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Cooperative systems theory and global stability of diffusion models. (English) Zbl 0665.92017

In this paper, it is proved that the single species linear diffusion model under patchy environment has always a positive and globally stable equilibrium point for any diffusion rate. Since the species is supposed to be able to survive in all the patches at a positive globally stable equilibrium point if the patches are isolated, or if the diffusion among patches is neglected and the species is confined to each patch, the results obtained in this paper show that no diffusion rate can change the global stability of the model. In order to show the global stability, the important properties of the model are the relationships the authors derived from their assumptions and the monotonicity of flows in cooperative systems.

When it is assumed that two patches are connected by diffusion and that each patch has only symbiotic interactions, Kamke’s theorem still holds but a property similar to the relationships derived is only true if the diffusion between the two patches is weak enough. Therefore, the global stability of such a model for weak diffusion can be proved.

Reviewer: Y.Sugai

MSC:
92D40Ecology
34D20Stability of ODE
92D25Population dynamics (general)
93D99Stability of control systems
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