Abstract
In this paper, a modified Leslie–Gower predator-prey model with Beddington–DeAngelis functional response and double Allee effect in the growth rate of a predator population is proposed. In order to consider memory effect on the proposed model, we employ the Caputo fractional-order derivative. We investigate the dynamic behaviors of the proposed model for both strong and weak Allee effect cases. The existence, uniqueness, non-negativity, and boundedness of the solution are discussed. Then, we determine the existing condition and local stability analysis of all possible equilibrium points. Necessary conditions for the existence of the Hopf bifurcation driven by the order of the fractional derivative are also determined analytically. Furthermore, by choosing a suitable Lyapunov function, we derive the sufficient conditions to ensure the global asymptotic stability for the predator extinction point for the strong Allee effect case as well as for the prey extinction point and the interior point for the weak Allee effect case. Finally, numerical simulations are shown to confirm the theoretical results and can explore more dynamical behaviors of the system, such as the bi-stability and forward bifurcation.
Highlights
Modeling interaction between prey and its predator has become a dominant topic in mathematical biology due to its ubiquitous existence and fundamentality in many biological systems
We found that the model has four types of biologically feasible equilibrium
The predator extinction point is always stable for the strong Allee effect case, but it is always unstable for the weak Allee effect case
Summary
Modeling interaction between prey and its predator has become a dominant topic in mathematical biology due to its ubiquitous existence and fundamentality in many biological systems. One of the crucial phenomenon in ecology that influences the per capita growth rate either in the predator or prey population is the Allee effect, which describes a condition where, at low population densities, the per capita growth rate of the population has a positive dependence with its density. In the strong Allee effect, there is a population threshold value named the Allee threshold, below which the per capita growth rate of the population is negative [1,2]. In terms of conservation biology, if the Allee threshold is larger, it places a population at higher risk of extinction in a low-density population
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