Bifurcation Analysis of a Predator-Prey System Involving Group Defence

A class of ODEs of generalized Gause type modeling predator-prey interaction is considered. The prey are assumed to exhibit a phenomenon called group defence, that is, predation is decreased or even eliminated due to the ability of the prey to defend or disguise themselves as their numbers increase. Using the carrying capacity of the environment as the bifurcation parameter, it is shown that the model undergoes a sequence of bifurcations that includes a homoclinic bifurcation as well as a Hopf bifurcation. Conditions (that hold even in the case of no group defence) that ensure a subcritical Hopf bifurcation and also the spontaneous appearance of a semistable periodic orbit that splits into a pair (one stable and one unstable) of periodic orbits are given. Ecological ramifications are considered. Unlike the classical model, sufficient enrichment of the environment combined with group defence leads to extinction of the predator (deterministically) for almost all initial conditions, providing strong support for the so-called paradox of enrichment.