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 nonconstant positive solutions. Our results show that if m1b < a <2m1b/(1−m1m2) when 0m1b when m1m2 1, 0 < d1 < (m1 ˜v − ˜u)/μ1, d2 > 0, d3 0 and d4 > 1/(m1 ˜v − ˜u), 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 nonconstant positive solutions are mainly created by the cross-diffusion d4. © 2006 Elsevier Inc. All rights reserved