Analysis of the stability of a predator-prey model including the memory effect, double Allee effect and Holling type-I functional response

Abstract

This study proposes and analyses a revised predator-prey model that accounts for a twofold Allee impact on the rate of prey population expansion. Employing the Caputo fractional-order derivative, we account for memory impact on the suggested model. We proceed to examine the significant mathematical aspects of the suggested model, including the uniqueness, non-negativity, boundedness, and existence of solutions to the noninteger order system. Additionally, all potential equilibrium points for the strong and weak Allee effect are examined under Matignon’s condition, along with the current state of conditions and local stability analysis. Analytical results are also provided for the necessary circumstances for the Hopf bifurcation initiated by the fractional derivative order to occur. We also demonstrated the global asymptotic stability for the positive equilibrium point in both the strong and weak Allee effect cases by selecting an appropriate Lyapunov function. This study’s innovation is its comparative investigation of the stability of the strong and weak Allee effects. To conclude, numerical simulations validate the theoretical findings and provide a means to investigate the system’s more dynamical behaviours.