Abstract
This paper deals with the study of the stability and the bifurcation analysis of a Leslie-Gower predator-prey model with Michaelis-Menten type predator harvesting. It is shown that the proposed model exhibits the bistability for certain parametric conditions. Dulac’s criterion has been adopted to obtain the sufficient conditions for the global stability of the model. Moreover, the model exhibits different kinds of bifurcations (e.g., the saddle-node bifurcation, the subcritical and supercritical Hopf bifurcations, Bogdanov-Takens bifurcation, and the homoclinic bifurcation) whenever the values of parameters of the model vary. The analytical findings and numerical simulations reveal far richer and complex dynamics in comparison to the models with no harvesting and with constant-yield predator harvesting.
1. Introduction
Marine life is a renewable natural resource that not only provides food to a large population of humans but also is involved in the regulation of the Earth’s ecosystem. The growing human needs for more food and more energy have led to increased exploitation of these resources which affects the Earth’s ecosystem. Thus, it is imperative to design harvesting strategies which aim at maximizing economic gains giving due consideration to the ecological health of the concerned Earth’s ecological system. Mathematical modeling in harvesting of species was started by Clark [1, 2]. There are mainly three types of harvesting according to Gupta et al. [3]: (i) , constant rate harvesting (see [4–7]), (ii) , proportionate harvesting (see [8, 9]), and (iii) (Holling type II), nonlinear harvesting (see [10–13]). Nonlinear harvesting is more realistic and exhibits saturation effects with respect to both the stock abundance and the effort level. Notice that as and as .
The objective of the present work is to study dynamical behaviors of a Leslie-Gower predator-prey model in the presence of nonlinear harvesting in predators depending on parameters of the model. There have been many papers on the Leslie-Gower predator-prey system with harvesting. For example, May et al. [14] proposed a Leslie-Gower predator-prey model in which two different kinds of constant-yield harvesting applied on both prey and predator species have been considered and this model was studied by Beddington and Cooke [15]. Beddington and May [16] proposed and studied Leslie-Gower predator-prey model when both the prey and predators were harvested with constant effort. Beddington and Cooke [15] studied a Leslie-Gower predator-prey model in which the preys are harvested at a constant-yield rate and predators are harvested with constant-effort rate. Zhu and Lan [17] studied a Leslie-Gower predator-prey model with constant-yield prey harvesting. Gong and Huang [18] studied the Bogdanov-Takens bifurcation for the model and showed that for different parameter values the model has a limit cycle or a homoclinic loop. Gupta et al. [3] discussed the bifurcation analysis of a Leslie-Gower prey-predator model in the presence of nonlinear harvesting in prey. Huang et al. [19] studied the effect of constant-yield predator harvesting on the dynamics of a Leslie-Gower type model and showed that the model has Bogdanov-Takens (BT) singularity of codimension or a weak focus of multiplicity two for some parameter values. They have shown that as the parameters change, the model exhibits saddle-node bifurcation, repelling and attracting B-T bifurcations, and supercritical, subcritical, and degenerate Hopf bifurcations.
This article is organized as follows. In Section 2, we describe the mathematical model in detail. In Section 3, we obtain the number and location of equilibrium points and the local and global stability of the equilibria are investigated. In Section 4, the existence of saddle-node, Hopf, and Bogdanov-Takens bifurcations is shown. In Section 5, we present several numerical simulations that support our theoretical results. Finally, we present some ramification of our results in Section 6.
2. Model Equations
2.1. Leslie-Gower Model
In general, the Leslie-Gower predator-prey model [20] is given as follows:where and are the prey density and predator density at time , respectively; , , , , and are positive parameters and represent the intrinsic growth rate of prey, intrinsic growth rate of predator, carrying capacity of prey in the absence of predator, maximal predator per capita consumption rate, and measure of the food quality that the prey provides for conversion into predator births. Model (1) has been studied by Hsu and Huang [21]. They showed that the unique positive interior equilibrium point of system (1) is globally asymptotically stable under all biologically admissible parameters.
2.2. Harvested Model
We assumed that only predator species is economically important and the nonlinear harvesting rate is considered. Model (1) reduces to the following:where the parameters and are positive and represent catchability coefficient and effort applied to harvest the individuals, respectively, and are suitable positive constants.
Model (2) is not well defined at . In order to define system (2) at , we improve the model as given in [22]; system (2) reduces toFor nondimensionalizing system (3), we introduce the following substitutions:System (3) reduces towhere , , , and . For the existence of the biological meaning of the variables in model, system (5) is studied in the closed first quadrant, , in -plane defined by .
