Positive steady states for prey-predator models with cross-diffusion. (English) Zbl 0863.35034

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\). The authors consider the stationary problem for \(u, v\geq 0\) in \(\Omega\): \[ \Delta[(1+\alpha v)u]+au(1-u-cv)=0\quad\text{in }\Omega,\quad u=0\text{ on }\partial\Omega, \]
\[ \Delta[(1+\beta u)v]+bv(1+du-v)=0\quad\text{in }\Omega,\quad v=0\text{ on }\partial\Omega, \] where \(\alpha\), \(\beta\) are nonnegative constants and \(a\), \(b\), \(c\), \(d\) are positive constants. The authors introduce functions \(U=(1+\alpha v)u\), \(V=(1+\beta u)v\) and reduce the problem to an equivalent one which can then be reduced to a suitable fixed point equation on which they use the theory of fixed point index in positive cones [see: E. N. Dancer, J. Differ. Equations 60, 236-258 (1985; Zbl 0549.35024) and L. Li, Trans. Am. Math. Soc. 305, No. 1, 143-166 (1988; Zbl 0655.35021)]. Sufficient conditions for the existence of positive solutions are determined as well as necessary and sufficient conditions when the cross diffusion effects are comparatively small. Conditions for nonexistence of positive solutions and for uniqueness \((N=1)\) are also proved.


35J55 Systems of elliptic equations, boundary value problems (MSC2000)
47H10 Fixed-point theorems
92D25 Population dynamics (general)
35J65 Nonlinear boundary value problems for linear elliptic equations