Stability analysis of positive solutions to classes of reaction-diffusion systems

Abstract

We analyze the stability of positive solutions to systems of the form \[ \begin{cases} - \Delta u_{i} = f_{i}(u_{1},u_{2},\dots,u_{m}) & \quad \mbox{ in }\ \Omega \\ u_{i} = 0 & \quad \mbox{ on }\ \partial \Omega \end{cases} \] where $ \Omega $ is a bounded region in $ {\mathbb R}^{n}\, (n \geq 1) $ with smooth boundary $ \partial \Omega $, and $ f_{i} : [0,\infty)^m \rightarrow {\mathbb R} $ are $ C^{1} $ functions for $ i=1,\dots, m $. In particular, we establish conditions for stability/instability when the system is cooperative and strictly coupled ($ \frac{\partial f_{i}}{\partial u_{j}} \geq 0, \ i \neq j,\ \sum_{j=1,j \neq i}^m(\frac{\partial f_i}{\partial u_j})^2 > 0$). When $ m=2, $ we extend this analysis for strictly coupled competitive systems ($ \frac{\partial f_{i}}{\partial u_{j}} < 0, \ i \neq j $). We apply our results to various examples, each one of different characteristics, and further analyze systems with unequal diffusion coefficients.