Forwards attractors for non-autonomous Lotka-Volterra cooperative systems: a detailed geometrical description (Q6615300)

The authors provide a detailed analysis on the asymptotic behavior of cooperative \(n\)-dimensional nonautonomous Lotka-Volterra systems \N\[\N\dot u_i=u_i\Big(a_i(t)-b_{ii}(t)u_i-\sum_{\substack{j=1\\ j\neq i}}^nb_{ij}(t)u_j\Big),\qquad 1\leq i\leq n,\N\]\Nfrom population dynamics. Assuming that the coefficient functions are continuous, sufficient conditions for the existence of a globally (forward) attractive solution existing on the entire time axis are given. The solution found is such that one species becomes extinct, or every species except one becomes extinct. In \(n=1\), \(n=2\) and \(n=3\) dimensions the precise geometrical structure of the forward attractor is obtained by establishing heteroclinic connections between the globally attractive solution and semi-stable solutions in case of the species permanence and extinction. \N\NThese results are particularly interesting, because time-variant equations can exhibit highly intricate dynamics, and in general it is rare to obtain corresponding invariant compact forward attracting sets. Moreover, only few papers in the literature explore the geometric structure of such sets.