3. Equilibria and Their Local Stability
The equilibrium points of system (5) are the nonnegative real solutions of the system . It is obvious that system (5) has the trivial equilibrium point and the axial equilibrium point and the abscissa of the positive interior equilibrium points are the roots of the quadratic equation:and the ordinance of the positive interior equilibrium points is given by , .
The quadratic equation (6) has two positive roots and whenever (say), a double positive root whenever , and no positive root whenever . The number and location of the equilibrium points of system (5) lying in the set are given in the following lemma.
Lemma 1. System (5) has (a)four equilibrium points, trivial equilibrium point , axial equilibrium point , and two interior equilibrium points and whenever , where , , , , and ;(b)three equilibrium points, trivial equilibrium point , axial equilibrium point , and a double interior equilibrium point whenever , where and ;(c)three equilibrium points, trivial equilibrium point , axial equilibrium point , and an interior equilibrium point whenever and , where and ;(d)three equilibrium points, trivial equilibrium point , axial equilibrium point , and an interior equilibrium point whenever , where and ;(e)two equilibrium points, trivial equilibrium point , and axial equilibrium point whenever and .
The number and location of interior equilibrium points have been depicted in Figure 1.
(a)
(b)
(c)
Now, we shall discuss the stability of the equilibrium points. The Jacobian matrix of system (5) at the equilibrium point cannot be calculated as the term is not defined at . We use the blow-up technique to analyze the stability of the equilibrium point as given in [23]. Let and ; then, system (5) reduces toSystem (7) has two equilibrium points and whenever . The Jacobian matrix of system (7) at the equilibrium point isThus, the equilibrium point of system (7) is an unstable point as . The Jacobian matrix of system (7) at the equilibrium point isThus, the equilibrium point of system (7) is always saddle as . Thus, we can conclude the discussion above as follows.
Theorem 2. The trivial equilibrium point of system (5) is a saddle point.
The Jacobian matrix of system (5) at the equilibrium point isThe Jacobian matrix of system (5) at the positive interior equilibrium point isThe determinant of the abovementioned Jacobian matrix and trace .
Theorem 3. (a) The equilibrium point is asymptotically stable whenever and a saddle whenever .
(b) The equilibrium point , if existent, is always a saddle point.
(c) The equilibrium point , if existent, is asymptotically stable whenever and is unstable whenever .
(d) The equilibrium point , if existent, is a degenerate singular point.
(e) The equilibrium point , if existent, is always asymptotically stable.
(f) The equilibrium point , if existent, is asymptotically stable whenever and is unstable whenever .
Proof. (a) The eigenvalues of the Jacobian matrix are and , so the equilibrium point is asymptotically stable whenever and a saddle whenever .
(b) The determinant , so the interior equilibrium point is a saddle point.
(c) The determinant and , so the interior equilibrium point is asymptotically stable whenever and unstable whenever .
(d) The determinant , so the interior equilibrium point is a degenerate singular point.
(e) The determinant as and always, so the interior equilibrium point is always asymptotically stable.
(f) The determinant and , so the interior equilibrium point is asymptotically stable whenever and unstable whenever .
From Lemma 1, if , system (5) has unique interior equilibrium point , and if , system (5) has unique interior equilibrium point . In Theorem 3, it is proved that these interior equilibrium points are always asymptotically stable. Now we show that these equilibrium points are globally asymptotically stable.
Theorem 4. The equilibrium points and , if existent, are globally asymptotically stable.
Proof. From Lemma 1, system (5) has one trivial equilibrium point , one axial equilibrium point , and one interior equilibrium point whenever . Further, it has one trivial equilibrium point , one axial equilibrium point , and one interior equilibrium point whenever . Also, from Theorem 3, the trivial equilibrium point is always a saddle point, axial equilibrium point is a saddle point whenever , and the interior equilibrium points and are asymptotically stable. We define the following function:where , , and .
After simplification, we obtain as or . Thus, by using Dulac’s criterion [24], system (5) will not have any nontrivial periodic orbit in . Note that -axis and -axis are the stable manifolds of the trivial and the axial equilibrium points, respectively. Using this in conjunction with the Poincare-Bendixson theorem [24] gives us that the interior equilibrium points and will be globally stable.
In Theorem 3, it is shown that the equilibrium point is a degenerate singular point. Now we study the property of this point.
