Post on 01-Feb-2021
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Lectures on
The Mathematics of Biological Invasions
Jonathan A. Sherratt
Department of Mathematicsand Maxwell Institute for Mathematical Sciences
Heriot-Watt University
University of Alberta, May/June 2013
These lectures can be downloaded from my web site
www.ma.hw.ac.uk/∼jas
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
Outline
1 Single-Species Invasions: Monostable Kinetics
2 Single-Species Invasions: Bistable Kinetics
3 Single-Species Invasions: Miscellaneous PDE Models
4 Invasions of Prey by Predators
5 Integrodifferential Equation Models for Invasions
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
Epidermal Wound Healing: A Cellular Invasion
Epidermal wounds are very shallow (no bleeding), e.g. blisters
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
Epidermal Wound Healing: A Cellular Invasion
Epidermal wounds are very shallow (no bleeding), e.g. blisters
Healing occurs by the cells
invading the wound, driven by:
cell movement
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
Epidermal Wound Healing: A Cellular Invasion
Epidermal wounds are very shallow (no bleeding), e.g. blisters
Healing occurs by the cells
invading the wound, driven by:
cell movement
increased cell division
near the wound edge
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
Epidermal Wound Healing: A Cellular Invasion
Epidermal wounds are very shallow (no bleeding), e.g. blisters
Healing occurs by the cells
invading the wound, driven by:
cell movement
increased cell division
near the wound edge
Cell division is
upregulated by
chemicals produced
by the cells
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
A Mathematical Model
Cell division is upregulated by chemicals produced by the cells
Model variables: n(x , t) and c(x , t).
Model equations:
∂n/∂t =
cellmovement︷ ︸︸ ︷
D∇2n +
celldivision︷ ︸︸ ︷
s(c)(N − n)n−
celldeath︷︸︸︷δn
∂c/∂t = Dc∇2c︸ ︷︷ ︸chemicaldiffusion
+An/(1 + αn2)︸ ︷︷ ︸production
by cells
− λc︸︷︷︸decay
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
A Mathematical Model
Cell division is upregulated by chemicals produced by the cells
Model variables: n(x , t) and c(x , t).
Model equations:
∂n/∂t =
cellmovement︷ ︸︸ ︷
D∇2n +
celldivision︷ ︸︸ ︷
s(c)(N − n)n−
celldeath︷︸︸︷δn
∂c/∂t = Dc∇2c︸ ︷︷ ︸chemicaldiffusion
+An/(1 + αn2)︸ ︷︷ ︸production
by cells
− λc︸︷︷︸decay
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
A Mathematical Model
Cell division is upregulated by chemicals produced by the cells
Model variables: n(x , t) and c(x , t).
Model equations:
∂n/∂t =
cellmovement︷ ︸︸ ︷
D∇2n +
celldivision︷ ︸︸ ︷
s(c)(N − n)n−
celldeath︷︸︸︷δn
∂c/∂t = Dc∇2c︸ ︷︷ ︸chemicaldiffusion
+An/(1 + αn2)︸ ︷︷ ︸production
by cells
− λc︸︷︷︸decay
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
A Mathematical Model
Cell division is upregulated by chemicals produced by the cells
Model variables: n(x , t) and c(x , t).
Model equations:
∂n/∂t =
cellmovement︷ ︸︸ ︷
D∇2n +
celldivision︷ ︸︸ ︷
s(c)(N − n)n−
celldeath︷︸︸︷δn
∂c/∂t = Dc∇2c︸ ︷︷ ︸chemicaldiffusion
+An/(1 + αn2)︸ ︷︷ ︸production
by cells
− λc︸︷︷︸decay
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
A Mathematical Model
Cell division is upregulated by chemicals produced by the cells
Model variables: n(x , t) and c(x , t).
Model equations:
∂n/∂t =
cellmovement︷ ︸︸ ︷
D∇2n +
celldivision︷ ︸︸ ︷
s(c)(N − n)n−
celldeath︷︸︸︷δn
∂c/∂t = Dc∇2c︸ ︷︷ ︸chemicaldiffusion
+An/(1 + αn2)︸ ︷︷ ︸production
by cells
− λc︸︷︷︸decay
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
A Mathematical Model
Cell division is upregulated by chemicals produced by the cells
Model variables: n(x , t) and c(x , t).
