Ecological implications of global bifurcations

1. Ecological implications of global bifurcations For the occasion of the promotion of George van Voorn 17 July 2009, Oldenburg

2. Overview • Laymen-friendly (hopefully) introduction • 2D Allee-model • 3D Rosenzweig-MacArthur model • 3D Letellier-Aziz-Alaoui model • Discussion

3. Ecology • Study of dynamics of populations of species • Interactions with other species and physical world • Obvious issues temporal and spatial scale

4. Modeling • Modeling can help in understanding • Common tool selection: • Ordinary differential equations (ODEs) + Ease in use and analysis, explicit in time − Homogeneous space

5. Example: Allee • Density-dependency affects population Variables (time-dependent): X(t) = # (= number of) Parameters: β = interspecific growth rate (no explicit nutrient modeling) K = carrying capacity (= maximum sustainable # of carrots) ζ = Allee threshold

6. Dynamics: Allee • Time dynamics of the model: # Too little carrots  extinction Enough carrots  growth to carrying capacity Too many carrots  decline to carrying capacity

7. Dynamics: Allee • Asymptotic behaviour: X = K # X = ζ X = 0 Stable equilibria: X = 0, X = K Unstable equilibria: X = ζ

8. Allee with predator x2 • We add a “predator” x1 x1 = prey population x2 = predator population l = extinction threshold, no fixed value (bifurcation parameter) k = carrying capacity, by default 1 c = conversion ratio, by default 1 m = predator mortality rate, no fixed value (bifurcation parameter) Note: dimensionless

9. Functional response • Predator-prey interaction Functional response  linear x1 = prey population x2 = predator population c = conversion ratio, by default 1

10. Analysis • Asymptotic behaviour (equilibria) • Stability (local info)  Jacobian matrix  eigenvalues • Variation of parameter (e.g, l and m) • Switch in asymptotic behaviour = bifurcation point • Numerical package AUTO

11. Equilibria • 2D Allee model has the following equilibria: E0 = (0,0), stable E1 = (l,0), unstable E2 = (k,0), with k≥l, depends E3 = (m,(m-l)(k-m)), depends

12. Analysis 2D Allee Two-parameter plot of equilibria depending on m vs l Mortality rate of rabbits Allee threshold for carrots Plot has several regions: different asymptotic behaviour

13. Analysis Equilibrium: Only prey m > 1 Equilibrium: Predator-prey Transcritical bifurcationTC2: transition to a positive equilibrium

14. Analysis Predator-prey Equilibrium Predator-prey Cycles Hopf bifurcationH3: transition from equilibrium to stable cycle

15. Periodic behaviour • Also: limit cycle, oscillations • Hopf bifurcation, also local info # #

16. Phase plot Orbits starting here go to (0,0)  Allee effect Attracting region # Bistability: Depending on initial conditions to E0 or E3/Cycle # l = 0.5, m = 0.74837

17. Problem… Time-integrated simulations  extinction of both species What do we miss? Local info not sufficient Predator-prey Cycles Extinction Prey AND predator !!

18. Extinction All orbits go to extinction! “Tunnel” # Bistability lost; Allee-threshold gone # l = 0.5, m = 0.735

19. What happens? ?

20. What happens is … Manifolds of two equilibria connect: Limit cycle “touches” E1/E2 # Heteroclinic orbit connecting saddle point to saddle point # l = 0.5, m = 0.73544235…

21. New phenomenon • Explains transition to extinction • NOT local info  global bifurcation • Heteroclinic connection between two saddle equilibria

22. Homotopy technique • Need new technique(s): global info • Take an educated guess • Formulate criteria • Convert fault to continuation parameter • Change parameter to match criteria  find connection

23. Method Δx1 = 0 ξ*w ε*v E1 E2 l = 0.5, m = 0.7 (shot in direction unstable eigenvector) l = 0.5, m = 0.7354423495 (connecting orbit)

24. Global bifurcation in Allee Using developed homotopy method • Regions: • Only prey • Predator –prey • 0. Extinct

25. Counter-intuitive Bizar: lower mortality rate kills the whole population… • Regions: • Only prey • Predator –prey • 0. Extinct Mortality rate of rabbits 2 Overharvesting or ecological suicide

26. Add another… • Rosenzweig-MacArthur 3D food chain model, no Allee-effect where (Holling type II) x = variable d = death rate note: dimensionless

27. Equilibria • There are 4 equilibria:

28. Chaos • New type of behaviour possible # A-periodic, but still “stable”

29. Bifurcation diagram Extreme values for top predator are plotted as function of one parameter # d1=0.25 d2

30. Bifurcation diagram Chaotic Extinct PeriodicStable coexistence # d1=0.25 d2

31. Global bifurcations Region of extinction marked by global bifurcation # d1=0.25 d2 Saddle limit cycle

32. New technique • This is a homoclinic cycle-to-cycle connection • No technique thusfar for detection and continuation • Formulation of new criteria • Adaptation of homotopy method

33. Global bifurcation • Using new technique: # d1=0.25 d2 = 0.0125 # # Connecting orbit from saddle limit cycle to itself

34. Bifurcation diagram Two parameters 0: no top predator SE: stable existence P: periodic solutions C: Chaos “Eye”: extinction 0 SE d2 P P C d1 Family of tangencies of connecting orbit  boundary of chaotic behaviour (boundary crisis)

35. Different model • Letellier & Aziz-Alaoui (2002)

36. Different model • Letellier & Aziz-Alaoui (2002) Identical to Rosenzweig-MacArthur Biological interpretation: - No dependence prey density - Different dependence predator density

37. One-parameter diagram c0 = 0.038 # a1 As compared to RM: two chaotic attractors Two different global bifurcations

38. One-parameter diagram c0 = 0.038 # a1 First globif bifurcation  boundary crisis No stable equilibrium, shift, but… survival

39. One-parameter diagram c0 = 0.038 # a1 Second global bifurcation  interior crisis Change of chaotic attractor

40. One-parameter diagram Chaos c0 = 0.038 # Low period limit cycle a1 Disappearance of one chaotic attractor Hysteresis (Scheffer) & simplification of system

41. Discussion • Connection types and ramifications • Allee: heteroclinic point-to-point  overharvesting • RM: homoclinic saddle cycle  chaos disappears, extinction top predator • L&AA: two homoclinic saddle cycle  hysteresis, persistence of top predator

42. Discussion • Global bifurcations mark different transitions than local • Required development new method • Implemented in AUTO • Essential for analysis • No obvious coupling connection type with biological consequences

43. Acknowledgements • Bas Kooijman, Bob Kooi • Dirk Stiefs, Ulrike Feudel, Thilo Gross • Yuri Kuznetsov, Eusebius Doedel • Martin Boer, Lia Hemerik • Funding: NWO

44. George.vanVoorn@wur.nl • www.biometris.wur.nl/ • www.bio.vu.nl/thb/research/project/globif/ Thank you for your attention!

45. Extra slides

46. Equilibria • The relevant equilibria now are E0 = (0,0,0) E1 = (1,0,0) E3 = (X1*,X2*,0)  No stable equilibrium all 3 species Default parameter values:

47. Proof: maps # # T+1 T+2 # # T T At the point where chaos disappears we plot the number of bears at time T+n as function of number at time T

48. Proof: maps # # T+1 T+2 # # T T First globif (upper chaotic attractor) is homoclinic period 1 Second (lower chaotic attractor) homoclinic period 2