Fractional-Order Model of Two-Prey One-Predator System

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.


Introduction
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 [1][2][3][4].These interactions may be among the individuals of the same species and among the individuals of different species or interactions against the environment, disease, and food supply.The most important one is the interaction among the individuals of different species which can be predatory, competitive, or mutualistic.A large number of simple mathematical models have been suggested to understand the predator-prey interaction.The work done independently by Lotka [5] and Volterra [6], which is known as Lotka-Volterra model, was the first stone in this field.Later, the model was extended to include density-dependent prey growth and a functional response [7].After that, a huge number of variants of this model were suggested.Initially, the authors proposed a system of two-dimensional coupled differential equations [8,9].Then, the scenario changed to discrete mathematical models which included a lot of complex dynamical behaviours [10,11].In the last two decades, the fractional-order differential equations appeared and began to study the predator-prey models in the fractional-order form [12][13][14][15][16][17][18][19].
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 [20][21][22][23][24][25][26][27][28][29][30].The fractional differential equations are naturally related to systems with memory.Also, they are closely related to fractals which are numerous in biological system.The definition of fractional derivative involves an integration which is nonlocal operator.This means that the fractional derivative is a nonlocal operator.Studies and results of solutions obtained using the fractional differential equations are more general and are as stable as their integer order counterpart.
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 [21,22,29,31].The fractional derivative is defined via the fractional integral operator.So, we will start by the definition of the fractional integral operator.The fractional integral operator of order  (>0) of a function () such that  > 0 is given by 2

Mathematical Problems in Engineering
The Riemann-Liouville derivative of order  (>0) is given by where  = ⌈⌉ and  = /.
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  (>0) is denoted by   * and is given in the following form: where  = ⌈⌉ and  > 0.
The following are some important properties of the fractional derivatives and integrals [18].Let ,  ∈ R + and  ∈ (0, 1); then we have the following: ( 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 2, we proved the existence and the uniqueness of the solutions of our model.In Section 3, we studied the local asymptotic stability of the equilibrium points of the system.The numerical solution of the fractional-order two-prey one-predator model is given in Section 4.

The Fractional-Order Prey-Predator Model
Let  1 () represent the first prey herds (gazelles) and  2 () be the second prey herds (buffalos) densities, respectively.Suppose  3 () represent the predator (lions) density.It is known that the logistic scenario is the most appropriate to describe the growth of the preys.So, the terms  1 ()(1 −  1 ()) and  2 ()(1 −  2 ()) are the growth of the two preys, where the positive parameters  and  are the intrinsic growth rate of them.The nature requires cooperations between preys against the predator to avoid the predation and to facilitate getting food.This cooperation is represented by the term  1 () 2 () 3 ().Also, the predation to the preys is represented by the terms  1 () 3 () and  2 () 3 ().Considering these assumptions, we get the following fractional-order two-prey one-predator system: with the initial values where  is the death rate of the predator, 0 <  ≤ 1, and  1 () ≥ 0,  2 () ≥ 0,  3 () ≥ 0. The constants , , , , and  are all positive.In the following, we studied the above model to understand the long time behaviour prey-predator interaction.

Lemma 3. The initial value problem (1, 2) has a unique solution in the region
] .Then, Since Then, we have Then where So, the continuous function (()) satisfies Lipschitz condition and has a unique solution [32].
A sufficient condition to say that an equilibrium point is a locally asymptotically stable is that all eigenvalues  satisfy |arg ()| > /2 [33].This condition implies that the characteristic polynomial of that point should satisfy Routh-Hurwitz conditions [17].Since  1 and  2 are both positive, then the stability of point  4 depends on the first bracket in (18).Thus  4 is locally asymptotically stable if  = − 2 /(+ ) 2 < 0.

Numerical Results
The Adams-type predictor-corrector method for the numerical solution of the fractional differential equations was discussed in [10].This method can be used for both linear and nonlinear problems.It may be extended to multiterm equations (involving more than one differential operator) too [10].
We used the trapezoidal method [34] to compare the results and plot some new figures; see Figures 4, 5, and 6 to show this comparison.

Conclusion
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  6 and  7 were stable equilibrium points under some conditions at the ordinary differential equation form of the model.But, in our fractional form, we found that the same points are stable without any conditions on the parameters.This means that the predators (lions) can live stably together with the two preys (gazelles and buffalos) without extinction of any of them.This is an example of the equilibrium point which is centre for the integer order system but locally asymptotically stable for its fractional-order counterpart.This means that the fractionalorder differential equations are, at least, as stable as their integer order counterpart.So, we recommend to restudy most of the complex models (biological, medical, etc.) in the fractional-order form.
The stability of approximate solutions  1 ,  2 and  3 Proposition 5.If the discriminant (()) is positive, then the Routh-Hurwitz conditions ( 1 > 0,  3 > 0 and  1  2 >  3 ) are the necessary and sufficient conditions for that all eigenvalues satisfy |arg ()| > /2.Proposition 6.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 The stability of approximate solutions  1 ,  2 , and  3 1,  2,  3, ) + The stability of approximate solutions  1 ,  2 , and  3 The stability of approximate solutions  1 ,  2 , and  3