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
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
In collaboration with:
Matthew Smith Xavier Lambin
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
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
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
Field Voles in Kielder Forest
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
Field Voles in Kielder Forest
Field voles in Kielder Forest are cyclic (period 4 years).
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
Field Voles in Kielder Forest
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?What Causes the Spatial Component of the Oscillations?
What Causes the Spatial Component of theOscillations?
Hypothesis: the periodic travelling waves are caused by the largecentral reservoir.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestA Standard Predator-Prey ModelBoundary Conditions in the Field Vole ExampleTypical Model Solution
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
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestA Standard Predator-Prey ModelBoundary Conditions in the Field Vole ExampleTypical Model Solution
Field Voles in Kielder Forest
For modelling, we need an assumption on the cause of thepopulation cycles.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestA Standard Predator-Prey ModelBoundary Conditions in the Field Vole ExampleTypical Model Solution
Field Voles in Kielder Forest
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestA Standard Predator-Prey ModelBoundary Conditions in the Field Vole ExampleTypical Model Solution
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:
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestA Standard Predator-Prey ModelBoundary Conditions in the Field Vole ExampleTypical Model Solution
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
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestA Standard Predator-Prey ModelBoundary Conditions in the Field Vole ExampleTypical Model Solution
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
Therefore we expect very high vole loss at the reservoiredge, implying a Robin boundary condition
∂
∂n
(
voledensity
)
= −(
largeconstant
)
·(
voledensity
)
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestA Standard Predator-Prey ModelBoundary Conditions in the Field Vole ExampleTypical Model Solution
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
Therefore we expect very high vole loss at the reservoiredge, implying a Robin boundary condition
∂
∂n
(
voledensity
)
= −(
largeconstant
)
·(
voledensity
)
To a good approx, vole density = 0 at the reservoir edge
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestA Standard Predator-Prey ModelBoundary Conditions in the Field Vole ExampleTypical Model Solution
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
Therefore we expect very high vole loss at the reservoiredge, implying a Robin boundary condition
∂
∂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
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestA Standard Predator-Prey ModelBoundary Conditions in the Field Vole ExampleTypical Model Solution
Typical Model Solution
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestA Standard Predator-Prey ModelBoundary Conditions in the Field Vole ExampleTypical Model Solution
Typical Model Solution
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestA Standard Predator-Prey ModelBoundary Conditions in the Field Vole ExampleTypical Model Solution
Typical Model Solution
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestA Standard Predator-Prey ModelBoundary Conditions in the Field Vole ExampleTypical Model Solution
Typical Model Solution
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestA Standard Predator-Prey ModelBoundary Conditions in the Field Vole ExampleTypical Model Solution
Typical Model Solution
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestA Standard Predator-Prey ModelBoundary Conditions in the Field Vole ExampleTypical Model Solution
Typical Model Solution
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestA Standard Predator-Prey ModelBoundary Conditions in the Field Vole ExampleTypical Model Solution
An Example of Irregular Pattern Generation
For some parameter values, obstacles with Dirichlet boundaryconditions generate irregular spatiotemporal patterns.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Field Voles in Kielder ForestA Standard Predator-Prey ModelBoundary Conditions in the Field Vole ExampleTypical Model Solution
Movie of Irregular Pattern Generation
Click here toplay the movie
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Regular and Irregular Patterns: the GoalStandard Theory of Periodic Travelling WaveOne-Dimensional ProblemThe Eigenvalue SpectrumBack to Wave Generation in 1-D Simulations
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
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Regular and Irregular Patterns: the GoalStandard Theory of Periodic Travelling WaveOne-Dimensional ProblemThe Eigenvalue SpectrumBack to Wave Generation in 1-D Simulations
Regular and Irregular Patterns: the Goal
Goal: to predict which parameter sets will give periodictravelling waves, and which will give spatiotemporal irregularity.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Regular and Irregular Patterns: the GoalStandard Theory of Periodic Travelling WaveOne-Dimensional ProblemThe Eigenvalue SpectrumBack to Wave Generation in 1-D Simulations
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
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Regular and Irregular Patterns: the GoalStandard Theory of Periodic Travelling WaveOne-Dimensional ProblemThe Eigenvalue SpectrumBack to Wave Generation in 1-D Simulations
Standard Theory of Periodic Travelling Wave
Mathematically, a periodic travelling wave is a soln of formU(x ± ct), with U(.) periodic.
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Regular and Irregular Patterns: the GoalStandard Theory of Periodic Travelling WaveOne-Dimensional ProblemThe Eigenvalue SpectrumBack to Wave Generation in 1-D Simulations
Standard Theory of Periodic Travelling Wave
Mathematically, a periodic travelling wave is a soln of formU(x ± ct), with U(.) periodic.
Some members of the periodictravelling wave family are stable assolutions of the partial differentialequations, while others areunstable.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Regular and Irregular Patterns: the GoalStandard Theory of Periodic Travelling WaveOne-Dimensional ProblemThe Eigenvalue SpectrumBack to Wave Generation in 1-D Simulations
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 < ∞.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Regular and Irregular Patterns: the GoalStandard Theory of Periodic Travelling WaveOne-Dimensional ProblemThe Eigenvalue SpectrumBack to Wave Generation in 1-D Simulations
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
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Regular and Irregular Patterns: the GoalStandard Theory of Periodic Travelling WaveOne-Dimensional ProblemThe Eigenvalue SpectrumBack to Wave Generation in 1-D Simulations
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=40000Conclusion: irregular patternsoccur when the (Dirichlet)boundary condition at x = 0generates a periodic travellingwave that is unstable.
