Dynamics of a generalized Gause-type predator-prey model with a seasonal functional response

We extend a previous Gause-type predator-prey model to include a general monotonic and bounded seasonally varying functional response. The model exhibits rich dynamical behaviour not encountered when the functional response is not seasonally forced. A theoretical analysis is performed on the model to investigate the global stability of the boundary equilibria and the existence of periodic solutions. It is shown that, under certain well-defined conditions, the Poincaré map of the model undergoes a Hopf bifurcation leading to the appearance of a quasi-periodic solution. Numerical results are given for the Poincaré sections and bifurcation diagrams for Holling-types II and III functional responses, using the amplitude of seasonal variation as bifurcation parameter. The model shows a rich variety of behaviour, including period doubling, quasi-periodicity, chaos, transient chaos, and windows of periodicity.