Non-constant positive steady states of a prey-predator system with cross-diffusions

In this paper, we study a strongly coupled elliptic system arising from a Lotka-Volterra prey-predator system, where cross-diffusions are included in such a way that the prey runs away from the predator and the predator moves away from a large group of preys. We establish the existence and non-existence of its non-constant positive solutions. Our results show that if m₁b<a<2m₁b/(1−m₁m₂) when 0<m₁m₂<1 or a>m₁b when m₁m₂?1, , d₂>0, d₃?0 and , then there exists (d₁,d₂,d₃,d₄) such that the stationary problem admits non-constant positive solutions. Otherwise, the stationary problem has no non-constant positive solution. In particular, the results indicate that its non-constant positive solutions are mainly created by the cross-diffusion d₄.