Bifurcations of higher codimension in a prey-predator model with generalist predator

Abstract

Consideration of generalist predators leads to relatively complex dynamics due to alternative food sources. Here, we propose and analyze a prey-predator model with a generalist predator. The availability of alternative food sources for the predator and a density-dependent growth rate induces not only bistability and tristability, but also more complicated dynamical behaviors. We have studied the possible number and geometric configurations of positive equilibria in detail. A systematic bifurcation analysis has revealed the existence of the degenerate Bogdanov-Takens bifurcation of codimension four and degenerate Hopf bifurcation of codimension three. We found that degenerate local bifurcations with a higher codimension are responsible for three limit cycles. Derivation of the analytical conditions for three limit cycles for a suitable range of parameters is a crucial finding of this work.