Stochastic Periodic Solution and Permanence of a Holling–Leslie Predator-Prey System with Impulsive Effects

Zhao, Jinxing; Shao, Yuanfu

doi:https://doi.org/10.1155/2021/6694479

Journal of Mathematics

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Abstract Introduction Preliminaries Examples Conclusions and Discussion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 6694479 | https://doi.org/10.1155/2021/6694479

Stochastic Periodic Solution and Permanence of a Holling–Leslie Predator-Prey System with Impulsive Effects

Jinxing Zhao¹and Yuanfu Shao²

Academic Editor: Yongqiang Fu

Received16 Oct 2020

Revised11 Dec 2020

Accepted18 Dec 2020

Published05 Jan 2021

Abstract

Considering the environmental effects, a Holling–Leslie predator-prey system with impulsive and stochastic disturbance is proposed in this paper. Firstly, we prove that existence of periodic solution, the mean time boundness of variables is found by integral inequality, and we establish some sufficient conditions assuring the existencle of periodic Markovian process. Secondly, for periodic impulsive differential equation and system, it is different from previous research methods, by defining three restrictive conditions, we study the extinction and permanence in the mean of all species. Thirdly, by stochastic analysis method, we investigate the stochastic permanence of the system. Finally, some numerical simulations are given to illustrate the main results.

1. Introduction

Predator-prey phenomenon is very popular in natural world, and recently more and more researchers pay attention to investigate the complicated dynamical behaviors between predator and prey species, which has been and is long to be one of the most important hot topics in the future [1]. Hsu and Huang [2] proposed the following Holling–Tanner model:where are the densities of prey and predator at time , respectively; is the intrinsic growth rate of prey or predator; and is the density-dependent coefficient; represents Holling type II functional response, where denote the capturing rate and half capturing saturation constant, respectively. Function is the Leslie–Gower term, which measures the loss in the predator population due to rarity of its favorite food, where is the carrying capacity. For more biological meanings of this model, see [3–5].

Moreover, in natural world, there are many kinds of functional responses. If the functional response is Holling type IV, then the model reads

Its dynamics has been sufficiently studied, and many better results have been obtained by Li and Xiao [6].

On the contrary, in practice, the environmental white noise almost exists everywhere. For ecological system, the growth rate of population is inevitably affected by the white noise. In order to reveal the effect of white noise, random disturbance is introduced in many mathematical models [7–12]. Meanwhile, due to the individual life cycle and seasonal variation and so on, the birth rate, death rate, carrying capacity of species, and other parameters always exhibit cycle changes [13–15]. Therefore, Jiang et al. [16] proposed the following nonautonomous stochastic model:where are all positive -periodic functions; and are independent standard Brownian motions defined on the probability space with a filtration satisfying the usual conditions (i.e., it is right continuous, and contains all -null set); and denotes the density of white noise. Parameter or means Holling type II or Holling type IV functional response, respectively.

However, for population system, the effect from natural or man-made factors is very popular, and hence, the growth of species will have to suffer from some discrete changes of relatively short time interval at some periodic times, such as stocking and harvesting. These effects are often modeled by impulsive parameters. In the last decades, many impulsive dynamical systems have been proposed and many better results have been reported, see, [17–23] and references therein. For example, the authors Zuo and Jiang [23] investigated the periodic solution and boundary periodic solution of the following impulsive model:

Inspired by the above discussion, considering some natural or man-made impulsive factors, we propose the following stochastic predator-prey model:where all coefficients are bounded, continuous, and periodic with period . The impulsive points satisfy , , and there exists an integer such that . Furthermore, by the biological meanings, we assume for . For the biological meanings of all parameters, refer to [2, 6, 16, 23].

Our main aim of this paper is as follows: firstly, for determinate system, the existence of equilibrium or periodic solution is an important topic for the dynamics of biological system [24–27]. Similarly, for stochastic system, it is very interesting to study whether there exists a periodic Markovian process or not.

