Many authors consider the effect of spatial factors, such as diffusion or migration among patches, in population dynamics. We suppose that the system is composed of several patches connected by diffusion and occupied by a single species. Furthermore, 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 problem considered in this paper is whether the equilibrium point, the value of which can be changed according to the strength of diffusion, continues to be positive and globally stable, if we increase the rates of diffusion.
Allen [1] proved by applying comparison techniques that the model of such a single species diffusion system remains weakly persistent if the strength of diffusion is small enough. The homotopy function technique was successfully applied by Beretta and Takeuchi [2,3] to show that small diffusion cannot change the global stability of the model. On the other hand, Hastings [6] proved that the positive equilibrium point of the model, if it exists, is locally stable for sufficiently large diffusion. These results are valid for general multiple patch models. For the model restricted to a two patch system, Freedman, Rai and Waltman [4] showed that there exists a positive equilibrium for any diffusion rate and that it is globally stable if it is unique. These known results may suggest that diffusion cannot change the global stability of the model and the purpose of this paper is to show that the model continues to be globally stable for any diffusion rate.