Abstract
A predator-prey model with Holling II functional response incorporating a prey refuge with impulse effect is considered in this paper. With the help of the Floquet theory of impulsive differential equations, local stability and global attractivity of the boundary periodic solution of the system are derived, and then sufficient conditions for global asymptotic stability of the boundary periodic solution are obtained. Next, the permanence of the system is proved by constructing a Lyapunov function. Further, by applying the bifurcation theory of impulsive differential equations, conditions under which the system has a positive periodic solution are obtained. Finally, numerical simulations are presented to illustrate the analytical results.
1. Introduction
The study of population dynamic system has always been the focus of scholars. Population dynamics behavior is always affected by many factors. Smith [1] pointed out that scholars often study the effects on the behaviors such as territorial behavior and migration. The challenge of reducing the negative environmental impacts of land use and maintaining economic and social benefits is considered [2]. Wildlife species are also affected by agricultural activities, such as farming, intercropping, drainage, rotation, grazing, and the widespread use of pesticides and fertilizers [3]. One of the more relevant behavioral traits that affect the dynamics of predator-prey systems is the use of spatial refuges by the prey. Some fractions of the prey population are partially protected against predators by using refuges. González-Olivares et al. [4] proposed a predator-prey system with Holling type II functional response incorporating a constant prey refuge:
They investigated the local stability of equilibria and the existence of limit cycle of the system. Chen et al. [5] discussed the instability and global stability properties of the equilibria and the existence and uniqueness of limit cycle of the above system. The predator-prey systems with prey refuge have been focused by several authors. Tripathi et al. studied the model with Beddington-DeAngelis type function response incorporating a prey refuge [6, 7], the model with time delay and prey refuge [8], and the model for prey with variable rates in protected areas [9]. Mondal and Samanta [10] investigated the model with prey refuge dependent on both species and constant harvest in predator [10] and the model with nonlinear prey refuge [11, 12]. Many models with prey refuge have been studied; one can also refer to [13–21] and the references cited therein.
It has been noticed that many dynamical systems may be disturbed by impulsive factors. With profound understanding of human nature, the theory of impulsive differential equations [22–24] becomes more perfect. The impulsive differential equation has become a widely concerned subject in recent years, and it is more appropriate to apply to biological systems for the actual reality, for example [25–35].
Shea and Amarasekare [33] pointed out that “conservation, harvesting, and pest control are three aspects of the same general problem: population management.” People seek to maintain exploited populations at productive levels by harvesting.
The results show that the population model with refuge is practical. Many periodic factors, such as seasonal periodic change, migration, and harvest, have a great impact on population dynamics. The model with refuge and impulsive effect has also been studied by many scholars, one can refer to [13, 34, 35]. Driven by the above reasons, we consider system (1) with prey subject to periodic impulse harvesting, the model is as follows:where and are the population density of prey and predator, respectively, represents the intrinsic growth rate, is the environment carrying capacity, is the capture rate of predator, is the death rate of the predator, is the conversion factor denoting the number of newly born predators for each captured prey, is a constant number of prey using refuges, which protect prey from predation, and is impulse capture rate of prey. We assume that the population density of prey is more than the constant , , , , , , , , and are all positive constants, , , and is the period of the impulse effect.
The rest of this paper is organized as follows. Some definitions and preliminary lemmas are introduced in Section 2. Existence and global attractivity of boundary periodic solution of system (2) are discussed in Section 3. Section 4 contributes to permanence of system (2), and Section 5 aims at discussion of positive periodic solution. Section 6 contains some numerical simulations and illustrates our analytical results. Some simple discussions are given in Section 7.
2. Definitions and Preliminary Lemmas
Let and . Denote the map defined by the right hand of system (2). Let : , then is said to belong to class , if(1) is continuous in and (2) is locally Lipschitzian in
Definition 1. If , then for , the upper right derivative of of system (2) is defined as .
It follows from the second equation of (2) that for , soTherefore, the following Lemma 1 is obvious.
Lemma 1. If is a solution of system (2) with , then for all .
Definition 2. System (2) is said to be permanent if there are constants , (independent of initial value) and a finite time , such that and when for all solutions , with all initial values and .
Lemma 2 (see [22]). Let and . Assume thatwhere satisfies is nondecreasing; let be the maximal solution of the following scalar impulsive differential equation:existing on . Then, implies thatwhere is any solution of system (2) existing on .
Now we first consider some basic properties of the following subsystem of system (2):
Lemma 3 (see [36]). System (7) has a positive periodic solution . Any solution of system (7) satisfies as . It is easy to see that
Therefore, is the positive periodic solution of system (7).
3. Global Attractivity of Boundary Periodic Solution
First of all, we give the extinction of predator.
Theorem 1. Let be any solution of system (2), ifthen the stability of boundary periodic solution is globally asymptotically stable, where and .
Proof. Firstly, we will prove the local stability of boundary periodic solution. Let and , then the solution of variation equation of system (2) iswhere satisfiesand , in which is the identity matrix. Hence, the fundamental solution matrix isThen, the third equation and the forth equation of system (2) are linearized to beStability of boundary periodic solution depends on the eigenvalue of byand the eigenvalues of areAccording to the Floquet theory of impulsive equation, the boundary periodic solution is locally asymptotically stable if and ; that is,Secondly, we will prove the global attractivity of boundary periodic solution . Let , so as to ; that is.where .
From the first equation of system (2), it is easy to obtainConsider the following comparison system:When is large enough, by Lemmas 2 and 3, we can obtain thatFor , we assume that , and then we haveIntegrating and solving the above system on , we can derive thatTherefore, , then as . When , we can easily know that , so as , where .
