A dynamical system with Hopf bifurcations and catastrophes

摘要

A well-known second-order positive dynamical system describing the dynamics of a renewable resource together with its consumers (prey-predator model) is first considered. Such a dynamical system is recognized to be a subtle Hopf bifurcation without catastrophes. Then, the model is slightly modified by including a control parameter which represents the indirect exploitation of the resource. This new system is shown to have a much richer and interesting dynamics: in particular, catastrophes of the fold type (hysteresis) may arise. Moreover, when the control value corresponding to the Hopf bifurcation is greater than the lower threshold of the hysteresis, one of the two transitions characterizing the catastrophe is a transition between a stableequilibrium and a limit cycle. The results are illustrated in the control phase portrait.