Model equations:
∂n/∂t =
cellmovement︷ ︸︸ ︷
D∇2n +
celldivision︷ ︸︸ ︷
s(c)(N − n)n−
celldeath︷︸︸︷δn
∂c/∂t = Dc∇2c︸ ︷︷ ︸chemicaldiffusion
+An/(1 + αn2)︸ ︷︷ ︸production
by cells
− λc︸︷︷︸decay
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
A Mathematical Model
Cell division is upregulated by chemicals produced by the cells
Model variables: n(x , t) and c(x , t).
Model equations:
∂n/∂t =
cellmovement︷ ︸︸ ︷
D∇2n +
celldivision︷ ︸︸ ︷
s(c)(N − n)n−
celldeath︷︸︸︷δn
∂c/∂t = Dc∇2c︸ ︷︷ ︸chemicaldiffusion
+An/(1 + αn2)︸ ︷︷ ︸production
by cells
− λc︸︷︷︸decay
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
Reduction to One Equation
∂n/∂t = D∇2n + s(c)(N − n)n − δn∂c/∂t = Dc∇2c + An/(1 + αn2)− λc
The chemical kinetics are very fast⇒ A, λ large.So to a good approximation c = (A/λ) · n/(1 + αn2).
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
Reduction to One Equation
∂n/∂t = D∇2n + s(c)(N − n)n − δn∂c/∂t = Dc∇2c + An/(1 + αn2)− λc
The chemical kinetics are very fast⇒ A, λ large.So to a good approximation c = (A/λ) · n/(1 + αn2).
⇒ ∂n/∂t = D∇2n + f (n)where
f (n) = s
(An
λ(1 + αn2)
)(N − n)n − δn
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
Typical Model Solution
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
Travelling Wave Solutions
During most of the healing, the solution moves with constant
speed and shape.
This is a “travelling wave solution”
n(x , t) = N(x − at)
where a is the wave speed. We will write
z = x − at . Then ∂n/∂x = dN/dz and∂n/∂t = −a dN/dz
⇒ D d2N
dz2+ a
dN
dz+ f (N) = 0
a
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
The Speed of Travelling Waves
We know that N → 0 as z →∞. Recall that
Dd2N
dz2+ a
dN
dz+ f (N) = 0
Therefore when N is small
Dd2N
dz2+ a
dN
dz+ f ′(0)N = 0
to leading order. This has solutions N(z) = N0eλz with
Dλ2 + aλ+ f ′(0) = 0⇒ λ = 12(−a±
√a2 − 4Df ′(0)
).
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
The Speed of Travelling Waves
We know that N → 0 as z →∞. Recall that
Dd2N
dz2+ a
dN
dz+ f (N) = 0
Therefore when N is small
Dd2N
dz2+ a
dN
dz+ f ′(0)N = 0
to leading order. This has solutions N(z) = N0eλz with
Dλ2 + aλ+ f ′(0) = 0⇒ λ = 12(−a±
√a2 − 4Df ′(0)
).
If a < 2√
Df ′(0) then λ is complex⇒ thesolutions oscillate about N = 0, whichdoes not make sense biologically
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
The Speed of Travelling Waves
We know that N → 0 as z →∞. Recall that
Dd2N
dz2+ a
dN
dz+ f (N) = 0
Therefore when N is small
Dd2N
dz2+ a
dN
dz+ f ′(0)N = 0
to leading order. This has solutions N(z) = N0eλz with
Dλ2 + aλ+ f ′(0) = 0⇒ λ = 12(−a±
√a2 − 4Df ′(0)
).
If a < 2√
Df ′(0) then λ is complex⇒ thesolutions oscillate about N = 0, whichdoes not make sense biologically
So we require
a ≥ 2√
Df ′(0).
