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

<H3> Global Stability of Competition in a Spatially Heterogeneous Environment </H3>

Preprint (1993), appeared in:  
<A HREF="http://www.dedsj.org/"> Differential Equations and Dynamical Systems </A> 
<strong> 4 </strong> (1996) 213-223. 
<P>
<EM>Abstract.</EM>
<BR>


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.
<BR>
We use the theory of monotone flows and index theory to prove our results.
<P>

Download: <A HREF="hosotak.dvi"> dvi </A>

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