Theorem 5. The equilibrium point , if existent, is(a)a saddle-node whenever ;(b)a cusp of codimension whenever .
Proof. First, we shall shift the equilibrium point to the origin by using the transformations and ; system (5) reduces into the formwhereIf , point is a saddle point. Now we consider ; that is, . Both eigenvalues of the Jacobian matrix are zero and system (13) reduces toNow we introduce the affine transformations , , to reduce system (15) aswhereNow, we consider the change of coordinates in the small vicinity of :Then, system (16) reduces toNow, we choose the change of coordinates in the small neighbourhood of :System (19) reduces toNow, we choose the final change of coordinates in the small neighbourhood of :System (21) reduces toIf (nondegeneracy condition), the origin of (23) is a cusp of codimension ; that is, the interior equilibrium point of system (5) is a cusp of codimension .
In Section 4, we shall study the bifurcations occurring in system (5).
4. Bifurcation Analysis
4.1. Hopf Bifurcation
In Theorem 3, it is shown that the interior equilibrium point is stable whenever and unstable whenever . Now, we consider the parametric condition . In this parametric condition, the equilibrium point is a weak focus or a center. Hence, system (5) may enter a Hopf bifurcation at the point . In this subsection, we consider the parameter as the Hopf bifurcation parameter and discuss the conditions under which the stability of will change and system (5) exhibits Hopf bifurcations.
Theorem 6. System (5) undergoes a Hopf bifurcation with respect to parameter around the equilibrium point , if existent, whenever . Moreover, (a)the equilibrium point is a weak focus of multiplicity if the parameter set is in or and is stable and unstable according to whether is in or ;(b)system (5) has at least one unstable limit cycle whenever is in and , and at least one stable limit cycle whenever is in and .
Proof. A critical magnitude of the bifurcation parameter exists as such that, at , and . In order to ensure the existence of a Hopf bifurcation, we have to check the transversality condition (see [24]). We have as . Hence, the transversality condition for a Hopf bifurcation is satisfied. To determine the direction of Hopf bifurcation and stability of , we compute the first Liapunov coefficient of system (5) at the equilibrium point .
Let and ; then, the equilibrium point is shifted to the origin and system (5) can be rewritten as where , , , , , , , , , , , , , , , , and . Hence, using the formula of the first Lyapunov number at the origin of (24), we have where . If , then the origin of (24) is a weak focus of multiplicity one: also origin is stable when and unstable when . Hence, in parameter space , there exist surfaces and , called subcritical and supercritical Hopf bifurcation surface, respectively, such that if the parameter set is in , the equilibrium point of system (5) is a weak focus of multiplicity and is unstable, and if the parameter set is in , the equilibrium point of system (5) is a weak focus of multiplicity and is stable.
From the discussion above, the equilibrium point of system (5) is a weak focus of multiplicity and is unstable if is in . Also from Theorem 3, the equilibrium point is stable whenever , that is, , and is unstable whenever , that is, . Thus, the equilibrium point generates an unstable limit cycle as the bifurcation parameter passes through the bifurcation value . From one side of the surface to the other side, system (5) can undergo a subcritical Hopf bifurcation. An unstable limit cycle arises in the small neighbourhood of the equilibrium point whenever is in and . Similarly, a stable limit cycle arises in the small neighbourhood of the equilibrium point whenever is in and .
4.2. Saddle-Node Bifurcation
In Section 3, it is shown that system (5) admits the double point whenever . In Theorem 5, it is shown that the point is a saddle node whenever . Now, we show that system (5) experiences a saddle-node bifurcation of codimension around the equilibrium point at the threshold value of the bifurcation parameter by means of Sotomayor’s theorem [24].
Theorem 7. System (5) undergoes a saddle-node bifurcation with respect to the parameter around the equilibrium point whenever and .
Proof. It has been shown that if and , one eigenvalue of the Jacobian matrix is zero and the other has nonzero real part. Suppose and are the eigenvectors corresponding to the zero eigenvalues of the Jacobian matrix and transpose matrix , respectively; then, and . We have , . Therefore, and . Since , . Thus, Sotomayor’s theorem confirms that system (5) experiences a saddle-node bifurcation of codimension around interior equilibrium point . This means that there are no equilibrium points for while there are two coexistence equilibrium points and for , one of which is saddle point and the other is a stable node.
