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Necessary-sufficient conditions for permanence and extinction in Lotka-Volterra system with distributed delays. (English) Zbl 1058.34101
The author considers the following autonomous two-species Lotka-Volterra competitive system with distributed delays $$ \frac{dx_{1}(t)}{dt} =x_{1}(t)\left( b_{1}-\sum_{j=1}^{2}\sum_{k=1}^{n}\int_{-\tau _{1j}^{(k)}}^{0}a_{1j}^{(k)}x_{j}(t+\theta )d\mu _{1j}^{(k)}(\theta )\right) ,$$ $$ \frac{dx_{2}(t)}{dt} =x_{\text{2}}(t)\left( b_{2}-\sum_{j=1}^{2}\sum_{k=1}^{n}\int_{-\tau _{2j}^{(k)}}^{0}a_{2j}^{(k)}x_{j}(t+\theta )d\mu _{2j}^{(k)}(\theta )\right) .$$ Necessary and sufficient conditions for the system to be permanent are given and it is shown that $x_{1}$ and $x_{2}$ extinct if and only if the coefficients $b_{1},b_{2}$ and $A_{ij}=\Sigma _{k=1}^{n}a_{ij}^{(k)}$, $i,j=1,2,$ in the system satisfy some inequalities, respectively.

MSC:
34K25Asymptotic theory of functional-differential equations
92D25Population dynamics (general)
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References:
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