## Taiwanese Journal of Mathematics

- Taiwanese J. Math.
- Volume 7, Number 3 (2003), 391-421.

### SINGULAR LIMIT OF A CLASS OF NON-COOPERATIVE REACTION-DIFFUSION SYSTEMS

D. Hilhorst, R. Weidenfeld, and M. Mimura

#### Abstract

We consider a two component reaction-diffusion system with a small parameter $\epsilon$ $$ \left \{ \begin{array}{l} u_t = d_u\Delta u + u^{\epsilon_n}(u^mv - au^n),\\ v_t = d_v\Delta v - \displaystyle{\frac{1}{\epsilon}} u^m v, \end{array} \right. $$ where $m$ and $n$ are positive integers, together with zero-flux boundary conditions. It is known that any nonnegative solution becomes spatially homogeneous for large time. In particular, when $n\gt m \geq 1$, $(u^{\epsilon},v^{\epsilon})(t)\to(0,0)$ as $t\to \infty$, while when $m\geq n \geq 1$, there exists some positive constant $v^\epsilon_\infty$ such that $(u^{\epsilon},v^{\epsilon})(t)\to(0,v^\epsilon_\infty)$ as $t\to \infty$. In order to find the value of $v^{\epsilon}_\infty$, we derive a limiting problem when $\epsilon\to 0$ under some conditions on the values of $m$, $n$ and on the initial functions $(u_0,v_0)$, by which an approximate value of $v^\epsilon_\infty$ can be obtained.

#### Article information

**Source**

Taiwanese J. Math., Volume 7, Number 3 (2003), 391-421.

**Dates**

First available in Project Euclid: 20 July 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.twjm/1500558394

**Digital Object Identifier**

doi:10.11650/twjm/1500558394

**Mathematical Reviews number (MathSciNet)**

MR1998758

**Zentralblatt MATH identifier**

1153.35344

**Subjects**

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations 35J55 92D25: Population dynamics (general)

**Keywords**

singular limit non-cooperative systems

#### Citation

Hilhorst, D.; Weidenfeld, R.; Mimura, M. SINGULAR LIMIT OF A CLASS OF NON-COOPERATIVE REACTION-DIFFUSION SYSTEMS. Taiwanese J. Math. 7 (2003), no. 3, 391--421. doi:10.11650/twjm/1500558394. https://projecteuclid.org/euclid.twjm/1500558394