Advances in Differential Equations

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

Kousuke Kuto

Abstract

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.

Article information

Source
Adv. Differential Equations, Volume 12, Number 2 (2007), 145-172.

Dates
First available in Project Euclid: 18 December 2012

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