Multistable Phenomena Involving Equilibria and Periodic Motions in Predator-Prey Systems

In this paper, we consider a number of predator-prey systems with various types of functional responses. Detailed analysis on the dynamics and bifurcations of the systems are given. Particular attention is focused on the complex dynamics due to bifurcation of limit cycles, which may generate bistable or tristable phenomena involving equilibria and oscillating motions. It is shown that predator-prey systems can exhibit such bistable or tristable phenomena due to Hopf bifurcation, giving rise to the coexistence of stable equilibria and stable periodic solutions. Explicit conditions on the system parameters are derived which can be used to determine the number of Hopf bifurcations, the stability of bifurcating limit cycles, and the parameter regime where the bistable or tristable phenomenon occurs. The method developed in this paper can be applied to study certain interesting patterns of complex dynamical behaviors in biological or other physical systems.