Secondly, for predator-prey system, the dynamical behaviors are another important topic [28–30]. By the comparison method, we establish some sufficient conditions assuring the extinction, permanence in the mean of all species, and the stochastic permanence of system (5).

The rest of the work of this paper is organized as follows: Section 2 begins with some notations, definitions, and important lemmas. Section 3 is devoted to the existence and uniqueness of the periodic Markovian process. Section 4 focuses on the extinction and permanence in the mean of species. Section 5 focuses on the stochastic permanence of system (5). Some numerical simulations are given to verify our main results in Section 6. Finally, we conclude this paper with a brief conclusion and discussion in Section 7.

2. Preliminaries

For an -dimensional stochastic differential equation [13],with initial value , where is an -dimensional standard Brownian motion. The differential operator associated (6) is defined by

For bounded and continuous function , setand if is integrable, then define

To investigate the dynamics of (5), we consider the following nonimpulsive system:where

It is easy to show that both are periodic functions with period (for details, see [27]). We assume the product equals unity if the number of factors is zero and stands for a sufficiently small positive constant whose value may be different at different places.

Now we present the definitions of periodic Markovian process and the solution of impulsive stochastic differential equation, and some auxiliary results of the existence of periodic Markovian process.

Definition 1. (see [13, 27]). A stochastic process is said to be periodic with periodic if for every finite sequence of numbers , the joint distribution of random variables is independent of , where .

Definition 2. (see [22, 27]). System (5) is stochastically permanent if for every , there is a pair of constants and such that for any initial data , the solution of (5) has the property thatwhere represents the probability of the events.

Definition 3. (see [20, 22]). For the following impulsive stochastic differential equationwith initial value , a stochastic process , is said to be a solution of the above system, if(i) is adapted and is continuous on and each interval , , where is all -valued measurable -adapted processes satisfying almost surely for all .(ii)For every , and exist, and with probability one.(iii)For all obeys the integral equationand for all obeys the following integral equation:

Lemma 1. (see [13, 27]). For the following Itô’s differential equationif all the coefficients are periodic in and satisfy the linear growing condition and the Lipschitz condition in every cylinder for , where and there exists a function which is twice continuously differentiable with respect to and once continuously differentiable with respect to in , T is periodic in and satisfies the following conditions:then there exists a solution of (16) which is -periodic Markovian process.

Lemma 2. (see [20]). Suppose that and .(a)If there exist two positive constants such that, for all , then(b)If there exist some constants such that, for all ,then

Lemma 3. Let ; then(1)if is the solution of (5), then is the solution of (10).(2)if is the solution of (10), then is the solution of (5).

Remark 1. The proof is similar with that of [20] and is omitted. Lemma 3 shows that the dynamics of (5) is equivalent to that of (10). Hence, in the later, we mainly consider (10) to reveal the dynamical properties of (5).
As to the existence of nontrivial positive solution of (5), we have Lemma 4.

Lemma 4. For any given initial value , system (5) has a unique solution on , and the solution remains in with probability one.

Remark 2. Lemma 3 implies that the existence of solution of (5) on is equivalent to the existence of the solution of (10). The proof of the existence of of (10) is similar with that of [23] and is omitted here.

3. Existence of Periodic Solution

In this section, we focus on the existence of periodic Markovian process of (5). Above all, we give the following assumption.

Theorem 1. Suppose the following condition holds,then there exists a solution for system (5), which is a -periodic Markovian process.

Proof. According to the equivalent property and existence of solutions (Lemmas 3 and 4), we only need to prove that, under , the solution of system (10) is a periodic Markovian process. Lemma 1 shows that it suffices to find a -function and a closed set such that all conditions of Lemma 1 hold. Definewhere are two constants defined later and and are the positive continuous function such thatIt is not difficult to verify that is -periodic on , andwhere . Hence is -periodic and satisfies the first condition of Lemma 1. Applying the Itô’s formula to and , thenLet , then we havewhere are defined as above, andTake , thenChoose any small positive and such thatDefine an open subset as follows:Obviously, is compact and its component , whereIt is easy to verify that when , or , or , or . Therefore, holds for any , which means the second condition of Lemma 1 holds. Using Lemma 1, then the existence of periodic solution of (10) is obtained. This completes the proof.

