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Zbl 0784.93092
Lu, Zhengyi; Takeuchi, Yasuhiro
Permanence and global stability for cooperative Lotka-Volterra diffusion systems.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 19, No.10, 963-975 (1992). ISSN 0362-546X

The boundedness and global stability of the Lotka-Volterra system $$\dot{x}\sb i=x\sb i(b\sb i+\sum\sb{j=1}\sp n a\sb{ij}x\sb j)+D\sb i(y\sb i-x\sb i) \quad \dot{y}\sb i=y\sb i(\bar b\sb i+\sum\sb{j=1}\sp n \bar a\sb{ij}y\sb j)+\overline D\sb i(x\sb i-y\sb i)$$ is considered. In particular, it is shown that this system has bounded solutions if $A=(a\sb{ij})$ and $\overline A=(\bar a\sb{ij})$ are $VL$-stable (i.e. there exist positive diagonal matrices $C$, $\overline C$ such that $CA+A'C$, $\overline C\overline A+\overline A'\overline C$ are negative definite). Moreover, if $a\sb{ij}>0$, $\bar a\sb{ij}>0$ for $i\ne j$ and det $A\ne 0$, det $A\ne 0$, then the condition is also necessary. Finally, it is shown that if $(x\sp*,y\sp*)$ is a positive equilibrium point and $$b\sb ix\sb i\sp*+D\sb i(y\sb i\sp*-x\sb i\sp*)\geq 0, \quad \bar b\sb iy\sb i\sp*+\overline D\sb i(x\sb i\sp*-y\sb i\sp*)\geq 0,$$ then $(x\sp*,y\sp*)$ is globally stable.
[S.P.Banks (Sheffield)]
MSC 2000:
*93D99 Stability of control systems
92D50 Animal behavior

Keywords: global stability; Lotka-Volterra system

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