Decay to Uniform States in Ecological Interactions

Weakly coupled parabolic equations describing systems undergoing diffusion and interaction in a spatial domain $\Omega $ are discussed. When there exists a critical point of the interaction dynamics that is the limit as $\tau \to 1$ of a continuous decreasing family of contracting rectangles $\sum {(\tau )} ,\tau \in [ 0,1 )$, the critical point is shown to be locally asymptotically stable. The result applies when either $\Omega $ is all of $\mathbb{R}^m $ or is a bounded domain in $\mathbb{R}^m $ and is also independent of the diffusion rates. Next, the classical Lotka–Volterra competition model with diffusion and a perturbation of the Lotka–Volterra predator-prey model with diffusion and crowding effects are considered. As applications of the above result, conditions are given for both models which guarantee the existence and global asymptotic stability of a critical point with all species coexisting. For the two-species competition interaction these conditions are seen to be necessary and sufficient, w...