Local exponential stability of four almost-periodic positive solutions for a classic Ayala-Gilpin competitive ecosystem provided with varying-lags and control terms

Abstract

Ayala-Gilpin ecosystem is one of the most famous differential equation models based on experimental and theoretical analysis. Most of the previous works on the almost periodic solution and its stability are mainly about the existence of an almost-periodic solution and its global stability. In fact, the real ecosystem presents multiple stable states due to the influence of various external factors. However, there are few researches on multiple almost-periodic solutions and local stability of Ayala-Gilpin ecosystem. Therefore, we focus on the existence and local exponential stability of multiple almost-periodic solutions for a classical nonlinear competitive Ayala-Gilpin ecosystem provided with varying-lags and control terms in this paper. Firstly, a class of functions having only two zeros in is investigated. Next, based on the existence of zeros of these functions, we obtain some sufficient conditions to ensure the existence of at least four almost-periodic positive solutions by utilising theorem of coincidence degree and some inequality techniques. Furthermore, we also conclude that this ecosystem exists at least two almost-periodic positive solutions under relatively weak conditions. Finally, we build some Lyapunov functionals to discuss the local exponential stability of each almost-periodic positive solution in its own region. As an application, a numerical simulation inspect the effectiveness of our major results.