Periodic Travelling Waves in Ecology: A Users Guidejas/talks/edinburgh_july08_voles.pdfAn Ecological Case Study Stage I: Modelling and Numerical Simulation Stage II: Predicting Regular

An Ecological Case StudyStage I: Modelling and Numerical Simulation

Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties

Conclusions and Limitations

Periodic Travelling Wavesin Ecology:

A Users Guide

Jonathan A. Sherratt

Department of MathematicsHeriot-Watt University

ECMTB08, Edinburgh, 3 July 2008

This talk can be downloaded from my web sitewww.ma.hw.ac.uk/∼jas

Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide




In collaboration with:

Matthew Smith Xavier Lambin





Outline

1 An Ecological Case Study

2 Stage I: Modelling and Numerical Simulation

3 Stage II: Predicting Regular vs Irregular Patterns

4 Stage III: Predicting Wave Properties

5 Conclusions and Limitations





Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?

Outline











Field Voles in Kielder Forest







Field voles in Kielder Forest are cyclic (period 4 years).







Field voles in Kielder Forest are cyclic (period 4 years).Spatiotemporal field data shows that the cycles are spatiallyorganised into a periodic travelling wave,speed 19km/year, direction 72◦ from N.






What is a Periodic Travelling Wave?

Population density is periodic in space, with peaks and troughsmoving across the domain at a constant speed.

Mathematically: a soln of form U(x ± ct), with U(.) periodic.






























































































































































































































































































































































































































































































































































































































































What Causes the Spatial Component of theOscillations?

Hypothesis: the periodic travelling waves are caused by the largecentral reservoir.





Field Voles in Kielder ForestA Standard Predator-Prey ModelBoundary Conditions in the Field Vole ExampleTypical Model Solution

Outline












For modelling, we need an assumption on the cause of thepopulation cycles.







For modelling, we need an assumption on the cause of thepopulation cycles.

We assume that vole cycles are caused by predation by weasels,and study using a standard predator-prey model.






A Standard Predator-Prey Model

predators

∂p/∂t = Dp∇2p︸︷︷︸

dispersal

+ akph/(1 + kh)︸︷︷︸

benefit frompredation

− bp︸︷︷︸

death

prey

∂h/∂t = Dh∇2h︸︷︷︸

dispersal

+ rh(1 − h/h0)︸︷︷︸

intrinsicbirth & death

− ckph/(1 + kh)︸︷︷︸

predation

Phase plane ofkinetics:






Boundary Conditions in the Field Vole Example

Voles are an important prey species for owls and kestrels

The open expanse of Kielder Water will greatly facilitatehunting at its edge

Short eared owl

Common kestrel









Therefore we expect very high vole loss at the reservoiredge, implying a Robin boundary condition

∂

∂n

(

voledensity

)

= −(

largeconstant

)

·(

voledensity

)










∂

∂n

(

voledensity

)

= −(

largeconstant

)

·(

voledensity

)

To a good approx, vole density = 0 at the reservoir edge










∂

∂n

(

voledensity

)

= −(

largeconstant

)

·(

voledensity

)

To a good approx, vole density = 0 at the reservoir edge

At the edge of the forest, a zero flux boundary condition is anatural assumption






Typical Model Solution




































An Example of Irregular Pattern Generation

For some parameter values, obstacles with Dirichlet boundaryconditions generate irregular spatiotemporal patterns.






Movie of Irregular Pattern Generation

Click here toplay the movie





Regular and Irregular Patterns: the GoalStandard Theory of Periodic Travelling WaveOne-Dimensional ProblemThe Eigenvalue SpectrumBack to Wave Generation in 1-D Simulations

Outline











Regular and Irregular Patterns: the Goal

Goal: to predict which parameter sets will give periodictravelling waves, and which will give spatiotemporal irregularity.






Standard Theory of Periodic Travelling Wave

Mathematically, a periodic travelling wave is a soln of formU(x ± ct), with U(.) periodic.

