Spatial Ecology: Lecture 4, Integrodifference

equations

II Southern-Summer school on Mathematical Biology

Integrodifference equations

Diffusion models assume growth and dispersal occur at the same time.

When reproduction and dispersal occur at discrete intervals an integrodifference equation is a more relevant formulation. E.g.

annual plants,

Many insects,

Migrating bird species

Integrodifference equation

The model:

and non-forest)

tH tH 1tHDynamicsDynamics DispersaltN tN 1tNDynamicsDynamics Dispersal

One generation (e.g.1 year)

The model:

dyyNyxKxN tt Dynamics

xy to fromdispersing of Prob.

)( ),()(1

Integrodifference equation

The model:

Dynamics

and non-forest)

tH tH 1tHDynamicsDynamics DispersaltN tN 1tNDynamicsDynamics Dispersal

One generation (e.g.1 year)

The model:

dyyNyxKxN tt Dynamics

xy to fromdispersing of Prob.

)( ),()(1

))(()( yNfyN tt

Kernels k(x,y)=k(x-y) when dispersal depends on distance only

Mechanistic derivation of dispersal kernels

Gaussian: a(t)=(t-T), stops at time T

Laplacian: a(t)=a>0, constant settling rate

settled Total

0

rateSettling

),()()(

)(),0( ,)(

dttxutaxk

xxuutaDuu xxt

x

Da

DaxkL exp

4)(

DTx

DTxkG 4

exp4

1)(2

Dispersal kernels from data

Travelling wave speeds

Assume

f is linearly bounded, f(N)<=f’(0)N (No Allee effect)

f is monotone

K(x) has a moment generating function (no ‘fat-tailed dispersal kernels)

Then we can linearly determine the asymptotic wave speed.

Look at behaviour near N*=0 (linearise there)

dyyNfyxKxN tt ))((),()(1

dyyNyxKfxN tt )(),()0(')(1

Travelling wave speeds

A travelling wave solution moves with constant shape and speed c, so )()(1 cxNxN tt

Travelling wave speeds

A travelling wave solution moves with constant shape and speed c, so

Then (assume distance dependent dispersal))()(1 cxNxN tt

dyyNyxKfcxN tt )(|)(|)0(')(

Travelling wave speeds

A travelling wave solution moves with constant shape and speed c, so

Then (assume distance dependent dispersal)

Look for solutions at the edge of the travelling wave which decay exponentially, so

M(s) is the moment generating function for k(y)

)()(1 cxNxN tt

dyyNyxKfcxN tt )(|)(|)0(')(

)exp()( sxxNt

)()0(')exp()()0(')exp( sMfdysyyKfsc

Asymptotic wave speed

Differentiating with respect to s, and noting that initial conditions with compact support lead to a minimum speed give c*

)0(')(ln1min* fsM

sc

s

Examples of wave speeds

Gaussian

Note if r=ln f’(0) and D=2 then the wave speed is the same as the PDE case:

Laplacian

We can’t find c explicitly, but since ML (s)>=MG (s) then cLaplace >cGaussian

DssM G 2 where),2/exp()( 222

aDs

sM L /2 where,2/1

1)( 222

)0('ln22 fc

Drc 2

Shape of the kernel greatly affects speed.

Fat tailed kernels

Spatial extent

Fat tailed kernels can give accelerating waves, we can’t calculate the speed, but we can measure the spatial extent of the wave at a given time.

Spatial extent= distance from source where population first falls below a threshold N.

Use Fourier Transforms

)()( ,)(),()0(')( 001 xNxNdyyNyxKfxN tt

0))(ˆ())0('()(ˆ

)(ˆ)( ,)()(ˆ

NwkfwN

dwewNxNdxexNwN

ttt

iwxtt

iwxtt

Hence,

Spatial extent

In the case of the Cauchy Kernel:

Its easy to find the inverse of the Fourier transform in this case so

More generally

|)|exp()(ˆ ,)(

)( 22 wwkx

xk

2022

0 )()( ,)(

)( tN

RtNtx

xttRN

xNt

f

t

t

1x provided ,)( ),()(0

10

tft

t RNNktxxkRNxN

Population spread and invasion Muskrat

House finch

Cheat grass

Linear expansionwith time

Slow initial spreadfollowed be linearexpansion (e.g.Alleeeffects)

Spread rate continually increaseswith time (e.g long distance dispersal)

House finch model

Spring, t

Summer

Fall

Reproduction

Survival

Survival

Form breeding pairs

Disperse

pairs Breeding)()()(dults)(

)springin old months 12(9 Juveniles)(

xAxJxNAxA

xJ

ttt

t

t

Reproduction

Average number of offspring produced

Competition for nesting sites

C= average number of offspring born that survive summer

, rate or pair formation

T, Time for pair formation

density of nest sites

/2)/(4)( 2

2

tt

tt NNT

cNNf

/2)/(4)( 2

2

tt

tt NNT

cNNf

Allee efffect!

Dispersal

dyyNxyKpdyyNfpxyKNfpsNfpxN

dyyNxyKpNfpsxA

dyyNfxyKpNfpxJ

tAAtJJtAtJt

tAAtAt

tJJtJt

)()())(()()()1()()1()(

)()()()1()(

))(()()()1()(

1

adults Dispersingsurvive which Adultsdispersing-Non

1

juveniles Dispersingjuvenilesdispersing-Non

1

Add the equations together to get an equation for breeders

Expected density of birdsat the Christmas Bird Countsin successive years

Dispersal kernels

Results: Range expansion

Slow initial spreadfollowed be linearexpansion

Invasion summary

Shape of the kernel significantly affects speed.

Travelling waves may exhibit accelerating spread if the dispersal kernels have ‘fat tails’ (not expontenially bounded)

Populations escaping an Allee effect may termporally accelerate before achieving a constant speed

References

M. Kot, M.A. Lewis, P van den Driessche. (1996) Dispersal data and the spread of invading organisms, Ecology 77:2027-2042

J.A.Powell,(2009)Spatiotemporal Models in Ecology:An Introduction to Integro-Difference Equations http://www.math.usu.edu/powell/wauclass/labs.html

Neubert, M.G., M. Kot and M.A. Lewis, 1995. Dispersal and pattern formation in a discrete-time predator-prey model. Theoretical Population Biology 48: 7–43.

R.R. Veit, M.A. Lewis(1996) Dispersal population growth and the Allee effect: Dynamics of the house finch invasion of Eastern North America, American Naturalist, 148(2), 255-274