A Modified Leslie–Gower Model Incorporating Beddington–DeAngelis Functional Response, Double Allee Effect and Memory Effect

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.

A Fractional-Order Eco-Epidemiological Leslie–Gower Model with Double Allee Effect and Disease in Predator

“…Recently, [28] studied a fractional-order predator-prey model by assuming that the predation follows the Beddington-DeAngelis functional response and the double Allee efect on the predator population. By the last assumption, there is always a bistability condition for the strong Allee efect case.…”

“…In other words, there is an extinction or existing condition for the predator depending on the choice of the initial condition. In this paper, we consider a predator-prey model in [28] by adding the assumption that there is a transmission of disease in the predator population. Biologically, the growth rates of a population must depend on the history of its previous conditions not only on the local conditions, and for this reason, the proposed model will use the fractional-order derivative which has a memory efect, to make it more accurate in predicting the future condition of the population.…”

Mathematical Model of Iteroparous and Semelparous Species Interaction

“…Then in 2017, Kon [1] examined the Leslie multispecies semelparous matrix model which has an arbitrary number of age classes. Then, there are also studies on multispecies but with other methods using the Rosenzweig-MacArthur model (See [14], [15]), the Leslie-Gower model (See [16], [17]), and the Lotka-Volterra model (See [18]- [20]).…”

Stability Analysis and Hopf Bifurcation for ODE System of Predator-Prey Model with Mutual Interference

“…It is assumed that the predator feeding rate decreases by higher predator density even when prey density is high, and therefore the effects of predator interfe-rence in the feeding rate remain important all the time whether an individual predator is handling or searching for a prey at a time. And in [17]. The authors consider a Leslie-Gower predator-prey model with a functional Crowley-Martin response describing predator mutual interference, which its takes the form:…”