Abstract
In this paper, we study a Leslie–Gower predator-prey model with harvesting effects. We carry out local bifurcation analysis and stability analysis. Under certain conditions, the model is shown to undergo a supercritical Hopf bifurcation resulting in a stable limit cycle. Numerical simulations are presented to illustrate our theoretic results.
1. Introduction
In this paper, we consider Leslie–Gower predator-prey model with harvesting effect,with and , where and are the prey and predator population densities, respectively, , and is the time.
Note that is Leslie–Gower term in which the carrying capacity of the predator’s environment is a linear function of the prey size . is the number of prey consumed by the predator in unit time which shows that when the number of the prey is severe scarcity, and the predators can switch over to other populations as food. Constants are the intrinsic growth rate of the prey and predator, respectively, and denote the harvesting efforts for the prey and predator, respectively.
Since the first prey-predator dynamical models which is the Lotka–Volterra model was built in the 1920s by Mathematician Lotka and Volterra, more and more researchers are interested in such issues, and they start from different angles to think the problem and many important results have been obtained [1–10]. In particular, in 2003, Aziz-Alaoui and Daher Okiye [11] considered the following Leslie–Gower predator-prey model:where is the numbers of prey and is the numbers of predators. Existence and stability of the fixed points were studied by using the Lyapunov function. In 2006, Lin and Ho [12] discussed the local and global stability for system (2) by using Poincaré–Bendixson theorem and Dulac’s criterion.
Harvesting is an effective way for humans to control the size of predators and prey so that the population has continued to develop healthily and produced good economic benefits [13–16]. Academically, researchers often only consider the harvesting of prey in order to control the size of the population. In 2010, Zhu and Lan [17] investigated the Leslie–Gower predator-prey systems:
In 2013, Gupta and Chandra [18] discussed the following Leslie–Gower predator-prey model with harvesting on the prey and the environment providing the same protection to both the predator and prey:
For ecological balance and healthy economic development, for fisheries, wildlife resources, etc., we not only need to consider the harvesting of prey, but also the predator. Therefore, in this paper, we study Leslie–Gower predator-prey model (1) with harvesting on the prey and predator.
In order to make the system dimensionless, we define new dependent variables by , and a new time variable by . Moreover, let , , , , , and . System (1) becomeswhere are all positive parameters.
Lemma 1 (see 19). If,, and, whereis a positive constant,, and, we have
Lemma 2 (see [20, 21]). Consider system and suppose that , Jacobian matrix has a simple eigenvalue with eigenvector , and the transpose of the Jacobian matrix has an eigenvector to the eigenvalue . Then, the system experiences a transcritical bifurcation at the equilibrium point as the control parameter passes through the bifurcation value if the following conditions are satisfied:
The rest of this paper is organized as follows. In Section 2, we study boundary of solutions. In Section 3, we discuss existence of equilibria points. In Section 4, we discuss stability of the equilibrium points.
2. Boundedness of Solutions
In this section, we prove that every solution of system (5) is positive and uniformly bounded with initial conditions and . Denote .
Theorem 1. Consider system (5). For any given initial conditions, the solutionof system (5) exists and is unique and positive and ultimate bounded.
Proof. Let function . Obviously, function is continuous differentiable on , so for any given the initial conditions , the solution of the system (5) exists and is unique. Furthermore, axis and axis are the solutions of the system (5); by the uniqueness of the solution, the solutions of the system (5) with the initial value cannot cross with axis and axis.
Next, we show the solutions of system (5) with the initial value which is ultimate bounded.
From system (5), we haveCombining Lemma 1, we haveSo, the following inequality is established:By Lemma 1, we havewhere .
3. Existence of Equilibria
In order to find the equilibrium points of system (5), we let , i.e.,
It is clear that equation (12) has a trivial solution . Furthermore, by calculation, we find other solutions of equation (12):
Remark 1. When and , the position of each point in (13) is shown in Figure 1.
Therefore, we have the following results.where .
Theorem 2. Consider system (5) admitsaxial only andaxial equilibria under following conditions.(i)The axial equilibrium, is a boundary equilibrium of system (5) if and only if(ii)The axial equilibrium, is a boundary equilibrium of system (5) if and only if
Theorem 3. Consider system (5) admits a unique positive equilibrium,if only if
Remark 2. Existence regions for equilibrium points of system (12) is shown in Figure 2. Equilibrium pointsexist, but positive equilibrium pointdoes not in region II and equilibrium points, andcoexist in region I. Moreover, if, then positive equilibriumbecomes boundary equilibrium.
