Modelling and Qualitative Analysis of Water Hyacinth Ecological System with Two State-Dependent Impulse Controls

Abstract

Water hyacinth and its ecological invasion have negative impacts on diversity of indigenous species and ecosystems, which becomes one of the hotspots in current ecological research. In this paper, a water hyacinth ecological system with two state-dependent impulse controls is studied. Firstly, we define the successor functions of semicontinuous dynamic system and give existence theorems of order-1 periodic solution and order-2 periodic solution of such system. Secondly, we analyze singular points of the system without impulsive state feedback control qualitatively and get the condition for focus point. Thirdly, we obtain the sufficient condition under which the system has an order-1 or order-2 periodic solution through the method of successor function and prove the stability of the order-1 or order-2 periodic solution by the analogue of Poincaré’s criterion. Furthermore, some examples and numerical simulations are given to illustrate our results.