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We propose a fractional-order model of the interaction within two-prey and one-predator system. We prove the existence and the uniqueness of the solutions of this model. We investigate in detail the local asymptotic stability of the equilibrium solutions of this model. Also, we illustrate the analytical results by some numerical simulations. Finally, we give an example of an equilibrium solution that is centre for the integer order system, while it is locally asymptotically stable for its fractional-order counterpart.

The branch of mathematics that concerned biology is called mathematical biology. Mathematical biology tries to model, study, analyze, and interpret biological phenomenon such as the interactions, coexistence, and evolution of different species [

Fractional calculus generalizes the concept of ordinary differentiation and integration to noninteger order. Fractional calculus is a fertile field for researchers to study very important real phenomena in many fields like physics, engineering, biology, and so forth [

There are a lot of approaches to define the fractional differential operator such as Grunwald-Letnikov, Riemann-Liouville, Caputo, and Hadamard. The Riemann-Liouville and Caputo approaches are the most widely used in applications [

In this paper, we used Caputo approach to define the fractional derivative. It is a modification to the Riemann-Liouville definition. The Caputo fractional derivative of order

The following are some important properties of the fractional derivatives and integrals [

If

If

If

In this paper, we proposed a fractional-order model to study the interaction of a system consists of two-prey and one-predator species. In Section

Let

The initial value problem (1, 2) can be written as the following matrix form:

Let

The initial value problem (1, 2) has a unique solution in the region

Let

Then, we have

So far there is no known method for solving nonlinear equations. Therefore, it is difficult and even impossible to find an analytical solution to system (

Solving the equations of system (

Substituting by the equilibrium point

For equilibrium point

A sufficient condition to say that an equilibrium point is a locally asymptotically stable is that all eigenvalues

Similarly, equilibrium point

The study of the coexistence points is more important. This study gives us the conditions that lead to coexistence between the prey and the predator. The Jacobian matrix for the first coexistence equilibrium point

For the equilibrium point

The discriminant

If the discriminant

From the above definition of the discriminant and the coefficients of the characteristic equation, we find that (with long calculations) all the four necessary and sufficient conditions are satisfied. Then equilibrium point

The Adams-type predictor-corrector method for the numerical solution of the fractional differential equations was discussed in [

In this paper, we used Adams-type predictor-corrector method for the numerical solution of our system. First, we will give the Adams-type predictor-corrector method for solving general initial value problem with Caputo derivative:

We used the step size to be

An example for the locally asymptotically stable equilibrium point

The approximate solutions between

The approximate solutions between

The approximate solutions between

The stability of approximate solutions

An example for the locally asymptotically stable of the equilibrium point

The approximate solutions between

The approximate solutions between

The approximate solutions between

The stability of approximate solutions

Example for the locally asymptotically stable equilibrium point

The approximate solutions between

The approximate solutions between

The approximate solutions between

The stability of approximate solutions

In Figures

We used the trapezoidal method [

An example for the locally asymptotically stable equilibrium point

The approximate solutions between

The approximate solutions between

The approximate solutions between

The stability of approximate solutions

An example for the locally asymptotically stable equilibrium point

The approximate solutions between

The approximate solutions between

The approximate solutions between

The stability of approximate solutions

Example for the locally asymptotically stable equilibrium point

The approximate solutions between

The approximate solutions between

The approximate solutions between

The stability of approximate solutions

We used the step size to be

In Figures

In this paper, we studied the existence, the uniqueness, the stability of the equilibrium points, and numerical solutions of a fractional-order two-prey one-predator model. The coexistence equilibrium points

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

The authors would like to express their gratitude to King Khalid University, Saudi Arabia, for providing administrative and technical support.