Stochastic Periodic Solution and Persistence of a Nonautonomous Impulsive System with Nonlinear Self-Interaction

Kong, Weili; Shao, Yuanfu

doi:https://doi.org/10.1155/2020/1053401

Discrete Dynamics in Nature and Society

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Abstract Introduction Preliminaries Examples Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 1053401 | https://doi.org/10.1155/2020/1053401

Stochastic Periodic Solution and Persistence of a Nonautonomous Impulsive System with Nonlinear Self-Interaction

Weili Kong¹and Yuanfu Shao²

Academic Editor: Nickolai Kosmatov

Received06 May 2019

Accepted13 Jan 2020

Published10 Feb 2020

Abstract

Considering periodic environmental changes and random disturbance, we explore the dynamical behaviors of a stochastic competitive system with impulsive and periodic parameters in this paper. Firstly, by use of extreme-value theory of quadratic function and constructing suitable functional, we study the existence of periodic Markovian process. Secondly, by comparison theory of the stochastic differential equation, we study the extinction and permanence in the mean of all species. Thirdly, applying an important lemma, we investigate the stochastic persistence of this system. Finally, some numerical simulations are given to illustrate the main results.

1. Introduction

In real world, the habitat space and food for species are relatively scare; hence, the interspecific competition phenomenon exists extensively. Usually, the competitive interaction is assumed to be linear, see [1, 2]. However, some experimental tests showed that the term of self-interaction may be nonlinear, so Ayala et al. [3] proposed the following model:

In practice, the environmental white noise is almost everywhere and inevitably affects the growth of species. Thus, in the progress of mathematical modeling, random disturbance is introduced to reveal the effect of white noise [4–9]. For ecological system, some discrete effects often appear at some short time interval, such as periodic spraying pesticides, harvesting, and stocking, which affect the growth of species and are often modeled by impulsive parameters. In last decades, many impulsive systems have been proposed and many good results have been reported, see, e.g., [10–15] and references cited therein. For example, Tan et al. [15] investigated the existence of solution, stochastic permanence of the following impulsive model:where represents the density of white noise and and are independent standard Brownian motions defined on the probability space , where denotes a filtration, which is right continuous, and all p-null set is contained in . For biological meanings of other parameters, refer to [15]. By constructing suitable functional and using inequality techniques, the stochastic permanence of (2) and extinction of were studied.

In natural world, due to individual life cycle and seasonal variation, the carrying capacity of species, birth rate, and other parameters always present periodic changes for population systems [16–18]. For the determinate biological system, the existence of periodic solution is a very important dynamical behavior [10, 19–22]. Similarly, for stochastic system, it is very interesting to study the existence of stochastic periodic solution (periodic Markovian process). On the other hand, the extinction and permanence in the mean and stochastic persistence in probability are all very important dynamical behaviors (see [12, 13, 23, 24]), but all these are not investigated in [15]. Hence, it is necessary for us to further explore these dynamical behaviors of (2). For this purpose, we give the following assumptions.

Assumption 1. All coefficients are bounded, continuous, and periodic functions with period T.

Assumption 2. The impulsive points satisfy , and there exists an integer q such that

Assumption 3. By the biological meanings, we assume for
For the following stochastic differential equation (see [21]),with initial data , we define the following differential operator:For any bounded and continuous function , we use the following notations:and if is integrable.
The rest of this paper is organized as follows. Section 2 begins with some definitions and lemmas. Section 3 focuses on the existence and uniqueness of the periodic Markovian solution. Section 4 is devoted to the extinction and permanence in the mean of species. The stochastic persistence of (2) is studied in Section 5. Some numerical examples are showed in Section 6 to validate the main results. Finally, a brief conclusion and discussion are given to conclude the paper in Section 7.

2. Preliminaries

In this section, we introduce the definitions of the periodic Markovian process, the solution of the impulsive stochastic differential equation, and some auxiliary results of the existence of the periodic Markovian process.

Definition 1. (see [21]). Let be a stochastic process and be a finite sequence of numbers. If the joint distribution of random variables is independent of h, where , then is said to be T-periodic stochastic process.