There is an extensive literature on periodic travelling wavesin oscillatory reaction-diffusion equations

∂u/∂t = Du ∂2u/∂x2 + f (u, v)

∂v/∂t = Dv ∂2v/∂x2 + g(u, v)︸︷︷︸

kinetics havea stable

limit cycle








Theorem (Kopell & Howard, 1973):An oscillatory reaction-diffusionsystem has a one-parameter familyof periodic travelling wave solutionsif the diffusion coefficients aresufficiently close to one another.








Some members of the periodictravelling wave family are stable assolutions of the partial differentialequations, while others areunstable.






One-Dimensional Problem

To simplify the field vole problem, solve on 0 < x < xmax with

h = p = 0 at x = 0 ↔ edge of reservoirhx = px = 0 at x = xmax ↔ edge of forest

In fact the condition at x = xmax plays no significant role, andwe can consider the equations on 0 < x < ∞.






Periodic Wave Generation in 1-D Simulations

Example of periodic wave generation by boundary conditionscorresponding to the reservoir edge in 1-D:

Pre

y, w

ith in

cr ti

me

c)

0 200 400 600 800 1000 1200 1400 1600 1800 2000space, x

Pre

y, w

ith in

cr ti

me

d)

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000








Pre

y, w

ith in

cr ti

me

c)

0 200 400 600 800 1000 1200 1400 1600 1800 2000space, x

Pre

y, w

ith in

cr ti

me

d)

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000Conclusion: irregular patternsoccur when the (Dirichlet)boundary condition at x = 0generates a periodic travellingwave that is unstable.

Therefore we mustinvestigate wavestability in detail.








Pre

y, w

ith in

cr ti

me

c)

0 200 400 600 800 1000 1200 1400 1600 1800 2000space, x

Pre

y, w

ith in

cr ti

me

d)

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000






The Eigenvalue Spectrum

Wave stability depends on the eigenvalue spectrum.

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Re(λ)

Im(λ

)

STABLE

Eckhausinstability

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Re(λ)

Im(λ

)

UNSTABLERecently methods have been developed that enable thespectrum to be calculated using numerical continuation.(Jens Rademacher, Björn Sandstede, Arnd Scheel. Physica D 229 166-183, 2007)






Back to Wave Generation in 1-D Simulations

Example of periodic wave generation by Dirichlet boundaryconditions in the predator-prey model:

Pre

y, w

ith in

cr ti

me

c)

0 200 400 600 800 1000 1200 1400 1600 1800 2000space, x

Pre

y, w

ith in

cr ti

me

d)

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000

t=30900

t=40000







From such simulations, we can easily calculate wave speed vsparameters







From such simulations, we can easily calculate wave speed vsparameters

Our stability calculationsexplain the surprisingresults from simulations ofperiodic wave generation





The Wave Selection ProblemStep 1: Reduction to Normal FormStep 2: Exact Solution for the Wave AmplitudeStep 3: Deduce Wave Properties from Amplitude

Outline











The Wave Selection Problem

An oscillatory reaction-diffusion system has a one-parameterfamily of periodic travelling waves.








Key Question:which member of the wavefamily is selected by theboundary condition at thereservoir edge?








Key Question:which member of the wavefamily is selected by theboundary condition at thereservoir edge?

This question can beanswered analytically whenDp = Dh, close to Hopfbifurcation in the kinetics.






Step 1: Reduction to Normal Form

Consider the case of Dp = Dh close to Hopf bifurcation in thekinetics. Then standard normal form analysis reduces thepredator-prey model to

ut = uxx + λ(r)u − ω(r)v

vt = vxx + ω(r)u + λ(r)v

where λ(r) = 1 − r2

ω(r) = ω0 − ω1r2.

Here ω0 = 2C(A−1)−(A+1)

»

A(A2−1)

B

–1/2

+h

A−1A(A+1)B

i1/2

ω1 = 4A2B2+(A2−1)(A2+5)AB+(A2

−1)2

6A5/2(A2−1)1/2B3/2











ω(r) = ω0 − ω1r2.

