Presentation on theme: "Chaos in Easter Island Ecology J. C. Sprott Department of Physics University of Wisconsin – Madison Presented at the Chaos and Complex Systems Seminar."— Presentation transcript:
Chaos in Easter Island Ecology J. C. Sprott Department of Physics University of Wisconsin – Madison Presented at the Chaos and Complex Systems Seminar in Madison, WI on January 25, 2011
η = 0.8 γ = 0.6 Basener-Ross Model Requires γ = 2η − 1 Structurally unstable
Poincaré-Bendixson Theorem In a 2-dimensional dynamical system (i.e. P,T), there are only 4 possible dynamics: 1. Attract to an equilibrium 2. Cycle periodically 3. Attract to a periodic cycle 4. Increase without bound Trajectories in state space cannot intersect
Invasive Species Model Four equilibria: 1. P = R = 0 2. R = 0 3. P = 0 4. coexistence
Conclusions Simple models can produce complex and (arguably) realistic results. A common route to extinction is a Hopf bifurcation, followed by period doubling of a limit cycle, followed by increasing chaos. Systems may evolve to a weakly chaotic state (“edge of chaos”). Careful and prompt slight adjustment of a single parameter can prevent extinction.