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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

x ˙ i =x i (b i + j=1 n a ij x j )+D i (y i -x i )y ˙ i =y i (b ¯ i + j=1 n a ¯ ij y j )+D ¯ i (x i -y i )

is considered. In particular, it is shown that this system has bounded solutions if A=(a ij ) and A ¯=(a ¯ ij ) are VL-stable (i.e. there exist positive diagonal matrices C, C ¯ such that CA+A ' C, C ¯A ¯+A ¯ ' C ¯ are negative definite). Moreover, if a ij >0, a ¯ ij >0 for ij and det A0, det A0, then the condition is also necessary. Finally, it is shown that if (x * ,y * ) is a positive equilibrium point and

b i x i * +D i (y i * -x i * )0,b ¯ i y i * +D ¯ i (x i * -y i * )0,

then (x * ,y * ) is globally stable.

93D99Stability of control systems
92D50Animal behavior