The Rosenzweig–MacArthur Graphical Criterion for a Predator-Prey Model with Variable Mortality Rate

We consider a general modified Gause type model of predation, for which the predator mortality rate can depend on the densities of both species, prey and predator. We give a graphical criterion for the stability of positive hyperbolic equilibria, which is an extension of the well-known Rosenzweig–MacArthur graphical criterion for the case of a constant predator mortality rate. We examine the occurrence of a Poincaré–Andronov–Hopf bifurcation and give an expression for the first Lyapunov coefficient. Our model generalizes several models appearing in the literature. The relevance of our results, i.e. the use of the graphical criterion and the expression for the first Lyapunov coefficient, is tested on these models. The global behavior of the system is illustrated by numerical simulations which confirm the local properties of the models near the equilibria.