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Persistence and global stability in Lotka-Volterra delay differential systems. (English) Zbl 1061.34057
Summary: Consider the persistence and the global asymptotic stability of models governed by the following Lotka-Volterra delay differential system $\frac {dx_i(t)}{dt}=x_i(t)\left\{c_i- a_ix_i(t)-\sum^m_{j=1}a_{ij}x_j(t-\tau_{ij}) \right\}, \quad t\geq t_0,\;1\leq i\leq n,$ $x_i(t)=\varphi_i(t)\geq 0,\quad t\leq t_0,\quad\text{and}\quad \varphi_i(t_0)>0,\quad 1\leq i\leq n,$ where each $$\varphi_i(t)$$ is a continuous function for $$t\leq t_0$$, each $$c_i$$, $$a_i$$, and $$a_{ij}$$ are finite and $a_i>0,\;a_i+a_{ii}>0,\;1\leq i\leq n,\quad\text{and} \quad \tau_{ij}\geq 0,\;1\leq i,j\leq n.$ Here, we obtain conditions for the persistence of the system, and extending a technique offered by Saito, Hara and Ma for $$n=2$$ to the above system for $$n\geq 2$$. We establish new conditions for the global asymptotic stability of the positive equilibrium which improve the well-known result of Gopalsamy for some special cases.

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