Remark 3. For system (5), if there is no impulsive effect, i.e., , by Theorem 1, we can obtain the sufficient conditions assuring the existence of -periodic solution, which is in accordance with Theorem 3 in Reference [16]. And if , i.e., the case of Holling type II functional response, Theorem 1 yields the same result as Theorem 3.3 in reference [23]. It is in this sense that we improved or generalized the main results in [16, 23].

4. Extinction and Permanence in the Mean

In this section, we discuss the extinction and permanence in the mean of system (5). Firstly, we provide a lemma on the presentation of the solution for an impulsive stochastic differential equation.

Lemma 5. For the following periodic impulsive differential equationlet be a solution with any given initial data . Then is a unique positive -periodic solution such that , where

Remark 4. The existence of -periodic solution can be obtained by Theorem 1. The presentation of and the global attractivity are referred to [21].

Lemma 6. For the solution of system (5), we havethat is, the solution of (5) is stochastically ultimately bounded.

Proof. From the first equation of (5), we haveConsider the following comparison system:By Lemma 5, the solution of (37) is positive and -periodic, right continuous, and globally attractive. Therefore, has maximum value and minimum value. Define , then by comparison theorem for stochastic equation, for any sufficiently small positive , we haveTherefore, is established.
On the contrary, using , we can obtain from (5) thatIn the same manner, we havewhere is the solution of the following stochastic comparison system:and . Therefore, . This completes the proof.
For convenience, denote

Theorem 2. For system (5), the following results hold:(i)If , then , that is, species is bounded in the mean.(ii)If , then and , that is, species is permanent in the mean.(iii)If , then , that is, species is extinct, and if , then , that is, species is extinct.(iv)If , then and . If , then and , that is, species is permanent in the mean, where are constants defined later in the proof.

Proof. Make use of Lemma 3, we only need to prove these conclusions hold for (10) for some constants . For system (10), by applying the Itô’s formula to and , we have(i)If , then integrating both sides of (43) from 0 to yields Since , then we obtain from Lemma 2 that(ii)Lemma 4 implies the solution of (10) satisfies , and hence there exists a positive constant such that . From (44), we have which leads to Under the condition , we obtain from Lemma 2 that On the contrary, combining Lemma 6, we have Integrating both sides of above inequality yields If , using Lemma 2 again, then Therefore, species is permanent in the mean.(iii)If , by Case (i), obviously . If , similarly, by the proof of Case (ii), we have .(iv)If , then Case (i) indicates . If , by use of Case (ii), we have , then (43) readsand hence,According to Lemma 2, thenIf , then obviously , and hence . Therefore,Furthermore, if , then from (33) and Lemma 6, we haveand hence,ThenCombining , then species is permanent in the mean.
To summarize, the above conclusions hold for system (10). Using Lemma 3, the required assertion is directly obtained. The proof is completed.

Remark 5. For system (5), if the impulsive is absent, then Theorem 2 implies the sufficient conditions of the extinction of species or , which is accordance with Theorem 2 in [16].

5. Stochastic Permanence

In this section, we consider the stochastic permanence of (5).

Theorem 3. If , then system (5) is stochastically permanent.

Proof. This proof is motivated by Reference [21]. Let , , then Lemma 6 implies that . Hence, for any , there exists such that for any .
On the contrary, by the assumption , we can choose a positive constant such thatFirstly, we prove that there exists , for any , we haveSuppose it is not true, i.e., there exist a solution with initial data , a positive integer , and a nonempty set with such that for any and . From (44), we haveThen Lemma 2 impliesOn the contrary,Integrating (64) from to yieldsThen we derive from (65) thatThen we conclude as , which contradicts with the boudedness of . Hence we have for any .
Secondly, we prove there exists a constant such thatWe claim that holds for all . If not, then there exists a sequence and a nonempty set with , and a positive integer such that for all , where is the solution of (10) with initial data . Using (64), for all , we haveAccording to the previous proof, there is a nonempty set where , and two positive integers and with such thathold for all . DefineThen , and for any and , we haveIntegrating both sides of (65) from to yieldswhich leads to as .