“λ–ω system”











ω(r) = ω0 − ω1r2.

The periodic travelling wave family is

u = r∗ cos[

ω(r∗)t ±√

λ(r∗)x]

v = r∗ sin[

ω(r∗)t ±√

λ(r∗)x]






Step 2: Exact Solution for the Wave Amplitude

For the λ–ω equations,the long term solution is

r(x , t) ≡√

u2 + v2 = R(x)

independent of time






Step 2: Exact Solution for the Wave Amplitude

For the λ–ω equations,the long term solution is

r(x , t) ≡√

u2 + v2 = R(x)

independent of time, where

R(x) = rptw tanh(

x/√

2)

with rptw =

12

»

1+q

1+ 89 ω2

1

–ff

−1/2






Step 3: Deduce Wave Properties from Amplitude

The periodic travelling wave amplitude is rptw =

12

»

1+q

1+ 89 ω2

1

–ff

−1/2

.

The wave solution is

u = rptw cos[ω(rptw )t ± λ(rptw )1/2x ]

v = rptw sin[ω(rptw )t ± λ(rptw )1/2x ]

(λ(r) = 1 − r2, ω(r) = ω0 − ω1r2).

Therefore: wavelength = 2π/√

1 − r2ptw

time period = 2π/(ω0 − ω1r2ptw )

speed = (ω0 − ω1r2ptw )/

√

1 − r2ptw





ConclusionsMathematical limitationsRelevant Papers and Software

Outline











Conclusions

The expected behaviour at the edge of Kielder Waterprovides a possible explanation for the periodic travellingwaves that are observed in field vole density.

For other parameter sets, the same mechanism generatesspatiotemporal irregularity. A detailed explanation of this ispossible via numerical calculation of wave stability.

Analytical periodiction of periodic travelling wave propertiesis possible close to Hopf bifurcation in the kinetics, bysolving the wave selection problem.






Conclusions (continued)

Since λ–ω equations are the normal form of any oscillatoryreaction-diffusion system close to Hopf bifurcation, andsince boundaries with Dirichlet conditions are common inapplications, we expect both periodic travelling waves andspatiotemporal irregularity to be a general feature of suchsystems.






Mathematical limitations

Mathematically, the major limitations are:

Analytical prediction of wave stability away from Hopfbifurcation is not currently possible.

There is not currently a solution of the wave selectionproblem away from Hopf bifurcation (either analytical ornumerical).






Relevant Papers and Software

J.A. Sherratt, M.J. Smith (2008) Periodic travelling waves incyclic populations: field studies and reaction-diffusion models.J. R. Soc. Interface 5, 483-505.

This paper is a review of periodic travelling waves in ecologicalfield data and in mathematical models of cyclic populations.The associated online material contains a detailed tutorial onnumerical calculation of periodic travelling wave stability,including computer code (in Fortran).

The paper and the online material are freely available from myweb site: www.ma.hw.ac.uk/∼jas






Relevant Papers and Software

J.A. Sherratt, X. Lambin, T.N. Sherratt (2003) The effects of thesize and shape of landscape features on the formation oftravelling waves in cyclic populations. Am. Nat. 162, 503-513.

This paper concerns ecological applications of periodictravelling wave generation by obstacles. The associated onlinematerial contains a detailed tutorial on the reduction of anoscillatory reaction-diffusion system to normal form close toHopf bifurcation, including computer code (in Maple).

The paper and the online material are freely available from myweb site: www.ma.hw.ac.uk/∼jas






List of Frames


Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?

What Causes the Spatial Component of the Oscillations?




Regular and Irregular Patterns: the Goal


One-Dimensional ProblemThe Eigenvalue Spectrum




5 Conclusions and LimitationsConclusionsMathematical limitationsRelevant Papers and Software


Documents

Periodic Travelling Waves in Ecology: A Users Guidejas/talks/edinburgh_july08_voles.pdfAn Ecological Case Study Stage I: Modelling and Numerical Simulation Stage II: Predicting Regular