On the asymptotic behaviour of solutions of an asymptotically Lotka-Volterra model

We make more realistic our model [Nonlinear Anal. 73(2010), 650–659] on the coexistence of fishes and plants in Lake Tanganyika. The new model is an asymptotically autonomous system whose limiting equation is a Lotka–Volterra system. We give conditions for the phenomenon that the trajectory of any solution of the original nonautonomous system “rolls up” onto a cycle of the limiting Lotka–Volterra equation as t → ∞, which means that the limit set of the solution of the non-autonomous system coincides with the cycle. A counterexample is constructed showing that the key integral condition on the coefficient function in the original non-autonomous model cannot be dropped. Computer simulations illustrate the results.