) of (3.1), we know (e.g., from the phase plane analysis) that u(x)is an increasing function on [0, L], and for x close to L, it is concave down(since u′(L) = 0). A natural question is whether u(x) is concave downthroughout the habitat [0, L], or u(x) has an inflection point x∗ ∈ [0, L].

We start by analyzing the cases of low, intermediate and high advec-tion.

Lemma 8.1. The solution u = u(x) of (3.2, 3.3) has an inflection pointif and only if u(0) > 1 − q2.

Proof. Let (u(x), v(x)) = (u(x), u′(x)). Then u(x) has an inflectionpoint if and only if its orbit in the uv-plane intersects the v-nullclinev = 1

qu(1 − u), which happens exactly when the point (u(0), v(0)) =

(u(0), qu(0)) lies above the v-nullcline. This is equivalent to

qu(0) >1

qu(0)(1 − u(0)),


oru(0) > 1 − q2,

as needed.

Proposition 8.2.

(i) For q > 1 every solution of (3.2, 3.3) has an inflection point.(ii) For q < 1/

√2 no solution of (3.2, 3.3) has an inflection point (see

Figure 8.1).

FIGURE 8.1: No inflection points for q < 1√2.

Proof. (i) By Lemma 8.1 and the fact that u(0) > 0 (upstream densityis positive).

(ii) If u(x) has an inflection point, by Lemma 8.1, we have u(0) >1 − q2 > 1

2 . But if an orbit of (3.4) starts at a point located above thev-nullcline and to the right from u = 1/2, from phase plane analysis wecan conclude that it will never cross the u-axis, hence will not satisfythe second boundary condition. Thus u(x) has no inflection point.

Remark 8.3. If 1√2< q < 1, then


• if νmax ≤ 1 − q2, then no solution has an inflection point;• if νmax > 1 − q2, then by lemma 8.1, the solutions with 1 − q2 <u(0) ≤ νmax have inflection points, and solutions with u(0) < 1 − q2

do not have inflection points; in other words, inflection points onlyoccur for large domains (equivalently, for high upstream/downstreamdensities, see Figure 8.2).

FIGURE 8.2: The case of intermediate advection ( 1√2

< q < 1).

In the case when upstream density is low, we can use linearizationaround the zero steady state to obtain the distance from the upstreamboundary to the inflection point (length of boundary layer). Note thatthe solution of the linear system will only have an inflection point ifq > 1 (otherwise the v-nullcline v = 1

qu will be above the first boundary

condition v = qu). The system linearized at the origin takes the form

(8.1)

{

u′ = v,

v′ = qv − u.

The general solution of the above system is given by

u(x) = eqx

2 (α cos θx+ β sin θx),


where

θ =

√

4 − q2

2.

Using the first boundary condition we obtain

v(x) = u′(x) = αeqx

2

(

q cos θx+

(

q2

4θ− θ

)

sin θx

)

.

Differentiating of the above expression gives

(8.2) u′′(x) =αq

2e

qx

2

(

q cos θx+

(

q2

4− θ

)

sin θx

)

+ αeqx

2

(

− qθ sin θx+

(

q2

4+ θ

)

θ cos θx

)

.

If u(x) has an inflection point at x = x∗, then u′′(x∗) = 0. Setting theright hand side of (8.2) equal to zero, we find the expression for x∗:

(8.3) x∗ =

1

θarctan

(

3q2

4−θ2

3qθ2

− q3

8θ

)

, 1 < q ≤√

3,

1

θ

(

π + arctan

( 3q2

4 − θ2

3qθ2 − q3

8θ

))

,√

3 < q < 2.

As we can see from Figure 8.3, for small upstream densities, for-mula (8.3) gives a good approximation of the inflection point of thesolution in nonlinear case (found numerically).