4.3. Bogdanov-Takens Bifurcation
In Theorem 5, we have proved that the interior equilibrium point is a cusp of codimension whenever and , which implies that there may exist the Bogdanov-Takens bifurcation in system (5). The parameters and are taken as the bifurcation parameters as they are important from biological point of view and by means of a series of transformations as given in Xiao and Ruan [5], we derive a normal form.
Theorem 8. System (5) undergoes a Bogdanov-Takens bifurcation with respect to the bifurcation parameters and around the double equilibrium point if , , and .
Proof. We consider that the parameters and vary in a small neighbourhood of the BT point . Let be a point of this neighbourhood, where are small. System (5) becomesTranslating the equilibrium point into the origin by using the transformations and and then using Taylor’s series expansion, system (26) reduces towhere , , , , , , and is a power series in with powers satisfying .
Making the affine transformations and , then system (27) reduces towhere , , , , , , and is a power series in with powers satisfying .
Consider the change of coordinates in the small neighbourhood of :then, system (28) reduces towhere , , , , , and and are the power series in with powers satisfying .
Consider the change of coordinates in the small neighbourhood of :then, system (30) reduces towhere , and are the power series in and with powers , and satisfying , , and , respectively.
Applying the Malgrange Preparation theorem [25], we have where and is a power series in whose coefficients depend on parameters .
The sign of is tedious to determine. So, we use numerical computation. Here, we consider as . We take the transformation then, system (32) reduces towhere is a power series in with powers satisfying and .
Applying the parameter dependent affine transformations and in system (35), we obtainwhere is a power series in with powers satisfying and .
By means of transformation, System (36) reduces towhere is a power series in with powers satisfying and .
The system above is topologically equivalent to the normal form of the Bogdanov-Takens bifurcation which is given byThe system undergoes a Bogdanov-Takens bifurcation if . When system (5) undergoes Bogdanov-Takens bifurcation at , three bifurcation curves in plane through the BT point are given by(1)saddle-node bifurcation curve: ;(2)Hopf bifurcation curve: ;(3)Homoclinic bifurcation curve: .
5. Numerical Simulation Results
In this section, we will present the numerical simulation results which will support our analytical findings.
(a) , , and . System (5) has two interior equilibrium points and , trivial equilibrium point , and an axial equilibrium point . If , the interior equilibrium point is asymptotically stable (see Figure 2(a)) and if , the interior equilibrium point is unstable (see Figure 2(b)). The axial equilibrium point is an asymptotically stable point and the interior equilibrium point is always a saddle point. If and , system (5) has only one positive interior equilibrium point which is globally asymptotically stable (see Figure 2(c)). If and , system (5) has only one positive interior equilibrium point which is also globally asymptotically stable (see Figure 2(d)). , , and . System (5) has no positive interior equilibrium points and the phase portrait diagram is shown in Figure 2(e). The trivial equilibrium point is always a saddle.
(a)
(b)
(c)
(d)
(e)
(b) , , , and then . System (5) undergoes Hopf bifurcation and the first Lyapunov coefficient , so an unstable limit cycle arises around the equilibrium point . Here, we take (see Figure 3(a)). , , , and then . System (5) undergoes Hopf bifurcation and the first Lyapunov coefficient , so a stable limit cycle arises around the equilibrium point . Here, we take (see Figure 3(b)).
(a)
(b)
(c) , , , and . System (5) has a double interior equilibrium point which is a saddle node (see Figure 4(a)). The phase portrait diagram is shown in Figure 4(a) and the saddle-node bifurcation is shown in Figures 4(b) and 4(c).
(a)
(b)
(c)
(d) and ; then, we obtain and . Thus, system (5) isSystem (40) has a unique positive interior equilibrium point .
Translating the equilibrium point into the origin by using the transformations and , by Taylor’s series expansion, system (40) reduces towhere , , , , , , and is a power series in with powers satisfying .
Making the affine transformations and , then system (41) reduces towhere , , , , , and .
Performing the change of coordinates , and , in the small neighbourhood of , system (42) reduces towhere , , , − , , and , and are the power series in and with powers , and satisfying , , and , respectively.
Applying the Malgrange Preparation theorem, we have where and is a power series in whose coefficients depend on parameters .
We have as . We take the transformation then, system (43) reduces towhere is a power series in with powers satisfying and .
Applying the parameter dependent affine transformations and in system (46), we obtainwhere is a power series in with powers satisfying and .
Consider the transformationThen, system (47) reduces towhere , , and is a power series in with powers satisfying and . Also . Thus, system (40) undergoes Bogdanov-Takens bifurcation.
