Repellers in systems with infinite delay (Q1123840)

A very important model for the interaction of species is a set of autonomous functional differential equations \[ (*)\quad x'(t)=f(x_ t),\quad t\geq 0, \] where x: \({\mathbb{R}}\to {\mathbb{R}}^ n\) and \(x_ t\) is defined for \(t\geq 0\) by \(x_ t(s)=x(t+s)\) for \(-\infty <s\leq 0\). Denote by \({\mathbb{C}}((-\infty,0],{\mathbb{R}}^ n)\) the set of continuous functions defined on (-\(\infty,0]\) with values in \({\mathbb{R}}^ n\) and assume there is a subset Y of \({\mathbb{C}}\) such that f: \(Y\to {\mathbb{R}}^ n\). Consider the initial value problem and suppose that initial functions in Y are prescribed. The authors study the long time survival of the species being interpreted in terms of so-called permanent coexistence. The system is said to be permanently coexistent if there exist constants m, M with \(0<m<M<\infty\) such that given any initial function from Y with \(x_ i(0)>0\), \(i=1,...,n\), there exists \(t_ 0\) (dependent on the initial values) such that \(m\leq x_ i(t)\leq M\), \(t\geq t_ 0\), \(i=1,...,n\). This criterion of permanent coexistence is based on the idea that it is enough if the boundary is a repeller in a strong sense. The authors present a theory which often resolves this problem even when the detailed asymptotics of the system are inaccessible to analysis. In addition, it is shown how to construct a space of initial functions so that when solutions are uniformly ultimately bounded, then there is a compact forward invariant set absorbing all solutions.