Stability and bifurcation analysis in a predator–prey system with Michaelis–Menten type predator harvesting
Introduction
Ever since the pioneering work of Lotka and Volterra who first proposed two differential equations that describe the relationship between predators and prey in 1925 and 1926, respectively [1], predators and prey models have been continuously researched over the last one hundred years due to its significance in many problems [2], [3], [4]. From the point of view of human needs, the exploitation of biological resources, the management of renewable resources, and the harvesting of populations are commonly human purpose of achieving the economic interest in fishery, forestry, and wildlife management [5], [6]. Hence, this is the motivation to introduce and to consider the harvesting of populations in predator–prey models. Predator–prey models with harvesting and the role of harvesting in the management of renewable resources are studied extensively by many authors [7], [8], [9], [10], [11], [12]. In 1979, May et al. [13] have proposed two types of harvesting regimes: (i) constant-yield harvesting, which is described as harvested biomass independent of the size of the population, and (ii) constant-effort harvesting, i.e., proportional harvesting, which is described as harvested biomass proportional to the size of the population.
In terms of predator–prey systems with constant-yield harvesting, Huang et al. [14] systematically studied the dynamical properties of a predator–prey model of Holling and Leslie type with nonzero constant-yield prey harvesting. They have shown that the harvested model can exhibit richer dynamics compared to the model with no harvesting, such as appearance of numerous kinds of bifurcations for the model, including saddle–node bifurcation, Hopf bifurcation, repelling and attracting Bogdanov–Takens bifurcations of codimensions 2 and 3. Sen et al. [7] focused on the global dynamics of a predator–prey system when predator is provided with additional food as well as harvested at a constant rate. Refs. [15], [16] also have paid great attention to study the effect of constant-yield harvesting in predator–prey models.
In terms of constant-effort harvesting, a ratio-dependent predator–prey model in which the prey is continuously being harvested at a linear function rate was studied by Xiao et al. [17]. They proved that the system has different behaviors for various parameter values. Particularly, there exist areas of coexistence in which both populations become extinct, and areas of “conditional coexistence” depending on the initial values. Makinde [18] developed an algorithm to approach the solution of the ratio-dependent predator–prey system with constant effort harvesting. In addition to these work mentioned above, the effect of constant-effort harvesting in predator–prey models has been studied in [19], [20], and the reference therein as well.
However, it is well-known that nonlinear type harvesting is more realistic from biological and economic points of view [21] and is better than the constant-yield harvesting and constant-effort harvesting [22], [23]. There are two main reasons. On the one hand, harvesting does not always occur with constant yield or constant effort [6]. On the other hand, constant-effort harvesting embodies several unrealistic features and limitations. Traditionally, the constant effort catch-rate function is taken into account in the form based on the catch-per-unit-effort hypothesis, where denotes effort and is a constant. We can see that tends to infinity as the effort tends to infinity if the population is finite and fixed, or as the population tends to infinity if the effort is finite and fixed [24]. The harvesting term proposed firstly by Clark [25] is the so-called Michaelis–Menten type functional form of catch rate, where is the catchability coefficient, is the external effort devoted to harvesting, and are constants. Now we have , and [26], [27]. Hence, these restrictive features which we have mentioned-above are largely removed. For more details about this kind of harvesting type one can see Ref. [27]. A modified Leslie–Gower predator–prey model with time delay and the Michaelis–Menten type prey harvesting was investigated by Yuan et al. [28]. They obtained the critical conditions for the saddle–node-Hopf bifurcation, and gave the completion bifurcation set by calculating a universal unfolding near the saddle–node-Hopf bifurcation point. In [29], Zhang et al. discussed a reaction–diffusion predator–prey model with non-local delay and Michaelis–Menten-type prey-harvesting. They revealed that the discrete and non-local delays are responsible for a stability switch in the model system, and a Hopf bifurcation occurs as the delays pass through a critical value.
