Permanence and global stability in a Lotka-Volterra predator-prey system with delays
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.
scientific article; zbMATH DE number 2075535
scientific article