On the nonautonomous \(n\)-competing species problem (Q2763892)

The authors study the asymptotic behaviour of \(n\) competing populations described by the nonautonomous Lotka-Volterra equations NEWLINE\[NEWLINE\dot u_i = u_i \left[ b_i(t) - \sum_{j=1}^n a_{ij}(t) u_j \right], \qquad 1\leq i \leq n,NEWLINE\]NEWLINE where \(b_i\) and \(a_{ij}\) are continuous and positive on \(\mathbb{R}\) and satisfy certain properties related to the behaviour of isolated populations. The authors establish conditions for permanence and existence of bounded solutions (including estimates) as well as global stability in the sense that any two positive solutions attract each other for \(t\to \infty\). The global stability conditions use iterative scheme based on results due to \textit{A. Tineo} [J. Differ. Equations 116, No. 1, 1-15 (1995; Zbl 0823.34048) and Nonlinear World 3, No. 4, 695-708 (1996; Zbl 0901.34050)].