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
The Speed of Travelling Waves (contd)
We require a ≥ 2√
Df ′(0)
Mathematical theory implies that in applications, waves will
move at the minimum possible speed, 2√
Df ′(0).
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
Applications to Wound Healing
For the wound healing model
f ′(0) = Ns(0)− δ
⇒ wave speed a = 2√
D(Ns(0)− δ)
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
General Results on Wave Speed
Consider the equation ∂u/∂t = D ∂2u/∂x2 + f (u) withf (0) = f (1) = 0, f (u) > 0 on 0 < u < 1, f ′(0) > 0 andf ′(u) < f ′(0) on 0 < u ≤ 1.For this equation, two important theorems were proved by
Kolmogorov, Petrovskii and Piskunov; a similar but slightly less
general study was done at the same time by Fisher.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
General Results on Wave Speed
Consider the equation ∂u/∂t = D ∂2u/∂x2 + f (u) withf (0) = f (1) = 0, f (u) > 0 on 0 < u < 1, f ′(0) > 0 andf ′(u) < f ′(0) on 0 < u ≤ 1.For this equation, two important theorems were proved by
Kolmogorov, Petrovskii and Piskunov; a similar but slightly less
general study was done at the same time by Fisher.
Theorem 1. There is a positive travelling wave solution for all
wave speeds ≥ 2√
Df ′(0), and no positive travelling waves forspeeds less than this critical value.
A proof of theorem 1 is in the supplementary material.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
General Results on Wave Speed
Consider the equation ∂u/∂t = D ∂2u/∂x2 + f (u) withf (0) = f (1) = 0, f (u) > 0 on 0 < u < 1, f ′(0) > 0 andf ′(u) < f ′(0) on 0 < u ≤ 1.For this equation, two important theorems were proved by
Kolmogorov, Petrovskii and Piskunov; a similar but slightly less
general study was done at the same time by Fisher.
Theorem 2. Suppose that u(x , t = 0) = 1 for x sufficiently largeand negative, and u(x , t = 0) = 0 for x sufficiently large andpositive. Then the solution u(x , t) approaches the travelling wavesolution with the critical speed 2
√Df ′(0) as t →∞.
The form of the approach to the travelling wave is known:
speed− 2√
Df ′(0) ∝ (log t)/t (Bramson, 1983).The proof of theorem 2 is extremely difficult.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
General Results on Wave Speed
Consider the equation ∂u/∂t = D ∂2u/∂x2 + f (u) withf (0) = f (1) = 0, f (u) > 0 on 0 < u < 1, f ′(0) > 0 andf ′(u) < f ′(0) on 0 < u ≤ 1.For this equation, two important theorems were proved by
Kolmogorov, Petrovskii and Piskunov; a similar but slightly less
general study was done at the same time by Fisher.
References:
Bramson, M.D. 1983 Convergence of solutions of the Kolmogorov
equation to travelling waves. Mem. AMS 44 no. 285.
Fisher, R.A. 1937 The wave of advance of advantageous genes. Ann.
Eug. 7, 355-369.
Kolmogorov, A., Petrovskii, I., Piskunov, N. 1937 Etude de l’équation de
la diffusion avec croissance de la quantité de matière et son application
à un problème biologique. Moscow Univ. Bull. Math. 1, 1-25.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
Therapeutic Addition of Chemical
Now return to the wound healing model and consider adding
extra chemical to the wound as a therapy.
The equation for the chemical changes to
∂c/∂t = Dc∇2c + An/(1 + αn2)− λc + χ
⇒ f (n) changes to
s
(χ
λ+
An
λ(1 + αn2)
)(N − n)n − δn
⇒ f ′(0) changes to Ns(χ/λ)− δ
⇒ observed (min poss) wave speed a = 2√
D(Ns(χ/λ)− δ)
Wavespeed
a
Chemical application rate χ
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
Deducing the Chemical Profile
Since we know c as a function of n, there is also a travelling
wave of chemical, whose form we can deduce. The chemical
profile has a peak in the wave front.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
Deducing the Chemical Profile
Since we know c as a function of n, there is also a travelling
wave of chemical, whose form we can deduce. The chemical
profile has a peak in the wave front.
c
n
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
Deducing the Chemical Profile
Since we know c as a function of n, there is also a travelling
wave of chemical, whose form we can deduce. The chemical
profile has a peak in the wave front.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
Deducing the Chemical Profile
Since we know c as a function of n, there is also a travelling
wave of chemical, whose form we can deduce. The chemical
profile has a peak in the wave front.