Therefore we mustinvestigate wavestability in detail.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Regular and Irregular Patterns: the GoalStandard Theory of Periodic Travelling WaveOne-Dimensional ProblemThe Eigenvalue SpectrumBack to Wave Generation in 1-D Simulations
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
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Regular and Irregular Patterns: the GoalStandard Theory of Periodic Travelling WaveOne-Dimensional ProblemThe Eigenvalue SpectrumBack to Wave Generation in 1-D Simulations
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)
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Regular and Irregular Patterns: the GoalStandard Theory of Periodic Travelling WaveOne-Dimensional ProblemThe Eigenvalue SpectrumBack to Wave Generation in 1-D Simulations
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
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Regular and Irregular Patterns: the GoalStandard Theory of Periodic Travelling WaveOne-Dimensional ProblemThe Eigenvalue SpectrumBack to Wave Generation in 1-D Simulations
Back to Wave Generation in 1-D Simulations
From such simulations, we can easily calculate wave speed vsparameters
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
Regular and Irregular Patterns: the GoalStandard Theory of Periodic Travelling WaveOne-Dimensional ProblemThe Eigenvalue SpectrumBack to Wave Generation in 1-D Simulations
Back to Wave Generation in 1-D Simulations
From such simulations, we can easily calculate wave speed vsparameters
Our stability calculationsexplain the surprisingresults from simulations ofperiodic wave generation
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
The Wave Selection ProblemStep 1: Reduction to Normal FormStep 2: Exact Solution for the Wave AmplitudeStep 3: Deduce Wave Properties from Amplitude
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
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
The Wave Selection ProblemStep 1: Reduction to Normal FormStep 2: Exact Solution for the Wave AmplitudeStep 3: Deduce Wave Properties from Amplitude
The Wave Selection Problem
An oscillatory reaction-diffusion system has a one-parameterfamily of periodic travelling waves.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
The Wave Selection ProblemStep 1: Reduction to Normal FormStep 2: Exact Solution for the Wave AmplitudeStep 3: Deduce Wave Properties from Amplitude
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?
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
The Wave Selection ProblemStep 1: Reduction to Normal FormStep 2: Exact Solution for the Wave AmplitudeStep 3: Deduce Wave Properties from Amplitude
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?
This question can beanswered analytically whenDp = Dh, close to Hopfbifurcation in the kinetics.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
The Wave Selection ProblemStep 1: Reduction to Normal FormStep 2: Exact Solution for the Wave AmplitudeStep 3: Deduce Wave Properties from Amplitude
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
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
The Wave Selection ProblemStep 1: Reduction to Normal FormStep 2: Exact Solution for the Wave AmplitudeStep 3: Deduce Wave Properties from Amplitude
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.
“λ–ω system”
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
The Wave Selection ProblemStep 1: Reduction to Normal FormStep 2: Exact Solution for the Wave AmplitudeStep 3: Deduce Wave Properties from Amplitude
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.
The periodic travelling wave family is
u = r∗ cos[
ω(r∗)t ±√
λ(r∗)x]
v = r∗ sin[
ω(r∗)t ±√
λ(r∗)x]
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
The Wave Selection ProblemStep 1: Reduction to Normal FormStep 2: Exact Solution for the Wave AmplitudeStep 3: Deduce Wave Properties from Amplitude
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
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
The Wave Selection ProblemStep 1: Reduction to Normal FormStep 2: Exact Solution for the Wave AmplitudeStep 3: Deduce Wave Properties from Amplitude
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
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
The Wave Selection ProblemStep 1: Reduction to Normal FormStep 2: Exact Solution for the Wave AmplitudeStep 3: Deduce Wave Properties from Amplitude
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
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
ConclusionsMathematical limitationsRelevant Papers and Software
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
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
ConclusionsMathematical limitationsRelevant Papers and Software
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
ConclusionsMathematical limitationsRelevant Papers and Software
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.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
ConclusionsMathematical limitationsRelevant Papers and Software
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).
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
ConclusionsMathematical limitationsRelevant Papers and Software
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
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
ConclusionsMathematical limitationsRelevant Papers and Software
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
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide
An Ecological Case StudyStage I: Modelling and Numerical Simulation
Stage II: Predicting Regular vs Irregular PatternsStage III: Predicting Wave Properties
Conclusions and Limitations
ConclusionsMathematical limitationsRelevant Papers and Software
List of Frames
1 An Ecological Case Study
Field Voles in Kielder ForestWhat is a Periodic Travelling Wave?
What Causes the Spatial Component of the Oscillations?
2 Stage I: Modelling and Numerical Simulation
Field Voles in Kielder ForestA Standard Predator-Prey ModelBoundary Conditions in the Field Vole ExampleTypical Model Solution
3 Stage II: Predicting Regular vs Irregular Patterns
Regular and Irregular Patterns: the Goal
Standard Theory of Periodic Travelling Wave
One-Dimensional ProblemThe Eigenvalue Spectrum
Back to Wave Generation in 1-D Simulations
4 Stage III: Predicting Wave Properties
The Wave Selection ProblemStep 1: Reduction to Normal FormStep 2: Exact Solution for the Wave AmplitudeStep 3: Deduce Wave Properties from Amplitude
5 Conclusions and LimitationsConclusionsMathematical limitationsRelevant Papers and Software
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Periodic travelling waves in ecology: a users guide