Continuation of global bifurcations using collocation technique

1. Continuation of global bifurcations using collocation technique In cooperation with: Bob Kooi, Yuri Kuznetsov (UU), Bas Kooijman George van Voorn 3th March 2006 Schoorl

2. Overview • Recent biological experimental examples of: • Local bifurcations (Hopf) • Chaotic behaviour • Role of global bifurcations (globif’s) • Techniques finding and continuation global connecting orbits • Find global bifurcations

3. Bifurcation analysis • Tool for analysis of non-linear (biological) systems: bifurcation analysis • By default: analysis of stability of equilibria (X(t), t ∞) under parameter variation • Bifurcation point = critical parameter value where switch of stability takes place • Local: linearisation around point

4. Biological application • Biologically local bifurcation analysis allows one to distinguish between: • Stable (X = 0 or X > 0) • Periodic (unstable X ) • Chaotic • Switches at bifurcation points

5. Hopf bifurcation • Switch stability of equilibrium at α = αH • But stable cycle  persistence of species Biomass time α < αH α > αH

6. Hopf in experiments Fussman, G.F. et al. 2000. Crossing the Hopf Bifurcation in a Live Predator-Prey System. Science 290: 1358 – 1360. Chemostat predator-prey system a: Extinction food shortage b: Coexistence at equilibrium c: Coexistence on stable limit cycle d: Extinction cycling Measurement point

7. Chaotic behaviour • Chaotic behaviour: no attracting equilibrium or stable periodic solution • Yet bounded orbits [X(t)min, X(t)max] • Sensitive dependence on initial conditions • Prevalence of species (not all cases!)

8. Experimental results Dilution rate d (day -1) Becks, L. et al. 2005. Experimental demonstration of chaos in a microbial food web. Nature 435: 1226 – 1229. 0.90 Chemostat predator-two-prey system 0.75 Brevundimonas 0.50 Pedobacter Tetrahymena (predator) 0.45 Chaotic behaviour

9. unstable equilibrium X3 Boundaries of chaos Example: Rozenzweig-MacArthur next-minimum map Minima X3 cycles

10. X3 Boundaries of chaos Example: Rozenzweig-MacArthur next-minimum map Possible existence X3 No existence X3

11. Boundaries of chaos • Chaotic regions bounded • Birth of chaos: e.g. period doubling • Flip bifurcation (manifold twisted) • Destruction boundaries  • Unbounded orbits  • No prevalence of species

12. Global bifurcations • Chaotic regions are “cut off” by global bifurcations (globifs) • Localisation globifs by finding orbits that: • Connect the same saddle equilibrium or cycle (homoclinic) • Connect two different saddle cycles and/or equilibria (heteroclinic)

13. Global bifurcations Example: Rozenzweig-MacArthur next-minimum map Minima homoclinic cycle-to-cycle

14. Global bifurcations Example: Rozenzweig-MacArthur next-minimum map Minima heteroclinic point-to-cycle

15. Localising connecting orbits • Difficulties: • Nearly impossible connection • Orbit must enter exactly on stable manifold • Infinite time • Numerical inaccuracy

16. Shooting method • Boer et al., Dieci & Rebaza (2004) • Numerical integration (“trial-and-error”) • Piling up of error; often fails • Very small integration step required

17. Shooting method Example error shooting: Rozenzweig-MacArthur model Default integration step X3 X1 X2 d1 = 0.26, d2 = 1.25·10-2

18. Collocation technique • Doedel et al. (software AUTO) • Partitioning orbit, solve pieces exactly • More robust, larger integration step • Division of error over pieces

19. Collocation technique • Separate boundary value problems (BVP’s) for: • Limit cycles/equilibria • Eigenfunction  linearised manifolds • Connection • Put together

20. Equilibrium BVP v = eigenvector λ = eigenvalue fx = Jacobian matrix In practice computer program (Maple, Mathematica) is used to find equilibrium f(ξ,α) Continuation parameters: Saddle equilibrium, eigenvalues, eigenvectors

21. Limit cycle BVP T = period of cycle, parameter x(0) = starting point cycle x(1) = end point cycle Ψ = phase

22. Eigenfunction BVP Wu w(0) T = same period as cycle μ = multiplier (FM) w = eigenvector Ф = phase Finds entry and exit points of stable and unstable limit cycles w(0) μ

23. Connection BVP T1 = period connection  +/– ∞ Truncated (numerical) Margin of error ε ν

24. X3 Case 1: RM model d1 = 0.26, d2 = 1.25·10-2 X3 Saddle limit cycle X2 X1

25. Case 1: RM model X3 Wu Unstable manifold μu = 1.5050 X2 X1

26. Case 1: RM model X3 Stable manifold Ws μs = 2.307·10-3 X2 X1

27. Case 1: RM model X3 Heteroclinic point-to-cycle connection Ws X2 X1

28. Case 2: Monod model Xr = 200, D = 0.085 X3 Saddle limit cycle X1 X2

29. Case 2: Monod model X3 Wu μs too small X1 X2

30. Case 2: Monod model X3 Heteroclinic point-to-cycle connection X1 X2

31. Case 2: Monod model X3 Homoclinic cycle-to-cycle connection X1 X2

32. Case 2: Monod model X3 Second saddle limit cycle X1 X2

33. Case 2: Monod model Wu X3 X1 X2

34. Case 2: Monod model X3 Homoclinic connection X1 X2

35. Future work • Difficult to find starting points • Recalculate global homoclinic and heteroclinic bifurcations in models by M. Boer et al. • Find and continue globifs in other biological models (DEB, Kooijman)

36. Supported by Thank you for your attention! george.van.voorn@falw.vu.nl Primary references: Boer, M.P. and Kooi, B.W. 1999. Homoclinic and heteroclinic orbits to a cycle in a tri-trophic food chain. J. Math. Biol. 39: 19-38. Dieci, L. and Rebaza, J. 2004. Point-to-periodic and periodic-to-periodic connections. BIT Numerical Mathematics44: 41–62.

37. Case 1: RM model • Integration step 10-3 good approximation, but: • Time consuming • Not robust X3 X1 X2