On the contrary, by the given condition, there exist two constants and such that for all and , we have

Proof. where .
For any and , we haveThe previous proof shows that there exists a positive integer such that for any and . Due to , then there exists a positive integer such that for any , and hencefor any and . Using (65) again, we havewhich leads toObviously it implies a contradiction, and our claim is obtained. ThereforeIn a similar way, we can derive that . Finally, making use of Lemma 3 yielding the required assertion. The proof is completed.

6. Examples and Simulations

In this section, we give some examples and apply the Melstein method [31] to illustrate our theoretical results and reveal the effects of random disturbance and impulsive factors. For system (5), except some special mentions later, we always take

For system (5), if , then we get the stochastic Holling–Tanner system. By computation,

Then condition holds, and Theorem 1 implies that the solution of the stochastic Holling–Tanner system is a -periodic Markovian process (Figure 1). Figures 1(a) and 1(b) are the time series graphs of and , respectively. Under the condition holding, Figures 1(c)–1(f) are the phase graphs of periodic solution of deterministic system , impulsive system , stochastic system , and stochastic impulsive system , respectively. Similarly, the conditions of Case (iv) of Theorems 2 and 3 hold, and the system is permanent in the mean and stochastically permanent, see Figures 2(a), 2(b), and 3, respectively. On the contrary, let all parameters are as before except or , then an easy computing yields ; therefore, and are both extinct, illustrated in Figures 4(a) and 4(b), respectively. If , then Theorem 2 indicates is always permanent (Figure 5(a)). If , then Theorem 2 implies that is permanent, but is extinct (Figure 5(b)). For system (5), if , then we get a stochastic periodic Holling type IV predator-prey system. It is clear that Theorem 1 implies that the solution is a periodic Markovian process (Figure 6). Figures 6(a)–6(d) are the phase graphs of periodic solution of deterministic system , impulsive system , stochastic system , and stochastic impulsive system , respectively. Similarly, the conditions of Case (iv) of Theorems 2 and 3 hold, and the system is permanent in the mean and stochastically permanent, see Figures 7(a), 7(b), and 8, respectively.

(a)

(b)

(c)

(d)

(e)

(f)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(c)

(d)

(a)

(b)

7. Conclusions and Discussion

In this paper, we study a stochastic predator-prey system with impulsive effects and Holling type II or Holling type IV functional responses, which contains many models such as those in [16, 23]. Theorem 1 gives the sufficient conditions of the existence of periodic Markovian process. Theorem 2 represents the extinction and permanence in the mean of predator and prey species. Theorem 3 shows the stochastic permanence of this system. Finally, by writing Matlab codes, some simulations (Figures 1 and 8) are provided to verify the main results. Our numerical examples reveal that impulsive and stochastic factors bring much influence to the dynamics of this system.

By comparison analysis, we give Remarks 3 and 5 to show that our main results improve or generalize the corresponding results in [16, 23]. We apply stochastic analysis techniques instead of constructing some suitable functionals to study the stochastic permanence, which is less applied and relatively new in some sense. In the process of our analysis, Holling-type functional responses bring some difficulties, and we apply inequality techniques to overcome them. In view of too many kinds of functional responses, then how to deal with other functional response such as Beddington-DeAngelis type? Further, time delays often appear in biological models, and how to discuss the effect of time delays? All these are necessary and very interesting for us to study in the future.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported partially by Natural Science Foundation of China (Grant nos. 11861027 and 11965014) and Inner Mongolia Natural Science Foundation (Grant no. 2018MS01017).

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Copyright © 2021 Jinxing Zhao and Yuanfu Shao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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