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

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: Stable equilibria: X = 0, X = K Unstable equilibria: X = ζ X = K X = ζ X = 0 #

8 Allee with predator We add a “predator” x 1 = prey population x 2 = 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 x2x2 x1x1

9 Functional response Predator-prey interaction Functional response  linear x 1 = prey population x 2 = 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: E 0 = (0,0), stable E 1 = (l,0), unstable E 2 = (k,0), with k ≥ l, depends E 3 = (m,(m-l)(k-m)), depends

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

13 Analysis Equilibrium: Only prey Equilibrium: Predator-prey Transcritical bifurcation TC 2 : transition to a positive equilibrium m > 1

14 Analysis Predator-prey Equilibrium Predator-prey Cycles Hopf bifurcation H 3 : transition from equilibrium to stable cycle

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

16 Phase plot l = 0.5, m = Attracting region Orbits starting here go to (0,0)  Allee effect Bistability: Depending on initial conditions to E 0 or E 3 /Cycle # #

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

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

19 What happens? ?

20 What happens is … Manifolds of two equilibria connect: Limit cycle “touches” E 1 /E 2 Heteroclinic orbit connecting saddle point to saddle point l = 0.5, m = … # #

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 l = 0.5, m = 0.7 (shot in direction unstable eigenvector) l = 0.5, m = (connecting orbit) ε*vε*v Δx 1 = 0 E1E1 E2E2 ξ*wξ*w

24 Global bifurcation in Allee Regions: 1.Only prey 2.Predator – prey 0.Extinct Using developed homotopy method

25 Counter-intuitive Regions: 1.Only prey 2.Predator – prey 0.Extinct Overharvesting or ecological suicide 2 Mortality rate of rabbits Bizar: lower mortality rate kills the whole population…

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 d2d2 d 1 =0.25 Extreme values for top predator are plotted as function of one parameter #

30 Bifurcation diagram d2d2 d 1 =0.25 # Chaotic Extinct Periodic Stable coexistence

31 Global bifurcations d2d2 d 1 =0.25 # Region of extinction marked by global bifurcation 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: d 1 =0.25 d 2 = Connecting orbit from saddle limit cycle to itself # # #

34 Bifurcation diagram d2d2 d1d1 Family of tangencies of connecting orbit  boundary of chaotic behaviour (boundary crisis) 0: no top predator SE: stable existence P: periodic solutions C: Chaos “Eye”: extinction Two parameters SE 0 P C P

35 Different model Letellier & Aziz-Alaoui (2002)

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

37 One-parameter diagram c 0 = # a1a1 As compared to RM: two chaotic attractors Two different global bifurcations

38 One-parameter diagram c 0 = # a1a1 First globif bifurcation  boundary crisis No stable equilibrium, shift, but… survival

39 One-parameter diagram c 0 = # a1a1 Second global bifurcation  interior crisis Change of chaotic attractor

40 One-parameter diagram c 0 = # a1a1 Disappearance of one chaotic attractor Hysteresis (Scheffer) & simplification of system Chaos Low period limit cycle

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

Thank you for your attention!

45 Extra slides

46 Equilibria The relevant equilibria now are E 0 = (0,0,0) E 1 = (1,0,0) E 3 = (X 1 *,X 2 *,0)  No stable equilibrium all 3 species Default parameter values:

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