The stability under perturbations of repulsive sets (Q1114842)

The author studies the system of differential equations (1) \(x'=f(x)\) where x belongs to \({\mathbb{R}}_+^ n\) and \(f_ i(x)=0\) if \(x_ i=0\). (1) arises as a model of interacting populations and the author's studies are motivated by considering the concept of permanence, i.e. the existence of solutions of (1) with all components positive for all positive time and for all initial values in \({\mathbb{R}}_+^ n\). He shows that permanence implies the repulsivity, in a suitale global sense, of \(S=the\) boundary of \({\mathbb{R}}_+^ n\). Then he gives sufficient conditions that S is repulsive when the right hand side of (1) is perturbed, using a converse Lyapunov function result proved earlier in the paper. Finally, he shows that if (1) is permanent, then what he calls an ``average Lyapunov function'' exists.