On the stability of non-autonomous perturbed Lotka–Volterra models

Abstract

The paper is devoted to an extended Lotka–Volterra system of differential equations of predator–prey model. The extension is proposed with perturbation terms, which are null for the positive equilibrium state. In the original Lotka–Volterra system, the equilibrium state is not asymptotically stable due to the fact that perturbations are periodic in time. The aim of the paper is to characterize a form of perturbation terms guaranteeing the asymptotic stability or instability of equilibrium state. The reason of the proposed model is that for large time scale, the Lotka–Volterra model is too simple to be realistic. In the paper, the non-autonomous perturbations do not change the equilibrium state but introduce functions of time as well as for additional perturbed terms as for the main part of the equations modified from Lotka–Volterra model. Theorems are proposed in a renormalized form of the differential equations for time and the two variables. The key point of the paper comes from the use of a Liapunov function introduced in Section 2 which allows to obtain conditions for the asymptotic stability (Section 3) and instability (Section 4) by using a Cetaiev instability theorem following conditions on the renormalized coefficients in time of System (6). An appendix recalls the main results of the Liapunov Direct Method for non-autonomous binary systems of ordinary differential equations.