Stability of a one-predator two-prey system governed by nonautonomous differential equations

TL;DR

This paper investigates the dynamics of a nonautonomous predator-prey system, establishing conditions for species survival, decay, and stability through mathematical proofs and numerical illustrations.

Contribution

It extends previous models by analyzing a non-periodic predator-prey system, proving existence, stability, and species survival conditions.

Findings

Existence of unique positive solutions

Presence of an invariant set indicating species survival

Predator decay when prey densities are low

Abstract

A non-periodic version of the one-predator two-prey system model presented in [L.T.H. Nguyen, Q.H. Ta, T.V. T\d{a}, Existence and stability of periodic solutions of a Lotka-Volterra system, SICE International Symposium on Control Systems, Tokyo, Japan, 712-4 (2015) 1-6] is considered. First, we prove existence of unique positive solutions to the model. Second, we show existence of an invariant set, which suggests the survival of all species in the system. On the other hand, we show that when the densities of two prey species are quite small, the predator falls into decay. Third, we explore global asymptotic stability of the system by using the Lyapunov function method. Finally, some numerical examples are given to illustrate our results.