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We analyze the dynamics of a fractional order modified Leslie-Gower model with Beddington-DeAngelis functional response and additive Allee effect by means of local stability. In this respect, all possible equilibria and their existence conditions are determined and their stability properties are established. We also construct nonstandard numerical schemes based on Grünwald-Letnikov approximation. The constructed scheme is explicit and maintains the positivity of solutions. Using this scheme, we perform some numerical simulations to illustrate the dynamical behavior of the model. It is noticed that the nonstandard Grünwald-Letnikov scheme preserves the dynamical properties of the continuous model, while the classical scheme may fail to maintain those dynamical properties.

The dynamical interaction of predator and prey is one of important subjects in ecological science. In recent years, one of the most important species interactions is predator-prey model [

One of other factors that influence the interaction of predator and prey is Allee effect, referring to a decrease in per capita fertility rate at low population densities. Allee effect may occur under several mechanisms, such as difficulties in finding mates when population density is low or social dysfunction at small population sizes. When such a mechanism operates, the per capita fertility rate of the species increases with density; that is, positive interaction among species occurs [

If

If

It is shown in system (

The Riemann-Liouville

Let

The Riemann-Liouville fractional derivative is historically the first concept of fractional derivative and theoretically well established. However, in the case of Riemann-Liouville fractional differential equation, the initial value is usually given in the form of fractional derivative, which is not practical. Consequently, one applies the Caputo fractional derivative which is defined as follows.

The Caputo fractional differential operator of order

From Definition

In this paper we reconsider system (

Consider the following autonomous nonlinear fractional order system:

Based on Theorem

The equilibria of system (

trivial equilibrium

two axial equilibria, that is, the prey extinction point

positive or coexistence equilibrium

where

The equilibria of system (

trivial equilibrium

three axial equilibria that are the prey extinction point

positive or coexistence equilibrium point

Let

If

Suppose that

If

If

If

Moreover, algebraic computations show that if (

To check the local stability of each equilibrium point, we linearize system (

Stability of trivial and axial equilibrium for weak Allee effect

the trivial equilibrium

the axial equilibrium

(i) The Jacobian matrix at

(ii) The Jacobian matrix at

Using the same argument as in the proof of Theorem

Stability of trivial and axial equilibrium for strong Allee effect

the trivial equilibrium

the axial equilibrium

The stability properties of positive (coexistence) equilibrium for the case of weak and strong Allee effect are stated in Theorems

Stability of coexistence equilibrium for weak Allee effect

suppose

The characteristics equation of

If

If

Similarly we have Theorem

Stability of coexistence equilibrium for strong Allee effect

suppose

Based on the above theorems it can be seen that the stability properties of both trivial and axial equilibrium points are not dependent on

To solve system (

The left hand side is approximated by the generalization of forward difference scheme

The nonnegative denominator function has to satisfy

The approximation of

By implementing the Grünwald-Letnikov approximation for the fractional derivative and the nonlocal approximation for the right hand side of system (

To verify our stability analysis as well as the effectiveness our numerical scheme, we perform some numerical simulations. First we use hypothetic values of parameters

Phase portrait of system (

Solutions of system (

Next, we set

Phase portrait of system (

Phase portrait of system (

Finally, we compare our numerical results obtained by the NSGL scheme to those obtained by the standard GL scheme using parameters

Phase portrait of system (

The dynamic of a fractional order modified Leslie-Gower model with Beddington-DeAngelis functional response and additive Allee effect has been analyzed. Our model has four types of equilibria that are the trivial (extinction of both prey and predator) equilibrium, two axial equilibria (the prey extinction point and the predator extinction point), and the interior (coexistence) point. The trivial and the predator extinction for both weak and strong Allee effects are always unstable. For the case of weak Allee effect, the prey extinction is conditionally stable while for that of strong Allee effect, the prey extinction is always stable. Our analysis also shows that the interior point for both weak and Allee effects is conditionally stable. The order of fractional derivative may influence the stability of interior point. Here, when the order

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the Directorate of Research and Community Service, The Directorate General of Strengthening Research and Development, and the Ministry of Research, Technology and Higher Education (Brawijaya University), Indonesia, Contract no. 137/SP2H/LT/DRPM/III/2016 dated March 10, 2016, and Contract no. 460.18/UN10.C10/PN/2017 dated April 18, 2017.