Next, we prove as . By , we can obtain . By Lemma 3, we can getLet , we have proved that for any when is large enough. Suppose that as . There exist and for such that . Assume that as , it follows by system (2)Consider the comparison systemBy equation (25), we can obtainwhere and .
By Lemmas 2 and 3, one can obtain that and , as . So, for any , there exists a , such that as . Let , then as . The theorem is completely proved.
4. Permanence of System (2)
Theorem 2. There exists a constant such that and for each solution of system (2) when is large enough.
Proof. Suppose that is any solution of system (2). Let and ; we calculate the upper right derivative of along system (2) and get the following inequality:Therefore, it is bounded. We choose an such thatIt is easy to know thatAccording to Lemma 2, we haveTherefore, is ultimately bounded. There exists a constant such that any solution of system (2) satisfies and for large enough.
Theorem 3. System (2) is permanent ifwhere and .
Proof. Suppose is any solution of system (2); we have proved there exists an such that and . Let ; we can know when is large enough. We will find a constant such that when is large enough, where and is a constant. Next, we consider the following two cases. Firstly, there exists a sufficient small such thatWe can prove is not satisfied for any ; otherwise,Consider the comparison systemIt is not hard to get ,whereTherefore, there exists a such that for . Let and as . It is easy to know thatIntegrating and solving the above system on , we can derive thatThen, ; it contradicts with . Thus, there is a such that .
Secondly, if , then . Otherwise, let , we can easily know when , choose such that , where . Let , then there is a such that . Otherwise, as . From system (25), we can obtainas , so .
Integrating and solving the above system on , we can derive that , so , it contradicts with the assumption . Let , we have . For , as , we discuss it in the same way, at last we obtain , for any .
5. Existence of Positive Periodic Solution
In this section, we will study the existence of positive periodic solution by bifurcation theory of impulsive differential equations. In order to be consistent with [37], we denote , as , , respectively. System (2) is
By formally deriving the equationwhich characterized the dynamics of the unperturbed flow associated to the first two equations in (2), we obtain that
This relation will be integrated in what follows in order to compute the components of explicitly. Firstly, it is clear that
Then, we derive that
The initial condition at being . Here is the identity matrix. It follows that
As , we can get for . We can obtain from (43) that
According to the initial condition, we obtain that
From (4), we obtain that ,which implieswhere , , and arewhere is the root of .
Firstly, we make sure the sign of , where
For determining , we must calculate and .
We have
Then, we have
Therefore, .
Secondly, we make sure the sign of , where
Therefore, we have to calculate ,
Then, we have
In order to make sure the sign of , let
Then,
As , we can obtain . Because is an increasing function, so . According to equation, we can know , thus . According to [37], the following conclusion can be derived.
Theorem 4. System (2) has a critical bifurcation; there exists a positive periodic solution of system (2) if , near .
6. Numerical Simulations
In this section, we investigate the effects of impulsive perturbations on system (2) by using numerical method to illustrate our theoretical results. For convenience, we assume keeping some parametric values of system (2) as , , , , , , and . By simple calculation, we can get for
Now, we choose different parameter to illustrate our main results by numerical simulation. Let , , and ; bounded periodic solution is globally asymptotically stable (see Figure 1); By Theorem 2, consider the comparison system, let , , and , and one can get system (2) is permanent (see Figure 2). By Theorem 4, let nearly to , , and ; system (2) has a critical bifurcation (Figure 3). Let , , and ; the number of predators population of system (2) decreases significantly (Figure 4). Further, take the smaller , , and , and we can obtain that the predator will be extinct (Figure 5).
We simulate the influence of the refuge constant on the system (see Figures 6 and 7) and the influence of pulse time on the system (see Figure 8).
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7. Discussion
We have studied a predator-prey model with Holling II functional response incorporating a prey refuge and impulsive effect. Bounded periodic solution is globally asymptotically stable if . System (2) is permanent if . By Figures 1 and 2, we show the facts above. System (2) has a critical bifurcation; there exists a positive periodic solution of system (2) if , nearby . The facts above are shown in Figure 3. By Theorem 1, one can know that the predator is extinct when is little smaller than , as shown in Figure 5. It is easily known that is a threshold. From the view point of biology, if the harvest period of the prey is too short , that is, the harvest frequency is relatively frequent, the number of the prey is bound to be small, which cannot guarantee the food supply of the predator, resulting in the final extinction of the predator. On the contrary, if the harvest period exceeds the threshold , the food supply of the predator is sufficient, and the system is persistent. This provides a good theoretical basis for how to select appropriate parameters to maintain ecological balance.
The influence of refuge constant on system (2) can be seen in Figures 6 and 7. This shows that the predator becomes extinct and the number of prey remains unchanged when the prey refuge constant reaches a certain value. This shows that when the number of prey entering the shelter reaches a certain amount, the prey that the predator can catch is less and insufficient, and the supply of prey is in short supply, resulting in the extinction of the predator. It is consistent with the reality. The impact of the change of pulse time on system (2) is shown in Figure 8, this shows that the selection of pulse period also has an obvious impact on the system. From of Figure 8, we can see that the predator is persistent when .
The effects of different pulse forms on the dynamic behavior of the system are also very different, so it is necessary for us to study the effects of these different pulse forms on the dynamic behavior of the system. The state feedback control of the system is also close to the real problem, we will study it in the following work.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
ZL and XY designed the study and carried out the analysis. ZL, XY, and SF wrote the paper. XY and SF performed numerical simulations. All authors read and approved the final manuscript.
Acknowledgments
This work was supported by the Natural Science Foundation of Chongqing (Grant no. cstc2021jcyj-msxmX0949) and the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant no. KJQN202101212).