Dynamical Behaviors of a Fractional-Order Predator-Prey Model: Insights Into Multiple Predators Competing for a Single Prey

Abstract

In this paper, we investigate the dynamical behaviors of a modified Bazykin-type two predator-one prey model involving the intra-specific and inter-specific competition among predators. A Caputo fractional-order derivative is utilized to include the influence of the memory on the constructed mathematical model. The mathematical validity is ensured by showing the model always has a unique, non-negative and bounded solution. Four kinds of equilibria are well identified which represent the condition when all populations are extinct, both two predators are extinct, only the first predator is extinct, only the second predator is extinct, and all populations are extinct. The Matignon condition is given to identify the dynamics around equilibrium points. The Lyapunov direct method, the Lyapunov function, and the generalized LaSalle invariant principle are also provided to show the global stability condition of the model. To explore the dynamics of the model more deeply, we utilize the predictor–corrector numerical scheme. Numerically, we find the forward bifurcation and the bistability conditions by showing the bifurcation diagram, phase portraits, and the time series. The biological interpretation of the analytical and numerical results is described explicitly when an interesting phenomenon occurs.