Definition 2. (see [13, 15]). The impulsive stochastic equation is given as follows:with . A stochastic process , is said to be a solution of the above system, if(i) is continuous on and and -adapted, and , and almost surely for all (ii) and exist, and with probability one, .(iii) satisfies the following integral equation:for and satisfiesfor .

Lemma 1 (see [13, 21]). For the following Itô’s differential equation,all coefficients of (9) satisfy linear growing condition and the Lipschitz condition in every cylinder and are -periodic in t. Furthermore, there exists a -periodic and once continuously differentiable function in t, which is twice continuously differentiable with respect to x and satisfies the following conditions:and then there exists a solution of (9) which is -periodic Markov process.

Lemma 2 (see [14]). Suppose that and (a)If there exist such that, for all , then(b)If there exist such that, for all , thenTo investigate (2), we consider the following nonimpulsive system:where

Assume that the product equals unity if the number of factors is zero. Obviously, are both T-periodic (for details, see [20]).

For (2) and (15), we have the following.

Lemma 3. Let :(1)If is solution of (2), then is solution of (15)(2)If is solution of (15), then is solution of (2).

Remark 1. The proof is similar to that of Theorem 3.1 in [20] and is omitted. Lemma 3 reveals the equivalence of (2) and (15), and hence, we will consider (15) later.

Lemma 4. For any given initial value if , then system (2) has a unique solution on and the solution remains in with probability one.

Remark 2. By reference [6], the existence of solutions of (15) can be derived; then, by Lemma 3, the required assertion is obtained.

3. Existence of Stochastic Periodic Solution

Theorem 1. Suppose the following condition holds:where and are defined later; then, there exists a periodic Markovian process for system (2).

Proof. Obviously, Lemma 4 implies the existence of positive solutions of (2); then, according to Lemma 3, it is only needed to prove the solution of (15) is a periodic Markovian process. By Lemma 1, it suffices to find a -function and a closed set such that all conditions of Lemma 1 hold for (15). Definewhere is a continuous differentiable function satisfyingLet , then it is not difficult to verify that is T-periodic function on andwhere . Hence, the first condition of Lemma 1 is satisfied. Now, we are in the progress of proving the second condition of Lemma 1. Using Itô’s formula on and , respectively, we obtainTherefore,DenoteThus,For any small positive define a closed setwhere is compact and its component in whichDiscuss LV as follows:(i)If , by , we choose small enough such that and hence, .(ii)If , by again, for small enough, we have Therefore, .(iii)If or , by the monotonicity, obviously, To summarize, the conditions of Lemma 1 are all satisfied and the required assertion is directly derived. This completes the proof.

Remark 3. Based on the existence theorem of periodic Markovian process and extreme-value theory of quadratic function, the sufficient conditions assuring the existence of stochastic periodic solution are established, which is not discussed in [15]. Theorem 1 shows that impulsive disturbance and stochastic disturbance affect the periodic behavior of (2), which is shown in Figures 1(g) and 1(h) in Section 6.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

4. Extinction and Permanence in the Mean

Lemma 5. The solution of (2) with initial value satisfiesfor any , where is a positive constant. Furthermore,that is, the solution of (2) is stochastically ultimately bounded.

Proof. By Lemma 3, it is only needed to study the equivalent system (15). Define for and . Applying Itô’s formula to , we obtainIntegrating both sides of the above inequality from 0 to t and taking expectation yields,By comparison theorem,which means the existence of constants and such thatHence, there exists ; for any , we have Furthermore, by the continuity of and , for any , there exist and such that and Let and thenhold for all . Applying Chebyshev inequality, we obtainwhich means (15) is stochastically ultimately bounded. This completes the proof.

Theorem 2. For system (2), let then the following results hold:(i)If then all species are extinct, i.e., (ii)If then is permanent in the mean and is extinct, i.e., (iii)If then is extinct and is permanent in the mean, i.e., and (iv)If then are both permanent in the mean, i.e., , where and are defined as before and , and are all constants defined later in the proof, .

