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

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

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

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

Hopf bifurcation Switch stability of equilibrium at α = α H But stable cycle  persistence of species time Biomass α < α H α > α H

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

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!)

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

Boundaries of chaos Example: Rozenzweig-MacArthur next-minimum map unstable equilibrium X 3 Minima X 3 cycles

Boundaries of chaos Example: Rozenzweig-MacArthur next-minimum map X3X3 No existence X 3 Possible existence X 3

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

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)

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

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

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

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

Shooting method X3X3 X2X2 X1X1 d 1 = 0.26, d 2 = 1.25·10 -2 Example error shooting: Rozenzweig-MacArthur model Default integration step

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

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

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

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

Eigenfunction BVP 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) w(0) μ WuWu

Margin of error ε Connection BVP ν T 1 = period connection  +/– ∞ Truncated (numerical)

Case 1: RM model X3X3 X2X2 X1X1 d 1 = 0.26, d 2 = 1.25·10 -2 Saddle limit cycle X3X3

Case 1: RM model X3X3 X2X2 X1X1 WuWu Unstable manifold μ u =

Case 1: RM model WsWs X3X3 X2X2 X1X1 Stable manifold μ s = 2.307·10 -3

Case 1: RM model Heteroclinic point-to-cycle connection X3X3 X2X2 X1X1 WsWs

Case 2: Monod model X3X3 X2X2 X1X1 X r = 200, D = Saddle limit cycle

Case 2: Monod model X3X3 X2X2 X1X1 WuWu μ s too small

Case 2: Monod model X3X3 X2X2 X1X1 Heteroclinic point-to-cycle connection

Case 2: Monod model X3X3 X2X2 X1X1 Homoclinic cycle-to-cycle connection

Case 2: Monod model X3X3 X2X2 X1X1 Second saddle limit cycle

Case 2: Monod model X3X3 X2X2 X1X1 WuWu

X3X3 X2X2 X1X1 Homoclinic connection

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)

Thank you for your attention! Primary references: Boer, M.P. and Kooi, B.W Homoclinic and heteroclinic orbits to a cycle in a tri-trophic food chain. J. Math. Biol. 39: Dieci, L. and Rebaza, J Point-to-periodic and periodic-to-periodic connections. BIT Numerical Mathematics 44: 41–62. Supported by

Case 1: RM model X3X3 X2X2 X1X1 Integration step  good approximation, but:  Time consuming  Not robust