Global asymptotic stability in Volterra's population systems (Q791473)

The author establishes the global asymptotic stability of the positive steady state in a population system of the form \[ dx_ i/dt=x_ i(t)(r_ i+ \sum^{n}_{j=1}a_{ij}x_ j(t)+ \sum^{n}_{j=1}b_{ij} \int^{t}_{-\infty}k_{ij} (t-s)x_ j(s)ds), \] \(t>0\), \(i=1,...,n\), under assumptions of diagonal dominance; i.e. \(a_{ii}<0,\) \[ | a_{ii}|> \sum^{n}_{j=1} | b_{ji}| \int^{\infty}_{0} | k_{ji}(s)| ds + \sum^{n}_{j=1,j\neq i} | a_{ji}|. \] Furthermore, a necessary condition for the persistence of positive solutions of the system is given.