4. Stability of of Equilibria
Theorem 4. Ifand, then the equilibriaandare unstable.
Proof. Firstly, we show the equilibria is a unstable equilibrium. The Jacobian matrix about is given bywith trace and determinant . From and , we have and . On the contrary, through direct calculation, we obtain . Therefore, is an unstable node point.
Secondly, we show the equilibria is a unstable equilibrium. The Jacobian matrix about is given bywith determinant . Therefore, is a saddle point which is unstable.
Theorem 5. (1) If, then the boundary equilibriumis a unstable equilibrium. (2) If, then the boundary equilibriumis locally asymptotically stable.
Proof. The Jacobian matrix about is given bywith trace and determinant .(1)When , we have , so is a saddle point which is an unstable.(2)When , we have and . Furthermore, a direct calculations gives Therefore, is locally asymptotically stable.
Theorem 6. (1) Ifandhold, then the positive equilibriumis unstable. (2) Ifandhold, then the positive equilibriumis locally asymptotically stable. Furthermore, assume that there is; then, the positive equilibriumis globally asymptotically stable.
Proof. The Jacobian matrix about is given bywith trace and determinant .
A direct calculation gives(1)From conditions and , we obtain Therefore, the positive equilibrium is unstable.(2)From conditions and , we have So, the positive equilibrium is locally asymptotically stable. Let . By calculations, we have Appling Dulac’s criterion and Theorem 1, we know that the positive equilibrium is globally asymptotically stable.
5. Local Bifurcation
hold, then system (5) undergoes a Hopf bifurcation with respect to bifurcation parameter around the equilibrium point . Furthermore, the direction of the Hopf bifurcation is subcritical and the bifurcation periodic solutions are orbitally asymptotically stable if
Theorem 7. If, then system (5) undergoes a transcritical bifurcation around.
Proof. From , we haveTherefore,We haveThe eigenvectors and associated to zero eigenvalues of matrixes and , respectively, areSo, .
In addition, equalsCombining (33) and (34), we haveThen, we compute as follows:Combining (33) and (36), we haveSince all three conditions of Lemma 2 are satisfied, system (5) undergoes a transcritical bifurcation around if .
Letand is of (40).
Theorem 8. Assume thatis satisfied. If
The direction of the Hopf bifurcation is supercritical and the bifurcation periodic solutions are unstable if
Proof. From (39), we haveSo, .
The Jacobian matrix of system (5) evaluated at the point is given byThe trace and the determinant of Jacobian matrix are given byBy and (40), we have and .
In addition, we haveSo, . Therefore, this guarantees the existence of Hopf bifurcation around .
We translate the equilibrium to the origin by the translation . For the sake of convenience, we still denote and by and , respectively. So, the system (5) becomesRewrite system (47) towhereDenote the eigenvalues of by with and .
Define matrixwhere and .
Obviously,and when , i.e., , we haveBy the transformation,system (48) becomeswherewithIn order to determine the stability of the periodic solution, we need to calculate the sign of the coefficient , which is given bywhere all partial derivatives are evaluated at the bifurcation point .
Combining , , , , and , we havewhere andCombining (41), we have . Therefore, according to Poincare–Andronow’s Hopf bifurcation theory, we have the direction of the Hopf bifurcation is subcritical and the bifurcation periodic solutions are orbitally asymptotically stable.
In addition, combining (42), we have . Therefore, according to Poincare–Andronow’s Hopf bifurcation theory, we have the direction of the Hopf bifurcation is supercritical and the bifurcation periodic solutions are unstable.
6. Numerical Illustrations
In this section, we perform numerical simulations about system (5). Figure 3 shows that is an unstable node point, is a saddle point, does not exist, and is asymptotically stable and every orbit tends to it. Figure 4 shows that is an unstable node point, is a saddle point, is unstable, is unstable, and there is a limit cycle around to which every orbit tends. Figure 5 shows that is an unstable node point, is unstable and is also unstable, but is asymptotically stable and every orbit approaches this equilibrium.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was partially supported by National Natural Science Foundation of P.R.China (no. 11661037).