lectures on the mathematics of biological invasionsjas/talks/sherratt-alberta-may2013.pdf · 2013....

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

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

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 movement

increased cell division

near the wound edge

Cell division is

upregulated by

chemicals produced

by the cells

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

Model equations:

∂n/∂t =

D∇2n +

s(c)(N − n)n−

by cells

Model equations:

∂n/∂t =

D∇2n +

s(c)(N − n)n−

by cells

Model equations:

∂n/∂t =

D∇2n +

s(c)(N − n)n−

by cells

Model equations:

∂n/∂t =

D∇2n +

s(c)(N − n)n−

by cells

Model equations:

∂n/∂t =

D∇2n +

s(c)(N − n)n−

by cells

Model equations:

∂n/∂t =

D∇2n +

s(c)(N − n)n−

by cells

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).

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

Typical Model Solution

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

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)

).

Dd2N

dz2+ a

dN

dz+ f (N) = 0

Dd2N

dz2+ a

dN

dz+ f ′(0)N = 0

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

Dd2N

dz2+ a

dN

dz+ f (N) = 0

Dd2N

dz2+ a

dN

dz+ f ′(0)N = 0

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).

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).

Applications to Wound Healing

For the wound healing model

f ′(0) = Ns(0)− δ

⇒ wave speed a = 2√

D(Ns(0)− δ)

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.

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.

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.

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

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

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).

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.

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

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?

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

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

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

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 ↓.

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

The spruce budworm is an important forestry pest in North

America.

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

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.

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.

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

∂u/∂t = D ∂2u/∂x2 + f (u)

f (u) = ru(1− u/q)−u2/(1 + u2)

• Outbreak spreads� Outbreak dies out

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.

There is no general result on the value of the unique wave

speed for a reaction-diffusion equation with bistable kinetics.

Basic Reaction-Diffusion Model

Reaction-Diffusion-Advection Model

Density-Dependent Dispersal

Invasions for Degenerate D-D Dispersal

Correlated Movement: The Telegraph Equation

Outline

Basic model: reaction-diffusion equation

∂u/∂t = D∂2u/∂x2 + f (u)

Note that this implies an infinite speed of propagation.

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

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.

Extension – density dependent dispersal: nonlinear diffusion

∂u/∂t = (∂/∂x)[D(u)∂u/∂x

]+ f (u)

Usually D′(u) > 0: a response to crowding.

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 .

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 ?

∂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).

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.

Predator-Prey Model

Simulating a Predator-Prey Invasion

What is a Wavetrain?

Why Does Dynamical Stabilisation Occur?

Oscillations and Chaos Behind Invasion

Outline

Predator-Prey Model

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

Predator-Prey Model

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)

Predator-Prey Model

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)

Predator-Prey Model

Example of a Cyclic Predator-Prey System

Voles in Fennoscandia are subject to predation by weasels.

Vole Weasel

Predator-Prey Model

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

Predator-Prey Model

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.

Predator-Prey Model

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).

Predator-Prey Model

Invasions for Oscillatory Kinetics

When the kinetic ODEs are oscillatory, spatiotemporal

oscillations occur behind the invasion front.

Wavetrain behind an invasion front

Predator-Prey Model

A wavetrain is a soln of form f (x − ct), with f (.) periodic.

Everyday example: Mexican wave

Predator-Prey Model

W

lectures on the mathematics of biological invasionsjas/talks/sherratt-alberta-may2013.pdf · 2013....

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