Global stability of Lotka-Volterra systems with delay
The authors investigate the system with infinite delay \[ x_i'(t)= x_i(t)\Biggl[r_i(t)+ \sum^n_{j=1} a_{ij}(t) x_j(t-\tau_{ij}(t))+ \sum^n_{j=1} b_{ij}(t) \int^t_{-\infty} k_{ij}(t- s)x_j(s)ds\Biggr], \] where \(i= 1,\dots,n\), and \(r_i\), \(a_{ij}\), \(\tau_{ij}\geq 0\), \(k_{ij}\geq 0\) are continuous functions with \(\tau_{ii}\equiv 0\), \(t-\tau_{ij}(t)\to\infty\) as \(t\to\infty\), and \(\int^\infty_0 k_{ij}(s)ds= 1\). Under certain conditions, involving in particular \(a_{ii}(t)<0\) and \(\int^\infty_0|a_{ii}(t)|dt=\infty\), it is shown that the positive equilibrium is globally asymptotically stable. Bounded delays are also considered.
scientific article; zbMATH DE number 1014367