Stability and Bifurcation Analysis of a Predator-Prey Model with Michaelis-Menten Type Prey Harvesting

In this paper, we investigated stability and bifurcation behaviors of a preda-tor-prey model with Michaelis-Menten type prey harvesting. Sufficient conditions for local and global asymptotically stability of the interior equilibrium point were established. Some critical threshold conditions for transcritical bifurcation, saddle-node bifurcation and Hopf bifurcation were explored analytically. Furthermore, It should be stressed that the fear factor could not only reduce the predator density, but also affect the prey growth rate. Finally, these theoretical results revealed that nonlinear Michaelis-Menten type prey harvesting has played an important role in the dynamic relationship, which also in turn proved the validity of theoretical derivation.


Introduction
The predator-prey model is one of the dominant population models, which has been researched extensively to comprehend interactions between various species in a fluctuant natural environment [1] [2] [3] [4] [5]. However, in reality, predators are unable to catch all prey because prey can decrease the risk of predation by using refuge [6] [7] [8]. So the prey refuge will more accord with authentic circumstance, and many scholars have achieved considerable process in this field [9] [10] [11] [12].
However, many researchers only consider directly killing of prey by predator, but ignore the impact of the presence of predator on prey. Zanette et al. [13] experimentally showed that the predation fears can reduce offspring production by 40%, which is even more intensively than the impact of direct hunting. In fact, all prey can exhibit different kinds of anti-predator behaviors, such as changes of habitat, physiological and foraging [14] [15] when they are confronted with predation risks. Many studies in this direction have been carried out and obtained numerous attractive results [16] [17] [18] [19] [20]. Particularly, Wang et al. [19] firstly proposed a predator-prey model with the cost of fear: where k is the level of fear, which causes anti-predator behaviors of prey. Biologically speaking, ( ) , f k v can reasonably obey the following conditions: They concluded that the cost of fear does not change the dynamical behaviors of this model and the unique equilibrium is globally asymptotically stable when the model with the linear functional response.
Inspired by the insightful work [19], Chakraborty [20] proposed a predator-prey model with fear factor, which is given by: where u and v represent prey and predator densities at time t, respectively. r is the intrinsic growth rate, k is the carrying capacity of prey, a is the predation rate, α is the conversion factor, m is the mortality rate of predator, θ is the Allee threshold. 1 1 fv + is the fear factor term. They showed that fear can dramatically lessen the per-capita growth rate, but cannot affect the equilibrium stability, but can generate richer dynamics such as bi-stability.
From the perspective of human needs and long-term progress, the exploitation of natural resources and the storage of renewable energy, harvesting are always one of the most crucial factors in the dynamics of predator-prey model [21]. We find that the predator-prey model with harvesting can lead to more complex properties than the model without it [22] [23] [24], which inspires us to take harvesting into account. Harvesting regimes can be classified into three  [38] proposed stochastic non-autonomous predator-prey models with and without impulse; they concluded that the model dynamics can be appreciably influenced by the stronger noises and nonlinear harvesting, which also can cause the extinction of the predator species.
Stimulated by the above review of literature, we will propose a predator-prey model with Michaelis-Menten type prey harvesting and prey refuge based on model (1), which can be expressed by the following equation, where q is the catchability coefficient, E is the effort used on harvesting the prey species, 1 m and 2 m are proper constants. The rest of the parameters have the same meanings as model (1). For simplicity, we will nondimensionalize the model ( The objective of this paper is to provide a stability and bifurcation analysis of model (3). The rest of the paper is arranged as follows. The existence of equilibria and their stability are presented in Section 2. Section 3 deals with the bifurcation, such as transcritical bifurcation, saddle-node bifurcation and Hopf bifurcation. In section 4, we analyze the impact of fear and prey refuge. Numerical simulations for model (3) are given in section 5 to illustrate the theoretical works.
Finally, a brief conclusion is drawn in section 6.

Existence and Stability of Equilibria
It is obvious that the possible boundary equilibria are the positive roots of the quadratic equation , the existence of boundary equilibria is concluded in Table 1.
2) If 1 c = , there exists an equilibrium point Evidently, the interior equilibrium point ( ) is the intersection of the nullclines. By simple calculation, we obtain the abscissa of interior equilibrium , and the ordinate * y is the positive root of the equa-Journal of Applied Mathematics and Physics

Stability of Equilibria
Now we analyze the local stability of equilibria identified above. The general Jacobian matrix of model (3) takes the form of where ( ) ( ) ( ) Proof. The Jacobian matrix of model (3) at the equilibrium point 0 E is given Hence, by Theorem 7.1 in [39], if the coefficient of 2 is a unstable node due to Proof. By replacing ( ) , x y in matrix (7) with SN E , The Jacobian matrix of Clearly, we find that the eigenvalues of ( ) We therefore consider the following two cases.  (9) can be rewritten as Proof. The Jacobian matrix of model (3) at 1 E is given by: We note that one of the eigenvalues of ( ) Proof. Similar to the proof of Theorem 3, the Jacobian matrix of model (3) at Then the eigenvalues of ( ) One can easily prove that the following two expres- hold on the basis of existence conditions of i E from Lemma 1, which means that the stability of ( ) Therefore, this theorem follows. □ Next, we study the local stability of the positive equilibrium ( ) , and give sufficient conditions on its global stability.
Theorem 5. The interior equilibrium point * E is locally asymptotically stable when 0 and unstable when Proof. The Jacobian matrix at * E is: Then we can easily gain the determinant of the matrix ( ) * J E is given by

