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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
##### Keywords:
Lotka-Volterra systems; permanence
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##### References:
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