Global stability of Volterra models with time delay (Q1178468)

The \(n\)-species Volterra system with discrete and continuous time delays \[ \begin{multlined} \dot x_ i(t)=x_ i(t)\left[e_ i+\sum^ n_{j=1}a_{ij}x_ j(t)+\sum^ n_{j=1}b_{ij}x_ j(t-\tau_ j)\right. +\\ \left. +\sum^ n_{j=1}\int^ r_ 0s_{ij}k_{ij}(\theta)x_ j(t-\theta)d\theta\right],\quad i=1,\dots,n\end{multlined} \] is studied under the assumption that the coefficients are non negative constants and \(k_{ij}\) are positive, decreasing and convex functions with compact support. The authors give conditions on the coefficients for the global asymptotical stability of a positive equilibrium solution both in the case of only discrete delays and in the general case by constructing a suitable Lyapunov functional.