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Mini-course bifurcation theory

Mini-course bifurcation theory. Part one: introduction, 1D systems. George van Voorn. Introduction. One-dimensional systems Notation & Equilibria Bifurcations Two-dimensional systems Equilibria Eigenfunctions Isoclines & manifolds. Introduction. Two-dimensional systems

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Mini-course bifurcation theory

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  1. Mini-course bifurcation theory Part one: introduction, 1D systems George van Voorn

  2. Introduction • One-dimensional systems • Notation & Equilibria • Bifurcations • Two-dimensional systems • Equilibria • Eigenfunctions • Isoclines & manifolds

  3. Introduction • Two-dimensional systems • Bifurcations of equilibria • Limit cycles • Bifurcations of limit cycles • Bifurcations of higher co-dimension • Global bifurcations

  4. Introduction • Multi-dimensional systems • Example: Rosenzweig-MacArthur (3D) • Equilibria/stability • Local bifurcation diagram • Chaos • Boundaries of chaos

  5. Introduction • Goal • Very limited amount of mathematics • Biological interpretation of bifurcations • Questions?!

  6. Systems & equilibria • One-dimensional ODE • Autonomous (time dependent) • Equilibria: equation equals zero

  7. Stability • Equilibrium stability • Derivative at equilibrium • Stable • Unstable

  8. Bifurcation • Consider a parameter dependent system • If change in parameter • Structurally stable: no significant change • Bifurcation: sudden change in dynamics

  9. Transcritical • Consider the ODE • Two equilibria

  10. Transcritical • Example: α = 1 • Equilibria: x = 0, x = 1 • Derivative: –2x + α • Stability • x = 0  f ’(x) > 0 (unstable) • x = α  f ’(x) < 0 (stable)

  11. Transcritical Transcritical bifurcation point α= 0

  12. Tangent • Consider the ODE • Two equilibria (α > 0)

  13. Tangent Tangent bifurcation point α= 0

  14. Application • Model by Rietkerk et al., Oikos 80, 1997 • Herbivory on vegetation in semi-arid regions P = plants g(N) = growth function b = amount of herbivory d = mortality

  15. Application Say, the model bears realism, then possible measurement points

  16. Application Would this have been a Nature article …

  17. Application But: T TC

  18. Application equilibrium bistability extinctie T TC

  19. Application Recovery from an ecological (anthropogenic) disaster: 1. Man wants more 2. Sudden extinction 3. Significant decrease in exploitation necessary 4. Recovery 1 4 2 3

  20. Application • If increase in level of herbivory (b) • Extinction of plants (P) might follow • Recovery however requires a much lower b • Bifurcation analysis as a useful tool to analyse models

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