Abstract
In this paper, the dynamical behavior of a three-dimensional fractional-order prey-predator model is investigated with Holling type III functional response and constant rate harvesting. It is assumed that the middle predator species consumes only the prey species, and the top predator species consumes only the middle predator species. We also prove the boundedness, the non-negativity, the uniqueness, and the existence of the solutions of the proposed model. Then, all possible equilibria are determined, and the dynamical behaviors of the proposed model around the equilibrium points are investigated. Finally, numerical simulations results are presented to confirm the theoretical results and to give a better understanding of the dynamics of our proposed model.
1. Introduction
The most influential theme in ecology and mathematical modeling is the dynamic of the relationship among species. Many authors extended or modified the work of the Lotka and Voltera model [1, 2], and they have investigated thse topics widely by using ordinary differential equations or deference equations, see [3–9], and references therein. Nowadays, authors formulate their mathematical models by fractional order differential equation due to their ability to give the precise description for various linear and nonlinear phenomena [10–15]. The increasing of mathematical models that based on fractional order differential equation has recently obtained popularity in the studying the behavior of biological models. Fractional-order differential equation has been successfully used and applied to model many areas of science, engineering, and phenomena that cannot be formulated by other types of equations [10, 16]. In [12, 16–21], authors have investigated the effects of the fractional order differential equation on a prey predator model as well as they also discussed the stability analysis of equilibrium points of fractional order model with and without harvesting, as well as the existence, uniqueness, and boundedness of the solutions that are proved.
There are several different types of definitions of fractional-order differential equation in the literatures [16, 22–25], for e.g., Caputo, Riemann-Litouville, Fabrizio, Marechand,Grunwald-Letnikov, wayl, and Riesz fractional-order derivatives. Throughout this work, we used the Caputo fractional-order derivatives because it is not necessary to define the initial conditions of fractional-order and its fractional-order derivative of constant function is zero, as well as the similarity of the initial conditions of fractional order differential and the integer order ones [8]. This work is organized as follows: in Section 2, some useful definitions and concept that concern to the fractional order are presented. In Section 3, a three-dimensional fractional-order prey-predator model is considered. The uniqueness and boundedness as well as the nonnegativity of its solution are proved. In Section 4, all the equilibrium points are determined, and the conditions to achieve its local stability are set. In Section 5, numerical simulations are given to confirm the theoretical results. Finally, conclusions are given in Section 6.
2. Preliminaries
Definition 1. [24, 26]: Caputo’s definition of fractional derivatives is given as follows:
,
where is the least integer which is not less than , and is the Riemann-Liouville integral operator of order which is given by
,
is the Euler Gamma function.
Throughout this work, we need some useful results for the fractional-order derivative. We start with the following results which are proved in [13, 16, 27].
Lemma 1. (1)Suppose that and with , then we have
where , . (2)Suppose and , . If, , then is a nondecreasing function , and if , , then is a nonincreasing function (3)Consider the Cauchy problem
With and , then the form of the solution is given by
While the solution to the problem is given by (4)Let be a continuous function on and satisfy
where , and is the initial time. Then, its solution has the form
For the existence a unique solution for the general fractional-order derivative, we have the next lemma that appeared in [28].
Lemma 2. Consider the system
,
with the initial condition , where, . If satisfies the locally Lipchitz condition with respect to , then there exists a unique solution of the above system on .
3. Model Formulation
Frist, the following three dimension prey-predator model with functional response of Holling type III is considered as follows:
where , are the density of prey species, middle species, and the top predator species. , , and are the intrinsic growth rates, the carrying capacity of the prey population, the death rates for middle predator population and top predator population, and capture rate of prey and middle predator, respectively. The parameteris the conservation rate of prey, and the parameters and are the conservation rate of middle predator to the top predator , respectively. The constants and are the predation rates and half saturation constants, respectively. The parameter denotes to the constant harvesting rate.
We use the following transformation to reduce the dimension of model (8). ,,, and , and then we obtain the following system:
where , , , , , , and .
Next, we introduce the fractional–order derivative in model (2) with the help of fractional order Caputo type derivative. Then, the system (9) becomes as follows:
Remark 3. Because of the biological significance, we are interesting solutions that are nonnegative and bounded only, so that we will prove the following theorem.
Theorem 4. Let and denotes all nonnegative real numbers in , and then all solutions of the system (10) with and are uniformly bounded and nonnegative.
Proof. we have to show that , assuming for considering that is not true. Then, there exists a constant such that
From the first equation of system (10), we have
According to part 1 in Lemma 1, we have , which contradicts the fact that , i.e., . Therefore, we have . By the same arguments, and can get .For the uniformly bounded, Let , and then we have
Now, for each , we have
Therefore,
, by taking min,
where .
Now by using part 4 in Lemma 1, we have
Thus, as and . Therefore, all solutions of system (10) that starts from are confined in the region.
