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

Year 2019, Volume: 23 Issue: 5, 1005 - 1011, 01.10.2019

İlknur Kuşbeyzi Aybar

https://doi.org/10.16984/saufenbilder.537485

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.

Keywords

predator-prey model, stability, Hopf bifurcation

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