Stability and Hopf Bifurcation in Three-Dimensional Predator-Prey Models with Allee Effect

Abstract

In this study, we perform the stability and Hopf bifurcation analysis for two population models with Allee effect. The population models within the scope of this study are the one prey-two predator model with Allee growth in the prey and the two prey-one predator model with Allee growth in the preys. Our procedure for investigating each model is as follows. First, we investigate the singular points where the system is stable. We provide the necessary parameter conditions for the system to be stable at the singular points. Then, we look for Hopf bifurcation at each singular point where a family of limit cycles cycle or oscillate. We provide the parameter conditions for Hopf bifurcation to occur. We apply the algebraic invariants method to fully examine the system. We investigate the algebraic properties of the system by finding all algebraic invariants of degree two and three. We give the conditions for the system to have a first integral.