Stochastic Hopf and Torus Bifurcations in Population Biology

摘要

In population biological systems often bifurcation sequences via period doubling are observed, especially in simplest models of epidemiology like the susceptible-infected-recovered (SIR) model with seasonal forcing in the infection rate. But other routes to complex behaviour can as easily be found, for example in multi-strain models of SIR-type without forcing, here after a Hopf bifurcation a torus bifurcation leads into chaos. However, these models are already relatively high dimensional. One of the simplest models in population biology, now in ecology, is the Rosenzweig-MacArthur model, displaying a Hopf bifurcation and with forcing also a torus bifurcation leading to more complex behaviour subsequently. Here we investigate an only slightly extended version which again can be interpreted as a stochastic process of a population dynamical model. Such stochastic models give insight into what can be observed ultimately in empirical data of the systems under investigation.