A strongly coupled diffusion effect on the stationary solution set of a prey-predator model

We study the positive solution set of the following quasilinear elliptic system: $$ \begin{cases} \Delta u+u(a-u-cv)=0 \ \ & \mbox{in} \ \ \Omega,\\ \Delta \Big [ \Big ( \mu+\dfrac{1}{1+\beta u} \Big ) v \Big ]+v(b+du-v)=0 \ \ & \mbox{in} \ \ \Omega,\\ u=v=0 \ \ & \mbox{on} \ \ \partial\Omega, \end{cases} $$ where $\Omega $ is a bounded domain in $\boldsymbol{R}^{N}$, $a, b, c, d$, and $\mu $ are positive constants, and $\beta $ is a nonnegative constant. This system is the stationary problem associated with a prey-predator model with the strongly coupled diffusion $\Delta (\frac{v}{1+\beta u})$, and $u$ (respectively $v$) denotes the population density of the prey (respectively the predator). In the previous paper by Kadota and Kuto \cite{KK}, we obtained the bifurcation branch of the positive solutions, which extends globally with respect to the bifurcation parameter $a$. In the present paper, we aim to derive the nonlinear effect of large $\beta $ on the positive solution continuum. We obtain two {\it shadow systems} in the limiting case as $\beta\to\infty $. From the analysis for the shadow systems, we prove that in the large $\beta $ case, the positive solutions satisfy $\| u\|_{\infty }=O(1/\beta)$ if $a$ is less than a threshold number, while the positive solutions can be approximated by a positive solution of the associated system without the strongly coupled diffusion if $a$ is large enough.