Periodicity and stability in a single-species model governed by impulsive differential equation

A periodic single-species model with periodic impulsive perturbations was investigated. By using Brouwer’s fixed point theorem and the Lyapunov function, sufficient conditions for the existence and global asymptotic stability of positive periodic solutions of the system were derived. Numerical simulations were presented to verify the feasibilities of our main results.     

Li-Yorke chaos in the system with impacts

The analogue of Li-Yorke chaos [T.Y. Li, J. Yorke, Period three implies chaos, Amer. Math. Monthly 87 (1975) 985-992] for a special initial value problem of a non-autonomous impulsive differential equation is developed.     

Dynamic Analysis of an Impulsive Chemostat Model with Microbial Competition and Nonlinear Perturbation

In this paper, we propose an impulsive chemostat model with microbial competition and nonlinear perturbation. First, thresholds for the extinction of both microoganisms are given. Second, we investigate the persistence in mean and boundedness of the chemostat system by constructing Lyapunov function. Moreover, we obtain the sufficient condition for the existence of an ergodic stationary distribution of the system. At last, numerical simulations are presented, and the results show that the competition between two species tends to make one species disappear from their common habitat, especially when the competition is concentrated in a single resource.