1. 1 Predator-Prey Oscillations in Space (again) Sandi Merchant D-dudes meeting November 21, 2005

2. 2 Review of the Model – ODE I ■ Starts with a standard ODE predator-prey model ● Logistic prey growth ● Constant predator death rate B ● Type II predator functional response ■ Behaviour of this model is fairly well- understood

3. 3 Review of the Model – ODE II ■ Equilibria and Stability: Extinction of both species Always unstable (saddle) Extinction of predator Prey at carrying capacity Stable node or unstable saddle Coexistence of predator and prey Can be any type of steady state

4. 4 Review of the Model – ODE III ■ Division of Parameter Space -- C=0.2 ■ Sample Parameter Path: fix A=0.5, vary B (predator death rate) Red = prey only eq. stable (coexistence unstable) Blue = stable coexistence (no limit cycle) White = stable oscillations (coexistence unstable) A B

5. 5 Review of the Model – ODE IV ■ B = 0.6 --- predator goes extinct (prey to carrying capacity)

6. 6 Review of the Model – ODE IV ■ B = 0.4 --- convergence to coexistence equilibrium

7. 7 Review of the Model – ODE IV ■ B = 0.345 --- damped oscillations to coexistence equilibrium

8. 8 Review of the Model – ODE IV ■ B = 0.33 --- small amplitude & high frequency oscillations

9. 9 Review of the Model – ODE IV ■ B = 0.1 --- large amplitude oscillations

10. 10 Review of the Model – ODE IV ■ B = 0.025 --- large amplitude & long period oscillations

11. 11 Review of the Model – PDE I ■ What happens if there is a biological invasion in this type of system? ● Invasion speed ● Spatial and temporal pattern after invasion ■ Sherratt et al. (1997) studied this. ● Added diffusion to make PDE model ● Numerically simulated invasion of predator into prey at carrying capacity

12. 12 PDE Model Behaviour ■ Sherratt et al. found surprising spatiotemporal patterns developed when simulating invasion in this way. ● Travelling wavetrains/plane waves ● Spatiotemporal “Chaos” ■ Implications of applying such a model to real systems ● Predators can cause prey populations to oscillate or even behave chaotically after invasion! ● ODE and PDE model predictions do not agree. ■ Sherratt et al. simply showed that these behaviours existed... no (compelling) explanation of why or how these patterns emerge.

13. 13 PDE Model Behaviour ■ I set out to understand this model better ■ Simulated the same system as Sherratt et al., for a variety of parameter values. ■ Example: Same parameter path as ODE graphs ■ Initial Condition for all simulations ● Prey density =1 everywhere ● Predator density = 0 everywhere, except ● Predator density = 0.5 for x in the interval [0,5]

14. 14 PDE Model Behaviour ■ B=0.35 ■ Ordinary Travelling Wavefront ■ Wavefront Speed ~ 0.494 Prey Density 0 x 1000 1.5 1

15. 15 PDE Model Behaviour ■ B=0.32 ■ Small Amplitude Wavetrains behind front ■ Wavefront Speed ~ 0.605 Prey Density 0 x 1000 1.5 1

16. 16 PDE Model Behaviour ■ B=0.25 ■ Large Amplitude Wavetrain behind damped oscillatory front ■ Wavefront Speed ~ 0.791 Prey Density 0 x 1000 1.5 1

17. 17 PDE Model Behaviour ■ B=0.1 ■ Behaviour getting a little less predictable ■ Wavefront speed ~ 1.088 Prey Density 0 x 1000 1.5 1

18. 18 PDE Model Behaviour ■ Can get quite chaotic-looking 0 x 1000 1.5 1

19. 19 Patterns Observed ■ Invasion Speed (speed of front) increases as B decreases

20. 20 Patterns Observed ■ More complex spatiotemporal pattern as B decreases ● Nothing ---> wavefront ---> wavetrain ---> chaos ● Counterintuitive? ● Invasion speed increases with increasing complexity ■ Wavetrain-type solutions – sub-patterns ● Fixed “damped oscillation” wavefront moving at constant speed forwards ● Wavetrains moving at different speed, usually in reverse direction. ● Amplitude and frequency of wavetrains increases as B is decreased

21. 21 Patterns Observed ■ Relationship between wavetrain and wavefront ● There appears to be some interaction between the “damping” behind the wavefront and the wavetrains ● Wavetrains originate at the “tail” of the wavefront ● More oscillatory wavefronts seem to produce larger amplitude wavetrains – why? ● Is there a relationship between the speed of the wavetrain and the speed of the wavefront? ● Have not measured wavetrain speed

22. 22 Two Big Questions (1) Are these patterns real? ● Always possible that simply result of numerical scheme/method of simulation ● Might be only transitional behaviour ● Could be result of boundary conditions (no-flux) (2) Where do these patterns come from? ● Can we show that certain solutions bifurcate from other solutions? ● Is the behaviour of the model predictable? (ie. relatively insensitive to parameters) ● Often in chaotic systems this is not the case ● Is there a predictable set of transitions before chaos? ● Otherwise, maybe useless for applications

23. 23 Plan to Answer (1) ■ Are these patterns real? ● Analytically finding these solutions is likely impossible ● Getting the same solutions using alternative numerical schemes would help verify their existence ● I plan to show that the wavetrains alone are solutions (in a non-invasion scenario) ● Make a new simulation with periodic boundary conditions and a domain length of one spatial period –see if same solutions arise. ● Use two different packages for the numerical simulation ● Matlab – as for invasion simulations ● AUTO – a numerical continuation package

24. 24 Plan to Answer (2) ■ Where do these patterns come from? ■ My Hypothesis: as B is decreased ● Prey only solution (no invasion) loses stability to coexistence eq. solution (ordinary wavefront) ● Coexistence solution loses stability to wavetrain solution (Hopf bifurcation) ● Wavetrain solution loses stability – eventually resulting in spatiotemporal chaos ■ How to test hypothesis ● If the solutions are real (question (1)), then examine their stability ● Compute the spectrum of the various “steady states” ● If the spectrum of a particular solution crosses the imaginary axis (real part zero), then it loses stability ● If this happens precisely where the new solution types appear, then it is evidence ● Pattern of the crossing may also lend support

25. 25 How to Compute the Spectrum? ■ Unlike ODE models, spectrum can be a continuous curve of values ■ Substantially more difficult to compute than eigenvalues for ODEs ■ Most common methods involve discretizing with finite differences and computing the eigenvalues of a HUGE matrix ■ I will use a new method developed by Rademacher et al (2005) ● Convert eigenvalue problem to a BVP ● Use AUTO to path-follow solution to the BVP ● Very efficient :) ● Involves programming in FORTRAN :(

26. 26 Spectrum of Wavetrain Solution ■ Wavetrain solution found using XPP/AUT ■ Unstable because crosses imaginary axis ■ I have yet to find a stable wavetrain solution Real part Imag. part

27. 27 Challenges ■ All the outlined methods for computing stability only determines the linear stability ● It is possible that true (nonlinear) stability does not necessarily follow ● Need to learn and apply PDE dynamical systems theory ■ What if my hypotheses are wrong? ● All of the above work may not end up telling us anything about how the spatial patterns emerge ● The application of the new method of computing essential spectra is still a novel and useful exercise

28. 28 Challenges ■ Need to be able to compute the speed and period of wavetrains accurately in order for AUTO work to be possible