Limiting characterization of stationary solutions for a prey-predator model with nonlinear diffusion of fractional type

Abstract

We consider the following quasilinear elliptic system: \begin{equation} \begin{cases} \Delta u+u(a-u-cv)=0 \ \ & \mbox{in} \ \ \Omega ,\\ \Delta \Big [ \Big ( 1+\dfrac{\gamma }{1+\beta u} \Big ) v \Big ]+v(b+du-v)=0 \ \ & \mbox{in} \ \ \Omega ,\\ u=v=0 \ \ & \mbox{on} \ \ \partial\Omega, \end{cases} \nonumber \end{equation} where $\Omega $ is a bounded domain in $\mathbb{R}^{N}$. This system is a stationary problem of a prey-predator model with non-linear diffusion $\Delta (\frac{v}{1+\beta u})$, and $u$ (respectively $v$) denotes the population density of the prey (respectively the predator). Kuto [15] has studied this system for large $\beta $ under the restriction $b>(1+\gamma )\lambda_{1}$, where $\lambda_{1}$ is the least eigenvalue of $-\Delta$ with homogeneous Dirichlet boundary condition. The present paper studies two {\it shadow systems} and gives the complete limiting characterization of positive solutions as $\beta\to\infty$ without any restriction on $b$.