Ecological consequences of global bifurcations

Ecological consequences of global bifurcations George van Voorn Vrije Universiteit, Amsterdam Edinburgh, ECMTB 2008 With: Bob Kooi, Lia Hemerik, Yuri Kuznetsov, Martin Boer & Eusebius Doedel

Overview • Allee model (2D) • What is an Allee model? • Analysis of the model, including bifurcation analysis • Conflicting results  global bifurcation • Develop new analysis techniques • Results • Rozenzweig-MacArthur model (3D) • Model equations • Existing connecting orbits • Results • Conclusion

Allee model • Density-dependency affects prey population • Below threshold  extinction (Allee, 1931) x1 = prey population x2 = predator population l = extinction threshold, no fixed value k = carrying capacity, by default 1 c = conversion ratio, by default 1 m = predator mortality rate, no fixed value Note: dimensionless

Equilibria • System has the following equilibria: • E0 = (0,0) • E1 = (l,0) • E2 = (k,0), with k≥l • E3 = (m,(m-l)(k-m))

Analysis Two-parameter bifurcation diagram: Predator mortality m vs Allee threshold l

Analysis Two-parameter bifurcation diagram: Predator mortality m vs Allee threshold l Equilibrium: Only prey m > 1 Equilibrium: Predator-prey Transcritical bifurcationTC2: transition to a positive equilibrium

Analysis Two-parameter bifurcation diagram: Predator mortality m vs Allee threshold l Predator-prey Equilibrium Predator-prey Cycles Hopf bifurcationH3: transition from equilibrium to stable cycle

Problem… Running time-integrated simulations result in extinction of both populations Predator-prey Cycles Extinction Prey AND predator !! Kent et al. (2003): “Allee effect does not support limit cycles.”

Problem… Running time-integrated simulations result in extinction of both populations What do we miss? Predator-prey Cycles Extinction Prey AND predator !! Kent et al. (2003): “Allee effect does not support limit cycles.”

Let’s take a look at the manifolds… 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

Connecting orbit Manifolds of two equilibria connect: Limit cycle “touches” E1/E2 l = 0.5, m = 0.7354423495

Extinction All orbits go to extinction! “Tunnel” Bistability lost l = 0.5, m = 0.735

Homotopy method • How can we find connecting orbit? • Take some value of m (l fixed) • Starting point at equilibrium E2 • Step ε in direction of unstable eigenvalue v • End point: x-value equal to x-value E1 – ζ w • Difference in y-value • Difference in vectors end point and E1

Homotopy method • Differences  homotopy (“dummy”) parameter • Define boundary conditions on starting and end points orbit (a.o. Beyn, 1990) • Implement in AUTO • Stepwise continuation, including m and homotopy parameter, until parameter = 0

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

Global bifurcation in Allee Continue with two bifurcation parameters m and l Curve G≠ Extremely close to Hopf  limit cycles are immediately destroyed Van Voorn, Hemerik, Boer, Kooi, Math. Biosci. 209 (2007), 451

Global bifurcation in Allee • Regions: • Only prey • Predator –prey • 0. Extinct Van Voorn, Hemerik, Boer, Kooi, Math. Biosci. 209 (2007), 451

Global bifurcation in Allee • Regions: • Only prey • Predator –prey • 0. Extinct Decrease in predator mortality m crossing global bifurcation Van Voorn, Hemerik, Boer, Kooi, Math. Biosci. 209 (2007), 451

Conclusions Allee • Global bifurcation is the interesting bifurcation • Kent et al. “No limit cycles” explained • Decrease in m extinction both populations • Overexploitation or ecological suicide • Observe: simple 2D system • How about 3D? G.A.K. van Voorn, L. Hemerik, M.P. Boer, and B.W. Kooi, Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect, Math. Biosci. 209 (2007) 451.

RM model • Three-dimensional example: Rozenzweig-MacArthur food chain model where (Holling type II) x = variable d = death rate note: dimensionless

Connections • There exist (at least) 2 types of connections: • Homoclinic connection of a limit cycle to itself • Heteroclinic connection of an equilibrium to a limit cycle • Connections are of codimension 0, meaning they exist in a parameter range (rather than just at one specific parameter value, as with Allee)

Approach • Homotopy method, in AUTO: • Collect data involved equilibria/limit cycles, cycle manifolds and approximate connecting orbits • Define as Boundary Value Problem (BVP) • Please, find details in: E.J. Doedel, B.W. Kooi, Y.A. Kuznetsov and G.A.K. van Voorn Continuation of connecting orbits in 3D-ODEs: (I) Point-to-cycle connections, Int. J. Bif. Chaos, in press (2008a) (II) Cycle-to-cycle connections, Int. J. Bif. Chaos, in press (2008b)

Cycle-to-cycle 3D representation of a connection at a1 = 5, a2 = 0.1, b1 = 3, b2 = 2, d1 = 0.25, d2 = 0.0125

Cycle-to-cycle • One pair of connections  Primary branch Region where connection exists

Primary branch • One-parameter diagram (Boer et al.) Stable limit cycle minimum Saddle limit cycle minimum

Primary branch • Boundary of chaos: homoclinic orbit disappears Chaos Global bifurcation Stable limit cycle minimum Saddle limit cycle minimum

Point-to-cycle 2D representation of the connection at a1 = 5, a2 = 0.1, b1 = 3, b2 = 2, d1 = 0.25, d2 = 0.0125

Point-to-cycle 2D bifurcation diagram: region where connection (actually two connections) exists, bounded by Tc and Thet

Point-to-cycle • Example of bistability structure Region of attraction to attractor with predator, d1 = 0.22, d2 = 0.0125

Conclusions • In Allee (2D): bifurcation of heteroclinic connection boundary of region where bistability exists • In RM (3D): bifurcation of heteroclinic connection also boundary of region where bistability exists, but: • Very complicated structure (Boer et al., 1999) • Depending on initial conditions convergence to attractor (x3 > 0) or extinction predator (x3 = 0)

Conclusions • Homoclinic cycle connection  boundary of chaos • Global bifurcation analysis vital to understanding of ODE model dynamics • e.g. overexploitation and regions of chaos

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Ecological consequences of global bifurcations