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Introduction to Bifurcation Theory: 1D Systems Mini-Course

Dive into the fundamentals of bifurcation theory with George van Voorn, exploring 1D and 2D systems, equilibria, limit cycles, chaos, and more. Unlock the power of bifurcation analysis for modeling real-world scenarios. Limited mathematics, biological interpretation, and practical applications included.

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Introduction to Bifurcation Theory: 1D Systems Mini-Course

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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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