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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 di® erential 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-di® usion (partial di® erential) system. In particular, we will show that even though the unique positive constant steady state is globally asymptotically stable for the ordinary-di® erential-equation dynamics, non-constant positive steady states exist for the partial-di® erential-equation model. This demonstrates that stationary patterns arise as a result of di® usion.

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