Local Bifurcation Analysis
In this section, we will discuss possible bifurcations of model (3), and derive the conditions for various bifurcations. The following lemma is given for proving saddle-node bifurcation and transcritical bifurcation.
Hence, when the bifurcation parameter µ through the thresholds value, that is, 0 µ µ = , the system undergoes a saddle-node bifurcation at 0 x .

2) Suppose
( ) Hence, when the bifurcation parameter µ through the thresholds value, that is, 0 µ µ = , the system undergoes a transcritical bifurcation at 0 x .

Transcritical Bifurcation
From Lemma 1 and Theorem 1, we find that 0 E is a saddle when h c < , and stable node when h c > , which indicates that 0 E changes its stability when parameter h over the threshold value TC h c = . Particularly, the boundary equi-  J E , respectively. We get and calculating, we have Using the expressions for v and w we get,

Saddle-Node Bifurcation
Furthermore, using the expression for F in Theorem 8, we can get Therefore, we can conclude that model (3) The conditions (1) and (2)  Furthermore, in order to investigate the stability (direction) of the limit cycle, we are going to calculate the first Lyapunov number σ at the equilibrium * E of model (3).
We firstly translate * E to the origin by using the transformation   a a b a b a b   a   b a a  a b  a b b  a a  a a a  b b   a  a In fact, if 0 σ > , the equilibrium * E loses its stability through a Hopf bifurcation; if 0 σ < , * E will obtain stability. Since the expression of σ is quiet complicated, we cannot differentiate the sign of σ . Thus, we have presented numerical example in Section 5. □

The Effects of the Fear Factor and the Prey Refuge
In this section, we will explore the influence of fear factor (measured by f) and prey refuge (measured by m) on population density of model (3).

The Impact of the Fear on Population Density
Using the expression of * x in Equation (6), we note that the density of prey at coexistence equilibrium is independent of f, so the fear effect cannot affect the prey density. However, we find that the prey growth rate is greatly influenced by the fear effect. On the other hand, differentiation of * y gives ( ) ( )

The Impact of the Prey Refuge on Population Density
We firstly denote that ( ) ( ) Using the expression in Equation (6)

Simulation Analysis
As we all know, the relationship between prey and predator in the natural environment is mutual restriction and influence. To better comprehend the dynamic relationship between prey and predator, we will perform numerical simulations to exhibit some complex dynamic behaviors of model (3). For convenience, we fix the parameters value as follows: In order to demonstrate how the nonlinear Michaelis-Menten type prey har-  will lose its stability, a stable limit cycle will be generated, and the periodic orbit is surrounded by a trajectory passer through the Journal of Applied Mathematics and Physics      In addition, in order to deeply investigate their impact mechanism of fear factor (measured by f), we have given the diagram of model (3) in Figure 5. In contrast to the model without fear factor, the increment of f will decrease the prey growth rate (seeing Figure 5(a)) and lessen the maximum of final size of predator density (seeing Figure 5(b)). Besides, the diagram of model (3) in Figure 6 manifests that the population density will change due to the variation of prey refuge (measure by m). Moreover, Figure 6(a) shows that the prey density will rise as the value of m increases, which implies that the increase of prey refuge can protect more prey from predation, Figure 6(b) exhibits that with the increase of m value, the predator density increases first, reaches the maximum value, and then decreases.

Conclusion
In this paper, we have studied the dynamics of a predator-prey model with non-Journal of Applied Mathematics and Physics linear Michaelis-Menten type prey harvesting. The main focus of this article is to explore the impact of a Michaelis-Menten type prey harvesting mathematically and numerically. Using an appropriate conversion, the original Michaelis-Menten type prey harvesting term becomes a nonlinear harvesting term with only two parameters. Based on relevant mathematical theory, we reveal that nonlinear prey harvesting term plays an important role in influencing the dynamics of model (3). Note that, the parameters h and c in nonlinear harvesting term can affect the number and stability of boundary equilibria, which can become a saddle point, stable node, unstable node or saddle node for different parameter values. This implies that some possible bifurcation dynamic behaviors will occur, such as saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation.
At the same time, comprehensive numerical simulation works have been carried out, which also, in turn, demonstrate the validity of these theoretical results.
Moreover, we also discuss the influence of fear factor and prey refuge on the population density, it is easy to obtain that the fear factor can not only reduce the predator density but also affect the prey growth rate, while the prey refuge can affect both prey and predator population density. We hope this type of investigations will be of great help in comprehending the dynamic complexity of ecological system or physical systems in the future when the nonlinear Michaelis-Menten type prey harvesting can interact with prey populations.