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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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