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A necessary and sufficient condition for permanence of a Lotka-Volterra discrete system with delays. (English) Zbl 0976.92031
Summary: We consider the permanence of the following Lotka-Volterra discrete competition system with delays \(k_1\), \(k_2\), \(l_1\), and \(l_2\): \[ \begin{aligned} x(n+1)&= x(n) \exp \{r_1 [1-x(n-k_1)- \mu_1 y(n-k_2)]\},\\ y(n+1)&= y(n) \exp \{r_2[1-\mu_2 x(n-l_1)- y(n-l_2)]\}. \end{aligned} \] We show the system is permanent for all nonnegative integers \(k_1\), \(k_2\), \(l_1\) and \(l_2\), if and only if \(\mu_1< 1\) and \(\mu_2< 1\).

MSC:
92D40 Ecology
39A11 Stability of difference equations (MSC2000)
39A10 Additive difference equations
39A12 Discrete version of topics in analysis
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