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

Abstract

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 non-autonomous system "rolls up"' onto a cycle of the limiting Lotka-Volterra equation as $t\to\infty$, 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.

Keywords

• asymptotically autonomous system
• limiting equation
• invariance principle
• lyapunov function