Global stability of periodic solution for a 3-species nonautonomous ratio-dependent diffusive predator-prey system

Global stability of periodic solution for a 3-species nonautonomous ratio-dependent diffusive predator-prey system

A 3-species nonautonomous ratio-dependent reaction-diffusive predator-prey system is considered in this article. Firstly, by utilizing a comparison principle and fixed point theorem, the existence of solution of the system which is space homogenous strictly positive and periodic is obtained. And the obtained conditions ensuring the existence of solution can be very easily verified. At the same time, we develop some new analysis techniques as a byproduct. Furthermore, with the help of the upper and lower solutions (UALS) approach for the parabolic partial differential equations and Lyapunov theory, we aim at the globally asymptotically stability problems of the solutions, and some judgment criteria are achieved. Finally, we give some numerical simulations results which validate the theoretical findings of this article.