Global asymptotic stability for a classical controlled nonlinear periodic commensalism AG-ecosystem with distributed lags on time scales

Abstract

Commensalism is a common phenomenon in nature. The Ayala-Gilpin (AG) dynamical system model is commonly used to describe the nonlinear interactions between species in ecosystems. Combining commensalism with AG-system models, the manuscript emphasizes on a classical controlled nonlinear periodic commensalism AG-ecosystem with distributed lags on time scales. In our model, the discrete and continuous cases are unified and generalized in the sense of time scale. Firstly, it is proved that a class of auxiliary functions have only two zeros in the real number field. Then, with the aid of these auxiliary functions, using the coincidence degree theory and inequality technique, we obtain some sufficient criteria for the existence of periodic solutions. Meanwhile, we prove that the periodic solution is globally asymptotically stable by applying Lyapunov stability theory. Finally, an example is numerically simulated with the help of MATLAB tools.