However, there are no
theorems on the speed of
wave fronts in systems of
reaction-diffusion
equations, except in a few
special cases (which do
not include this model).
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
The Spread of Rabies Amongst Foxes
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
The Spread of Rabies Amongst Foxes
Biological facts:
Rabies is transmitted by direct
contact (saliva)
Rabies is always fatal
Healthy foxes are highly territorial
but rabid foxes lose this sense of
territoriality and wander randomly
over significant distances.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
The Spread of Rabies Amongst Foxes
Biological facts:
Rabies is transmitted by direct
contact (saliva)
Rabies is always fatal
Healthy foxes are highly territorial
but rabid foxes lose this sense of
territoriality and wander randomly
over significant distances.
Healthy ∂h/∂t = −Khr
Rabid ∂r/∂t = Khr − Ar + D∂2r/∂x2
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
The Spatial Spread of Rabies
Travelling waves: h(x , t) = H(z), r(x , t) = R(z), z = x − ct
⇒ cH ′ − KHR = 0DR′′ + cR′ + KHR − AR = 0
Questions:
How fast does the disease spread?
What proportion of the fox population is killed by the
disease?
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
The Speed of the Epidemic
Write as 3 first order equations:
H ′ = KHR/c R′ = W W ′ = −(c/D)W−(K/D)HR+(A/D)R
Stability matrix
R/c KH/c 00 0 1
−KR/D (A− KH)/D −c/D
Subs (H,R,W ) = (H0, 0, 0)⇒
∣∣∣∣∣∣
−λ KH0/c 00 −λ 10 (A− KH0)/D −λ− c/D
∣∣∣∣∣∣= 0
⇒ λ = 0,(−c ±
√c2 + 4D(A− KH0)
)/2D
⇒ c ≥ cmin = 2√
D(KH0 − A) for non-oscillatory solutionsJonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
The Speed of the Epidemic
Write as 3 first order equations:
H ′ = KHR/c R′ = W W ′ = −(c/D)W−(K/D)HR+(A/D)R
Stability matrix
R/c KH/c 00 0 1
−KR/D (A− KH)/D −c/D
Subs (H,R,W ) = (H0, 0, 0)⇒
∣∣∣∣∣∣
−λ KH0/c 00 −λ 10 (A− KH0)/D −λ− c/D
∣∣∣∣∣∣= 0
⇒ λ = 0,(−c ±
√c2 + 4D(A− KH0)
)/2D
⇒ c ≥ cmin = 2√
D(KH0 − A) for non-oscillatory solutionsJonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
The Impact of the Epidemic
cH ′ − KHR = 0DR′′ + cR′ + KHR − AR = 0
⇒ DR′′ + cR′ + cH ′ − AcH ′/(RH) = 0⇒ DR′ + cR + cH − (Ac/K ) log H = constant
Before and after wave, R = R′ = 0
⇒ cH1 − (Ac/K ) log H1 = cH0 − (Ac/K ) log H0
⇒ (K/A)H1 − {(K/A)H0 − log H0} = log H1
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Epidermal Wound Healing: A Cellular Invasion
The Spread of Rabies Amongst Foxes
The Impact of the Epidemic (continued)
(K/A)H1 − {(K/A)H0 − log H0}= log H1
Recall: speed = 2√
D(KH0 − A).
