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Permanence and global stability in a Lotka-Volterra predator-prey system with delays. (English) Zbl 1054.34127

The author considers permanence and global asymptotic stability of models governed by the following Lotka-Volterra-type system \[ \dot x_{i}(t)=x_{i}(t)\{r_{i}-\alpha_{i}x_{i}(t)-a_{i}x_{i-1}(t- \tau_{i,i-1})-b_{i}x_{i}(t-\tau_{i,i})-c_{i}x_{i+1}(t- \tau_{i,i+1})\}, \] \(t\geq t_{0},1\leq i\leq n,\) with the initial conditions \[ x_{i}(t)=\phi_{i}(t)\geq 0,\;t\leq t_{0},\;\text{ and } \phi_{i} (t_{0})>0,\;1\leq i\leq n, \] where \(x_{0}(t)=x_{n+1}(t)\equiv 0\) and \(\phi_{i}(t),\;1\leq i\leq n\), are bounded continuous functions on \([t_{0},+\infty)\) and \(r_{i}>0\), \(\alpha_{i}>0\), \(c_{i}>0\), \(\tau_{i,j}\geq 0\), for all relevant \(i,\;j\). The author derives sufficient conditions for permanence and global asymptotic stability of solutions.

MSC:

34K25 Asymptotic theory of functional-differential equations
34K20 Stability theory of functional-differential equations
92D25 Population dynamics (general)
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References:

[1] Saito, Y.; Hara, T.; Ma, W., Necessary and sufficient conditions for permanence and global stability of a Lotka-Volterra system with two delays, J. Math. Anal. Appl., 236, 534-556 (1999) · Zbl 0944.34059
[2] Gopalsamy, K., Global asymptotic stability in Volterra’s population systems, J. Math. Biol., 19, 157-168 (1984) · Zbl 0535.92020
[3] Saito, Y., The necessary and sufficient condition for global stability of Lotka-Volterra cooperative or competition system with delays, J. Math. Anal. Appl., 268, 109-124 (2002) · Zbl 1012.34072
[4] Xu, R.; Chen, L., Persistence and global stability for a delayed nonautonomous predator-prey system without dominating instantaneous negative feedback, J. Math. Anal. Appl., 262, 50-61 (2001) · Zbl 0997.34070
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