## Differential and Integral Equations

- Differential Integral Equations
- Volume 22, Number 7/8 (2009), 725-752.

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

Kousuke Kuto and Yoshio Yamada

#### 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 $b$.

#### Article information

**Source**

Differential Integral Equations, Volume 22, Number 7/8 (2009), 725-752.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356019545

**Mathematical Reviews number (MathSciNet)**

MR2532118

**Zentralblatt MATH identifier**

1240.35157

**Subjects**

Primary: 35J66: Nonlinear boundary value problems for nonlinear elliptic equations

Secondary: 35J60: Nonlinear elliptic equations 35K52: Initial-boundary value problems for higher-order parabolic systems 35K55: Nonlinear parabolic equations 35Q92: PDEs in connection with biology and other natural sciences 47J15: Abstract bifurcation theory [See also 34C23, 37Gxx, 58E07, 58E09] 58E07: Abstract bifurcation theory 92D25: Population dynamics (general)

#### Citation

Kuto, Kousuke; Yamada, Yoshio. Limiting characterization of stationary solutions for a prey-predator model with nonlinear diffusion of fractional type. Differential Integral Equations 22 (2009), no. 7/8, 725--752. https://projecteuclid.org/euclid.die/1356019545