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Permanence and global stability for cooperative Lotka-Volterra diffusion systems. (English) Zbl 0784.93092

The boundedness and global stability of the Lotka-Volterra system \[ \dot{x}_ i=x_ i(b_ i+\sum_{j=1}^ n a_{ij}x_ j)+D_ i(y_ i-x_ i) \quad \dot{y}_ i=y_ i(\bar b_ i+\sum_{j=1}^ n \bar a_{ij}y_ j)+\overline D_ i(x_ i-y_ i) \] is considered. In particular, it is shown that this system has bounded solutions if \(A=(a_{ij})\) and \(\overline A=(\bar a_{ij})\) 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 \(a_{ij}>0\), \(\bar a_{ij}>0\) for \(i\neq j\) and det \(A\neq 0\), det \(A\neq 0\), then the condition is also necessary. Finally, it is shown that if \((x^*,y^*)\) is a positive equilibrium point and \[ b_ ix_ i^*+D_ i(y_ i^*-x_ i^*)\geq 0, \quad \bar b_ iy_ i^*+\overline D_ i(x_ i^*-y_ i^*)\geq 0, \] then \((x^*,y^*)\) is globally stable.

MSC:

93D99 Stability of control systems
92D50 Animal behavior
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