Order-Induced Bifurcation Analysis for a Three-Species Prey–Predator Model with Fear Effect Under Fractional PD Control

Abstract

This paper aims to investigate the stability and Hopf bifurcation for a three-species prey–predator model with Holling type-II and type-III functional responses under fractional Proportional–Derivative (PD) control. The prey’s fear of the middle predator and the super predator as well as gestation delay is also incorporated in our proposed model for its wider applicability. Based on the stability theorem of linear system, Hopf bifurcation theorem and implicit function array solution method, some sufficient order-induced bifurcation conditions are given. The application and effectiveness of the presented theoretical results are illustrated through a simulation example. Especially, the impact of the feedback control gains as well as the fear effect on the stability and Hopf bifurcation is explored.