Analysis for a Delayed Three-Species Predator-Prey Model with Feedback Controls and Prey Diffusion

In this paper, we study a class of 3-species multidelay Lotka–Volterra ratio-dependent predator-prey model with feedback controls and prey diffusion. By using the theory of delay differential inequalities and developing some new analysis methods as well as constructing a suitable Lyapunov function, some sufficient conditions are obtained to guarantee the permanence of the system and the global attractivity of the positive solution for the predator-prey system. Furthermore, the corresponding periodic system is discussed, and some conditions are established about the existence, uniqueness, and stability of the positive periodic solution for the periodic system by using the fixed-point theory and theoretical analysis. In additional, some numerical solutions of the equations describing the system are given to verify the obtained criteria. Finally, the corresponding stochastic time-delay predator-prey model with multiplicative noise sources is solved numerically, and some interesting new dynamics behavior is obtained.