Existence and non-existence of steady states to a cross-diffusion system arising in a Leslie predator-prey model

This paper is concerned with a cross-diffusion system arising in a Leslie predatorprey population model in a bounded domain with no flux boundary condition. We investigate sufficient condition for the existence and the non-existence of non-constant positive solution. We obtain that if natural diffusion coefficient of predator is large enough and cross-diffusion coefficients are fixed, then under some conditions there exists non-constant positive solution. Furthermore, we show that if natural diffusion coefficients of predator and prey are both large enough, and cross-diffusion coefficients are small enough, then there exists no non-constant positive solution. Copyright (c) 2012 John Wiley & Sons, Ltd.