Permanence and positive periodic solutions for Kolmogorov competing species systems (Q1193223)

The non-autonomous (competitive case) Kolmogorov system \(x_ i'=x_ i f_ i(t,x_ 1,\cdots,x_ n)\), \(i=1,2,\dots,n\) is considered, where the vector function \((f_ i)\) is taken to be \(T\)-periodic in the \(t\)- variable. The existence of periodic solutions in a compact attractor in \((R^ +)^ n\) is established under certain conditions on \((f_ i)\). The theory is applied to generalized Lotka-Volterra systems with periodic coefficients, and also extended to differential delay equations. The theory of persistence is well-developed for autonomous systems, but less so in the non-autonomous case. This paper makes a useful contribution to the latter, as the (sufficient) conditions, though complicated by the nature of the problem, are not impossibly difficult to verify for a specific problem.