Coexistence of species in a defensive switching model

We consider a simple mathematical model of two-predators and one-prey system which has the defensive switching property of predation-avoidance. We assume that the prey remains vigilant against relatively abundant predator species and guards against it by switching to another (relatively rare) predator species. We analyze how the intensity of defensive switching affects the stability of the model system. It is seen that the system generally has a stable three species coexisting equilibrium state. In the special case that the intensity of defensive switching equals one and the two predators have the same mortality rates, it is shown that the system asymptotically settles to a Volterra's oscillation in three-dimensional space. It is observed that a sufficiently small or sufficiently large value of intensity of defensive switching can make the system unstable. Finally, it is shown that the handling time may have a stabilizing effect on predator-prey systems with defensive switching.