Now, the existence and uniqueness of the solution of the considered system (10) are shown in the next theorem:
Theorem 5. Let be a sufficiently large, and then for each , there exists a unique solution of the fractional-order system (10) with initial condition , which is defined for all .
Proof. Let , , and consider a mapping , such that For any , we have Since and and , we can have where .
Therefore, satisfies the Lipchitz condition. By Lemma 2, the proof is finished.
4. The Equilibrium Points and Local Stability Analysis
In this section, the existence and the local stability of equilibrium points of the considered system (10) are studied and investigated. Before that, we need the following theorem.
Theorem 6 (see [29]). Consider the fractional–order differential system ; , with and . The equilibrium points of the above system are solutions to the equation . An equilibrium point is locally asymptotically stable if all the eigenvalues of the Jacobian matrix evaluated at the equilibrium satisfy .
Proof (see [29]). To obtain the equilibrium points of the considered system (10), we solve the following simultaneous equations: Thus, the considered system (10) has the following possible equilibrium points: (1)The trivial equilibrium point and the equilibrium point without any restriction on the parameters(2)The top predator free equilibrium point exists if and , where and (3)The interior equilibrium point exists if and where , , is the positive root of the equation and The Jacobian matrix of system (10) at is given by where and .
The next theorems give the local stability of , , and , respectively.
Theorem 7. The trivial equilibrium point of the system (10) is always unstable point.
Proof. The Jacobian matrix at is given by The characteristic equation of the Jacobian matrix is given by Therefore, the roots of the above characteristic equation are , , and . Then, , and also . Hence, the trivial equilibrium point is always unstable point.
Theorem 8. The equilibrium point of system (10) is locally asymptotically stable if .
Proof. The Jacobian matrix at is given by The characteristic equation of the Jacobian matrix is given by The roots of the above characteristic equation are , and . Then, and . Hence, the equilibrium point is locally asymptotically stable if . Otherwise, it is unstable point.
Theorem 9. The equilibrium point is locally asymptotically stable if one of the following holds: (i), , and (ii) and and where , , , and
Proof. The Jacobian matrix at is given by The characteristic equation of the Jacobian matrix is given by It is clear that is negative if so that . The other two roots and are . Hence, if . then the eigenvalue and the eigenvalue are taken negative value; thus, . Therefore, the point is locally asymptotically stable. Also, if and if hold, then the equilibrium point is asymptotically stable. Otherwise, it is unstable point.
In order to discuss the local stability of the interior point, we need the following definition and lemma.
Definition 10 (see [30]). The discriminant of a polynomial is defined by , where is the derivative of and where and is an determinant. If , then is given by
Lemma 11 (see [30]). Let be a discriminant of the cubic polynomial equation, , then (1)If D, then (2)If , , and , then for(3)If , , , and , then (4)If , , , and , then .
Proof (see [30]).
Theorem 12. The interior equilibrium point is locally asymptotically stable if one of the following conditions holds: (1)If and , for (2)If , , for (3)If , , for where , , and .
Proof. The Jacobian matrix of system (10) at is given by Therefore, the characteristic equation of is given by Thus, the discriminant of the cubic polynomial is where .
Then, all results can be obtained by Lemma 6.
5. Numerical Simulation
All previous theoretical results in Section 4 are confirmed by giving numerical simulations. At different set of values of parameters, the local stability of , , and of the considered system (10) is investigated numerically.
First, for the point , the following set of values of the parameters is chosen , and (0.1,0.5,0.9) is the initial value. Therefore, the condition in Theorem 4 is satisfied. Figure 1 shows that the point is locally stable.
(a)
(b)
For the point , we choose the values of parameters as following , ; ; ; ; ; ; and 5, and (0.5,2.5,03) is the initial value. According to Theorem 9, the local stability of is satisfied. Figure 2 illustrates the local stability of the point .
(a)
(b)
To show the local stability of the unique positive equilibrium point , these values of parameters are chosen ; ; ; ; ; ; , and 5, (0.82,1.95,0.9) is the initial value so that according to Theorem 12, the equilibrium point is local stability point, and the trajectories of prey and predator are shown in Figure 3.
(a)
(b)
6. Conclusions
Fractional-order differential equations have recently been successfully used and applied to model many areas of science, especially, phenomena that cannot be formulated by other types of equations, so that this work concerns a study of a three-dimensional fractional-order prey-predator model with Holling type III functional response and constant rate harvesting. It is shown that the model possesses the existence, nonnegativity, boundedness, and uniqueness of the solutions, as desired in any population dynamics. The local stability of all possible equilibrium points of the considered system is investigated analytically then confirms by numerical verifications.
Data Availability
All simulations in the article are used by the authors to confirm the theoretical results. We have also cited references that were used, and we gave the DOI in the references.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors are very much thankful to the reviewers for their valuable suggestions and comments that lead to improve the quality of this manuscript.