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Non-constant positive steady states of a prey-predator system with cross-diffusions. (English) Zbl 1168.35350
Summary: 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 $m_1b<a<2m_1b/ (1-m_1m_2)$ when $0<m_1m_2<1$ or $a>m_1b$ when $m_1m_2\ge1$, $0<d_1<(m_1\tilde v-\tilde u)/\mu_1$, $d_2>0$, $d_3\ge 0$ and $d_4>1/(m_1\tilde v-\tilde u)$, then there exists $(d_1,d_2,d_3,d_4)$ 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 $d_4$.

##### MSC:
 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35Q80 Applications of PDE in areas other than physics (MSC2000) 92D25 Population dynamics (general) 35J50 Systems of elliptic equations, variational methods
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