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

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.