On Liénard's equation and the uniqueness of limit cycles in predator-prey systems

Abstract

We study a system of ODE's modelling the interaction of one predator and one prey dx/dt = xg(x) - yp(x), dy/dt = gamma y[- delta - nu y - alpha y2 + h(x)]. This system defines a two-species community which incorporates competition among prey in the absence of any predators as well as a density-dependent predator specific death rate. This system is investigated under ecologically natural regularity conditions and assumptions on g, p and h to ensure the existence and uniqueness of limit cycles. The proof uses the standard Hopf-Andronov bifurcation theory and the technique of Liénard's equation.