Conclusion: KH0 < A: no epidemicKH0 > A: epidemic; faster and more severe as K ↑ and A ↓.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Spruce Budworm Dynamics
Travelling Waves in the Spruce Budworm Model
Direction of Wave Propagation
Existence of Travelling Waves
The Value of the Wave Speed
Outline
1 Single-Species Invasions: Monostable Kinetics
2 Single-Species Invasions: Bistable Kinetics
3 Single-Species Invasions: Miscellaneous PDE Models
4 Invasions of Prey by Predators
5 Integrodifferential Equation Models for Invasions
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Spruce Budworm Dynamics
Travelling Waves in the Spruce Budworm Model
Direction of Wave Propagation
Existence of Travelling Waves
The Value of the Wave Speed
Spruce Budworm Dynamics
The spruce budworm is an important forestry pest in North
America.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Spruce Budworm Dynamics
Travelling Waves in the Spruce Budworm Model
Direction of Wave Propagation
Existence of Travelling Waves
The Value of the Wave Speed
Spruce Budworm Dynamics
The spruce budworm is an important forestry pest in North
America.
A simple model is
∂u/∂t = D ∂2u/∂x2 + f (u)
where
f (u) = ru(1− u/q)︸ ︷︷ ︸logisticgrowth
− u2/(1 + u2)︸ ︷︷ ︸predationby birds
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Spruce Budworm Dynamics
Travelling Waves in the Spruce Budworm Model
Direction of Wave Propagation
Existence of Travelling Waves
The Value of the Wave Speed
Travelling Waves in the Spruce Budworm Model
A simple model is
∂u/∂t = D ∂2u/∂x2 + f (u)
where
f (u) = ru(1− u/q)︸ ︷︷ ︸logisticgrowth
− u2/(1 + u2)︸ ︷︷ ︸predationby birds
In applications, waves travelling between u = u1 (“refuge state”)and u = u3 (“outbreak state”) are important. Both of thesesteady states are locally stable. Therefore the direction of wave
propagation is not obvious.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Spruce Budworm Dynamics
Travelling Waves in the Spruce Budworm Model
Direction of Wave Propagation
Existence of Travelling Waves
The Value of the Wave Speed
Direction of Wave Propagation
Travelling wave solutions U(z) (z = x − ct) satisfy
DU ′′ + cU ′ + f (U) = 0
DU ′U ′′ + cU ′2 + f (U)U ′ = 0
⇒[
12DU
′2]+∞−∞
+ c
∫ +∞
−∞U ′2dz +
∫ +∞
−∞f (U)U ′dz = 0
We know that U ′(±∞) = 0. Therefore for a wave front withU(−∞) = u1 and U(+∞) = u3, c has the same sign as−∫ u3
u1f (U)dU.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Spruce Budworm Dynamics
Travelling Waves in the Spruce Budworm Model
Direction of Wave Propagation
Existence of Travelling Waves
The Value of the Wave Speed
Direction of Wave Propagation
For a wave front with U(−∞) = u1 and U(+∞) = u3, c has thesame sign as −
∫ u3u1
f (U)dU.
For the spruce budworm model, this can be used to predict
whether a local outbreak will die out or spread.
Outbreak dies out Outbreak spreads
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Spruce Budworm Dynamics
Travelling Waves in the Spruce Budworm Model
Direction of Wave Propagation
Existence of Travelling Waves
The Value of the Wave Speed
Direction of Wave Propagation
∂u/∂t = D ∂2u/∂x2 + f (u)
f (u) = ru(1− u/q)−u2/(1 + u2)
• Outbreak spreads� Outbreak dies out
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Spruce Budworm Dynamics
Travelling Waves in the Spruce Budworm Model
Direction of Wave Propagation
Existence of Travelling Waves
The Value of the Wave Speed
Existence of Travelling Waves
Theorem [Fife & McLeod, 1977]. Consider the equation
∂u/∂t = D ∂2u/∂x2 + f (u) with f (u1) = f (u2) = f (u3) = 0,f ′(u1) < 0, f
′(u2) > 0, f′(u3) < 0. This equation has a travelling
wave solution u(x , t) = U(x − ct) with U(−∞) = u1 andU(+∞) = u3 for exactly one value of the wave speed c.A proof of this theorem is in the supplementary material.
Reference:Fife, P.C. & McLeod, J.B. 1977 The approach of solutions of nonlinear
diffusion equations to travelling front solutions. Arch. Rat. Mech. Anal.