The local representations of the bifurcation curves are as follows:(a)(b)(c)These bifurcation curves are sketched in a small neighbourhood of the origin in the plane (see Figure 5(a)). The bifurcation curves divide the parameter plane into four parts: , , , and . The saddle-node bifurcation curve (SN) is the horizontal axis, that is, axis, and the homoclinic bifurcation curve (HL) and Hopf bifurcation curve (H) lie in the fourth quadrant. When , then system (5) has a unique interior equilibrium point which is a cusp of codimension (see Figure 5(b)). When the parameters and vary and lie in region , then system (5) has no interior equilibrium point and all the solution trajectories go to the axial equilibrium point; that is, the axial equilibrium point is globally stable (see Figure 5(c)). Hence, in this case, the predator will go extinct and prey will approach the carrying capacity. When the parameters and pass the SN bifurcation curve and lie in region , the cusp of codimension breaks into a hyperbolic saddle and an unstable focus (see Figure 5(d)). When the parameters and lie in region , the cusp of codimension breaks into a stable focus and a hyperbolic saddle. The change of stability of the focus yields an unstable limit cycle (see Figure 5(e)). Further, when the parameters and lie in region , the cusp of codimension breaks into a saddle point and a stable focus (see Figure 5(f)). Hence, an open set of initial population densities exists for which both predator and prey approach a stable steady state.
(a)
(b)
(c)
(d)
(e)
(f)
6. Discussion and Conclusion
In this paper, a Leslie-Gower predator-prey model has been analyzed in the presence of nonlinear predator harvesting. It is shown that the system has at most four equilibrium points in , in which the trivial equilibrium point and the axial equilibrium point always exist while the positive interior equilibrium point changes from two to zero. The qualitative properties of solutions in the vicinity of have been studied by using a blow-up technique, and it is found that the origin will never be stable; ecologically speaking, the two species cannot go to extinction together. The axial equilibrium point is either asymptotically stable or globally stable or a saddle depends on the parametric conditions. Thus, ecologically speaking, the prey species never goes to extinction whatever the choice of initial population density. If two interior equilibrium points exist, then one is always a saddle point while the other is either asymptotically stable or unstable or the system undergoes a Hopf bifurcation around this point; that is, for certain parametric domain, the system exhibits a bistability phenomenon as well as oscillatory coexistence of both the populations. The stability of limit cycles has been studied and validated through numerical simulations by calculating the first Lyapunov number. It is also found that for certain parametric domain the proposed system has a unique interior equilibrium which is globally asymptotically stable.
It is shown that the system can have zero, one, or two interior equilibrium points through saddle-node bifurcation as the bifurcation parameter crosses a certain threshold value. By means of Sotomayor’s theorem, the existence of saddle-node bifurcation has been shown. Ecologically speaking, a maximum threshold value of exists, below which both species coexist and above which predator species suddenly collapse to extinction. The proposed system undergoes Bogdanov-Takens bifurcation near degenerate equilibria. We have considered the parameters and as bifurcation parameters and reduced the system into normal form. Ecologically speaking, a small perturbation in the bifurcation parameters is a cause of extinction, coexistence, and oscillation.
The dynamical analysis of model (5) shows complex and rich dynamics as compared to the model with no harvesting [21] and with constant-yield predator harvesting [19]. The model without harvesting has a unique interior equilibrium point which is globally asymptotically stable [21] while model (5) has zero to two interior equilibrium points and model (5) undergoes a series of bifurcations (Hopf bifurcation, saddle-node bifurcation, and Bogdanov-Takens bifurcation). The model with constant-yield predator harvesting has no axial equilibrium point, and the solutions once touching the -axis will leave the first quadrant [21]. But the model with nonlinear predator harvesting has one trivial equilibrium point and one axial equilibrium point and the solutions touching the -axis will go to the axial equilibrium point . Further, if no interior equilibrium point exists, then the axial equilibrium point is globally asymptotically stable (see Figure 2(e)). Ecologically speaking, in the absence of predator species, the prey density approaches the carrying capacity. Although the model with constant-yield predator harvesting undergoes a series of bifurcations, like Hopf bifurcation, saddle-node bifurcation, and Bogdanov-Takens bifurcation [19], the model with nonlinear harvesting gives more general parametric conditions for the occurrence of these bifurcations. Thus, the results of nonlinear harvesting model can explain the real life situation in a more effective and realistic manner.
Competing Interests
The authors declare that they have no competing interests.