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 m1bm1b 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.