65, 335-361.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Spruce Budworm Dynamics
Travelling Waves in the Spruce Budworm Model
Direction of Wave Propagation
Existence of Travelling Waves
The Value of the Wave Speed
The Value of the Wave Speed
There is no general result on the value of the unique wave
speed for a reaction-diffusion equation with bistable kinetics.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Basic Reaction-Diffusion Model
Reaction-Diffusion-Advection Model
Density-Dependent Dispersal
Invasions for Degenerate D-D Dispersal
Correlated Movement: The Telegraph Equation
Outline
1 Single-Species Invasions: Monostable Kinetics
2 Single-Species Invasions: Bistable Kinetics
3 Single-Species Invasions: Miscellaneous PDE Models
4 Invasions of Prey by Predators
5 Integrodifferential Equation Models for Invasions
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Basic Reaction-Diffusion Model
Reaction-Diffusion-Advection Model
Density-Dependent Dispersal
Invasions for Degenerate D-D Dispersal
Correlated Movement: The Telegraph Equation
Basic Reaction-Diffusion Model
Basic model: reaction-diffusion equation
∂u/∂t = D∂2u/∂x2 + f (u)
Note that this implies an infinite speed of propagation.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Basic Reaction-Diffusion Model
Reaction-Diffusion-Advection Model
Density-Dependent Dispersal
Invasions for Degenerate D-D Dispersal
Correlated Movement: The Telegraph Equation
Reaction-Diffusion-Advection Model
Extension – advection: reaction-diffusion-advection equation
∂u/∂t = D∂2u/∂x2 − χ∂u/∂x + f (u)
Note the − sign in front of the advection term.Flux = −D∂u/∂x + χu
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Basic Reaction-Diffusion Model
Reaction-Diffusion-Advection Model
Density-Dependent Dispersal
Invasions for Degenerate D-D Dispersal
Correlated Movement: The Telegraph Equation
Invasion Speed for Reaction-Diffusion-Advection
Model
∂u/∂t = D∂2u/∂x2 − χ∂u/∂x + f (u)Travelling waves u(x , t) = U(z), z = x − ct satisfy
DU ′′ + (c − χ)U ′ + f (U) = 0
Therefore the advection simply adds χ to the invasion speed.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Basic Reaction-Diffusion Model
Reaction-Diffusion-Advection Model
Density-Dependent Dispersal
Invasions for Degenerate D-D Dispersal
Correlated Movement: The Telegraph Equation
Density-Dependent Dispersal
Extension – density dependent dispersal: nonlinear diffusion
∂u/∂t = (∂/∂x)[D(u)∂u/∂x
]+ f (u)
Usually D′(u) > 0: a response to crowding.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Basic Reaction-Diffusion Model
Reaction-Diffusion-Advection Model
Density-Dependent Dispersal
Invasions for Degenerate D-D Dispersal
Correlated Movement: The Telegraph Equation
Invasion Speed for Density Dependent Dispersal
We will look just at monostable kinetics:
∂u/∂t = (∂/∂x)[D(u)∂u/∂x
]+ ku(1− u)
Travelling waves u(x , t) = U(z), z = x − ct satisfy[D(U)U ′
]′+ cU ′ + kU(1− U) = 0
Linearise about U = 0:
D(0)U ′′ + cU ′ + kU(1− U) = 0
Invasion speed = 2√
D(0)k .
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Basic Reaction-Diffusion Model
Reaction-Diffusion-Advection Model
Density-Dependent Dispersal
Invasions for Degenerate D-D Dispersal
Correlated Movement: The Telegraph Equation
Invasion Speed for Density Dependent Dispersal
We will look just at monostable kinetics:
∂u/∂t = (∂/∂x)[D(u)∂u/∂x
]+ ku(1− u)
Travelling waves u(x , t) = U(z), z = x − ct satisfy[D(U)U ′
]′+ cU ′ + kU(1− U) = 0
Linearise about U = 0:
D(0)U ′′ + cU ′ + kU(1− U) = 0
Invasion speed = 2√
D(0)k .