9 Conclusions and biological implications In this work, weconsidered a reaction-advection-diffusion equation with logistic growthterm on a bounded domain with Danckwert’s boundary conditions. Weproved the existence, uniqueness and stability of a positive steady statefor domain lengths above a critical threshold. Below this threshold, thezero steady state is stable. In particular, a transcritical bifurcation oc-curs at this threshold. The positive steady state solution is an increasingfunction of the spatial variable. For large enough advection, the solutionhas an inflection point; we gave an approximate formula for the distanceof this inflection point from the upstream boundary. We also showedthat the solution is a decreasing function of advection speed.


FIGURE 8.3: Distance from upstream boundary to the inflection pointvs. advection as given by numerical simulation (thick) and formula (8.3)(thin), for upstream density u(0) = 0.001.

These results complement and contrast results on reaction-diffusionequations without advection, but with hostile boundary conditions; see,e.g., [10]. Those models also exhibit a transcritical bifurcation at the“critical domain size.” Loss of individuals in that case is entirely dueto diffusion, while in our case, it is purely due to advection. Also, thepositive steady state in those models have no inflection point.

While this work focuses on mathematical analysis, our model is in-spired by the biological question of population dynamics in advectiveenvironments, such as streams and rivers. Speirs and Gurney [21] firstapplied a linear analogue of our model (albeit with slightly differentboundary conditions) to study the persistence conditions of planktonand insects in small streams in Southeast England. They demonstratedthat, under certain conditions, diffusive movement of individuals cancounterbalance the loss of individuals through directed movement withthe water and thereby lead to population persistence.

For example, their model explains the absence of plankton organismsin Broadstone Stream (Southeast England), which is relatively shortand shallow with significant advection. The authors argue that due toits shallowness and the nature of plankton, the organisms would have


to be present throughout the water column, and would be subjectedto average advection, which exceeds the critical value for any realisticgrowth rate and diffusivity of plankton. On the other hand, stonefliesare actually found in the creek. Their nymphs are primarily benthicorganisms, and enter the stream for only 0.01% of time. Therefore, theyexperience an effective advection speed that is reduced by a factor of10−4, well below the critical advection value for this species. Thus, eventhough the time scale of the growth dynamics is typically much slowerthan that of the advective movement, the latter is signifcantly smallerfor predominantly benthic species and is actually comparable with theformer.

Stoneflies have winged adult stages that can easily move upstream foregg-deposition. It is unclear whether this movement can be accuratelycaptured by the diffusion operator. Potentially more realistic could bea discrete-time model with a dispersal kernel as in [13]. The reaction-advection-diffusion approach does, however, seem appropriate for manybenthic species that never emerge from the water, for example, Dreis-sena polymorpha (zebra mussels), meroplanktonic copepod (Caullanacanadensis), and mysid shrimp (Neomysis integer), where a similar re-duction of the actual advection speed occurs. Benthic algae can beconsidered within this framework as well. Some “best guesses” for pa-rameter values were given in [12]

Our model makes some testable hypotheses. Is it true that the abun-dance of these organisms increases downstream from an insurmountablebarrier? And does the location of the inflection point (8.3) predict thespatial scale over which the influence of this upstream barrier is present?

The reduction of population density at the upstream end can also bea mechanism for coexistence of two competing species that would notcoexist in the absence of advection. This has been demonstrated numeri-cally in [12]. We are currently working on a more detailed mathematicalanalysis of the reaction-advection-diffusion model with two competingspecies. Effects of advection on predator-prey systems have recentlybeen studied by Hilker and Lewis [7]. In the future, we plan to analyzethe positive steady state(s) of an extension of our model that include abenthic compartment explicitly; see [17].

Acknowledgements We thank an anonymous referee for sugges-tions to significantly shorten several proofs. OV acknowledges supportfrom MITACS. FL is grateful for an NSERC discovery grant and foran Early Researcher Award from the Ontario Ministry of Research andInnovation.


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Department of Mathematics and Statistics, University of Ottawa,

585 King Edward Avenue, Ottawa, ON K1N 6N5

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