May et al. [13] proposed the following model to describe the interaction of predators and their prey subjected to various harvesting regimes: where and represent the prey and predators densities at time , respectively; and describe the intrinsic growth rate and the carrying capacity of the prey in the absence of predators, respectively; is the maximum value at which per capita reduction rate of the prey can attain; is the intrinsic growth rate of predators; takes on the role of a prey-dependent carrying capacity for predators and is a measure of the quality of the food for predators. and describe the effect of harvesting on the prey and predators, respectively. In [30], the authors have studied the case of constant-yield harvesting on the predators only of system (1) as well as given a brief research history of this system, including: (a) unharvested system [3], (b) constant-effort harvesting on both the prey and predators [31], (c) constant-yield harvesting on the prey and constant-effort harvesting on predators [16], (d) constant-yield harvesting on both the prey and predators [16], (e) constant-yield harvesting on the prey only [32].
Inspired by the work mentioned-above, the present paper aims to focus on questions about how predators and prey change when system (1) has a nonlinear harvesting, the Michaelis–Menten type functional form of catch rate in predator. Here, we have assumed that the prey in system (1) is not of commercial importance. This study can be thought as a supplement to system (1). To the best of our knowledge, system (1) containing nonlinear harvesting in prey or predator has not been studied.
In this paper, we give a detailed analysis for system (1) with the Michaelis–Menten type functional form of catch rate in predators, i.e., . The number of equilibria, local asymptotic stability, codimension one bifurcations, such as saddle–node bifurcation, transcritical bifurcation and Hopf bifurcation, codimension 2 bifurcation, such as Bogdanov–Takens bifurcation are shown in system (1) with Michaelis–Menten type harvesting term in predators. More specifically, we derive the parameter conditions for the existence of equilibria. The stability of the positive equilibria and boundary equilibrium are determined and illustrated graphically. Specially, the stability of some positive equilibria are determined by using numerical method due to that the equilibria themselves and the corresponding determinant and trace of the Jacobian matrix for these equilibria are very complex. The conditions for saddle–node bifurcation, transcritical bifurcation are derived by using Sotomayor’s theorem. In order to determine the stability of limit cycle of Hopf bifurcation, the first Lyapunov number is calculated and then illustrated graphically for a numerical example. Choosing two parameters of the system as bifurcation parameters, we prove that the system exhibits Bogdanov–Takens bifurcation of codimension 2 by calculating a universal unfolding near the cusp. Numerical simulations including Bogdanov–Takens bifurcation diagram of codimension 2, and phase portraits are carried out to demonstrate the validity of theoretical results.
The layout of this paper is as follows. A predator–prey model with nonlinear predator harvesting (Michaelis–Menten type) is described in Section 2. The existence of equilibria and their stability are discussed in Section 3. Section 4 deals with the bifurcations. We show that this system not only exhibits codimension one bifurcations, such as saddle–node bifurcation, transcritical bifurcation, Hopf bifurcation but also have codimension 2 bifurcation, for instant, Bogdanov–Takens bifurcation. Numerical simulations are carried out for illustrating the theoretical results. Finally, a conclusion is presented in Section 5.
Section snippets
The system
In this paper, we consider system (1) with the Michaelis–Menten type functional form of catch rate in predators, i.e.,
In order to simplify system (2), we take the following transformations as [30] dropping the bars, then the system (2) can be rewritten as where , and are positive constants. Note that after the rescaling transformation, the harvesting term in
Existence of equilibria
In order to obtain the equilibria of system (3), we consider the prey nullcline and predator nullcline of this system, which are given by:
It is obvious that the equilibria are the intersections of these nullclines. We easily see that system (3) possesses a unique boundary equilibrium given by . For the possible positive equilibria, we only need consider the positive solutions of the following equations:
For positive
Bifurcation analysis
In this section, we will discuss various possible bifurcations of system (3). The conditions for saddle–node bifurcation, transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation are derived.
Conclusion
In this paper, we have studied the dynamics of a predator–prey system where predator is provided with harvesting at Michaelis–Menten type rate. To our knowledge, there is little literature that considers the effect of Michaelis–Menten type predator harvesting on system (1). By an appropriate scaling, the original Michaelis–Menten type predator harvesting term becomes a nonlinear harvesting term with only two parameters. Qualitative analysis reveals that Michaelis–Menten type predator harvesting
Acknowledgments
This work is supported by National Natural Science Foundation of China (NSFC) under Project No. 11171017 and the Fundamental Research Funds for the Central Universities under Project No. 2015YJS175.
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