What happens if D(0) = 0, e.g. D(u) = u ?
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Basic Reaction-Diffusion Model
Reaction-Diffusion-Advection Model
Density-Dependent Dispersal
Invasions for Degenerate D-D Dispersal
Correlated Movement: The Telegraph Equation
Invasions for Degenerate D-D Dispersal
∂u/∂t = (∂/∂x)[D(u)∂u/∂x
]+ ku(1− u)
If D(0) = 0 then the invasion speed is not determined by simplelinearisation about u = 0. In fact the invading wave is of “sharpfront” type.
For D(U) = U, the speed of the sharp front wave is√
k/2(see supplementary material for details).
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Basic Reaction-Diffusion Model
Reaction-Diffusion-Advection Model
Density-Dependent Dispersal
Invasions for Degenerate D-D Dispersal
Correlated Movement: The Telegraph Equation
Correlated Movement: The Telegraph Equation
Extension – correlated movement steps: telegraph equation
∂u/∂t = (V 2/2λ)∂2u/∂x2−(1/2λ)∂2u/∂t2+ f (u)+(1/2λ)∂f/∂t
Here V is the velocity and λ is the rate of changing direction.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
Outline
1 Single-Species Invasions: Monostable Kinetics
2 Single-Species Invasions: Bistable Kinetics
3 Single-Species Invasions: Miscellaneous PDE Models
4 Invasions of Prey by Predators
5 Integrodifferential Equation Models for Invasions
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
A Model for Predator-Prey Interactions
predators
∂p/∂t = Dp∇2p︸ ︷︷ ︸dispersal
+ akph/(1 + kh)︸ ︷︷ ︸benefit frompredation
− bp︸︷︷︸death
prey
∂h/∂t = Dh∇2h︸ ︷︷ ︸dispersal
+ rh(1− h/h0)︸ ︷︷ ︸intrinsic
birth & death
− ckph/(1 + kh)︸ ︷︷ ︸predation
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
Predator-Prey Kinetics
For some parameters, the kinetic ODEs have a stable
coexistence steady state.
predators
∂p/∂t = akph/(1 + kh)
−bp
prey
∂h/∂t = rh(1− h/h0)−ckph/(1 + kh)
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
Predator-Prey Kinetics
For other parameters, the coexistence steady state is unstable,
and there is a stable limit cycle.
predators
∂p/∂t = akph/(1 + kh)
−bp
prey
∂h/∂t = rh(1− h/h0)−ckph/(1 + kh)
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
Example of a Cyclic Predator-Prey System
Voles in Fennoscandia are subject to predation by weasels.
Vole Weasel
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
Example of a Cyclic Predator-Prey System
Voles in Fennoscandia are subject to predation by weasels.
Vole Weasel
Removal of weasels⇒ loss of multi-year cyclesImplication: vole cycles are caused by predation by weasels
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
Simulating a Predator-Prey Invasion
Initially we set the prey to the
prey-only equilibrium throughout
the domain.
Initially we set the predators to
zero except near the left hand
boundary.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
Simple Predator-Prey Invasions
When the kinetic ODEs have a stable coexistence steady state,
invasions have a simple form.
The invasion speed can be determined by linearising about the
prey-only steady state (ahead of the invasion).
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
Invasions for Oscillatory Kinetics
When the kinetic ODEs are oscillatory, spatiotemporal
oscillations occur behind the invasion front.
Wavetrain behind an invasion front
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Everyday example: Mexican wave
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
Why Does Dynamical Stabilisation Occur?
Oscillations and Chaos Behind Invasion
What is a Wavetrain?
A wavetrain is a soln of form f (x − ct), with f (.) periodic.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Lectures on The Mathematics of Biological Invasions
Single-Species Invasions: Monostable Kinetics
Single-Species Invasions: Bistable Kinetics
Single-Species Invasions: Miscellaneous PDE Models
Invasions of Prey by Predators
Integrodifferential Equation Models for Invasions
Predator-Prey Model
Simulating a Predator-Prey Invasion
What is a Wavetrain?
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