Qualitative analysis of a ratio-dependent predator–prey system with diffusion

Abstract

Ratio-dependent predator–prey models are favoured by many animal ecologists recently as they better describe predator–prey interactions where predation involves a searching process. When densities of prey and predator are spatially homogeneous, the so-called Michaelis–Menten ratio-dependent predator–prey system, which is an ordinary differential system, has been studied by many authors. The present paper deals with the case where densities of prey and predator are spatially inhomogeneous in a bounded domain subject to the homogeneous Neumann boundary condition. Its main purpose is to study qualitative properties of solutions to this reaction-diffusion (partial differential) system. In particular, we will show that even though the unique positive constant steady state is globally asymptotically stable for the ordinary-differential-equation dynamics, non-constant positive steady states exist for the partial-differential-equation model. This demonstrates that stationary patterns arise as a result of diffusion.