Josef Hofbauer, Joseph W.-H. So, Yasuhiro Takeuchi:

Global Stability of Competition in a Spatially Heterogeneous Environment

Preprint (1993), appeared in: Differential Equations and Dynamical Systems 4 (1996) 213-223.

We consider the problem whether a system composed of multiple competitive patches, which are connected by species dispersion, is globally stable or not, if each isolated patch is globally stable. It is known under some biologically reasonable assumptions that global stability continues to hold if all the patch dynamics are identical and the dispersion is species-independent. It turns out that this need not be the case when the environment is heterogeneous, that is, when the dynamics are patch dependent. When the number of the patches is two or three, it will be shown that global stability still holds under some particular types of the species-dispersion.
We use the theory of monotone flows and index theory to prove our results.

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The result was extended to four patches in Lu and Wang, Computers and Math. with Appl. 38 (1999) 19-27.