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

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

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

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

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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

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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

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

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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

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9 Functional response Predator-prey interaction Functional response linear x 1 = prey population x 2 = predator population c = conversion ratio, by default 1

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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

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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

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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

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

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

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

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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 # #

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

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

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19 What happens? ?

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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 = … # #

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

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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

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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

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

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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…

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

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27 Equilibria There are 4 equilibria:

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

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

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

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31 Global bifurcations d2d2 d 1 =0.25 # Region of extinction marked by global bifurcation Saddle limit cycle

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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

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33 Global bifurcation Using new technique: d 1 =0.25 d 2 = Connecting orbit from saddle limit cycle to itself # # #

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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

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35 Different model Letellier & Aziz-Alaoui (2002)

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

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

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

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

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

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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

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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

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

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44 Thank you for your attention!

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45 Extra slides

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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:

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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

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

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