Proof. By Lemma 3, it suffices to prove that there exist some positive constants such that these conclusions hold for system (15). Above all, applying Itô’s formula to , we haveIntegrating both sides of (37) from 0 to t, we obtainSince , under using comparison theorem of stochastic differential equation, we obtainOn the other hand, by (15), we haveIn the same manner, under condition , we have(i)If , then it directly follows from (39) and (41) that . That is, all species of (15) are extinct.(ii)If , then . Using (37) again, we have Applying Lemma 2 yields Therefore, holds, that is, species is permanent in the mean.(iii)If , then . From (40), we have By assumption , we obtain(iv)If then using (37) again, we haveLemma 2 impliesSimilarly, we derive from (41) thatUsing Lemma 2 yieldsCombining (39) and (41), holds for .
To summarize the above discussion and combine Lemma 3, the required results are obtained. The proof is completed.

Remark 4. The result of extinction for all species (case (i)) is in accordance with Theorem 3.2 of [15], but other dynamics such as permanence in the mean for all species (case (ii)–case (iv)) is not studied in [15], which reveals richer dynamical behaviors of this system.

5. Stochastic Persistence in Probability

Firstly, we give the following lemmas.

Lemma 6. The solutions of (2) are uniformly continuous.

Proof. Using the property of expectation and Lemma 5, we haveBy stochastic integral inequality, for and , we obtainHence, when , and we havewhere Therefore, the solution of (15) is uniformly continuous. Similarly, we can obtain the uniform continuity of . Lemma 3 implies the required assertion. This completes the proof.

Lemma 7 (see [25]). Let f be a nonnegative function defined on such that f is integrated and uniformly continuous, then .

Theorem 3. Supposethen the solution of (2) is globally attractive. Furthermore, for any other solution , we have

Proof. Similarly, we only prove the conclusion for (15). Define . Applying Itô’s formula and computing the right derivative of function along the solution of (15) yieldwhere is the of and , i.e., . According to , there exist and such that for ; hence, for . Integrating both sides of it from to t leads toHence,That is,Therefore, the global attractiveness of (15) is obtained by Lemmas 6 and 7.
On the other hand,and hence, is uniformly continuous. The uniform continuity of can be similarly obtained. According to (58) and Barbalat’s conclusion [25],This completes the proof.
Finally, we discuss the stochastic persistence in probability of (2). For the systemwhere , is an m-dimensional standard Brownian motion. Suppose is a set such that for . Denote the set of the states where the size of at least one species is less than or equal to η by . stands for the states where the size of one or more populations is 0. Let be the occupation measure of , where represents a Dirac measure at .

Definition 3. (see [26, 27]). For any , if there exists such that with a.s., then we say (61) is stochastically persistent.

Remark 5. The definition of shows one or more populations have a density less than η; then, stochastic persistence means that all populations spend an arbitrarily small fraction of time at arbitrarily low densities. It is more appropriate than permanence in the mean and Definition 2 of [15], see [27].

Lemma 8 (see [26, 27]). Let System (61) is said to be stochastically persistent in probability if there exists a unique invariant probability measure μ such that and the distribution of converges to μ as whenever .

Theorem 4. Suppose and , then (2) is stochastically persistent in probability, where are defined in Theorem 2.

Proof. Firstly, the proof of stochastic persistence in probability is motivated by [20]. Let be the solution of (15) with initial data , be the transition probability of , and be the probability of events . The stochastic boundedness of implies there exists a compact subset such that for any given , that is, is tight. represents the probability measure on ; then, we prove the series is Cauchy in equation.
For any , define the metric of and as follows:whereFor any and ,where . Because of the tightness of , one can find sufficiently large K such that for any . Then,By the global attractivity (Theorem 3), for arbitrarily given and sufficiently large t, we haveConsequently, for any and sufficiently large t,Hence, for any initial data, is Cauchy in with the metric and there exists a probability measure such that, for fixed initial data Finally, by the triangle inequality, we haveIn view of the global attractivity, we can deduceTherefore, for any , that is, for any , there exists unique probability measure such that the transition probability of converges weakly to as . Therefore, (15) is asymptotically stable in distribution.
Meanwhile, Theorem 2 impliesand hence, (15) is stochastically persistent in probability by Lemma 8. Applying Lemma 3 yields the required assertion. This completes the proof.

6. Examples and Simulations

In this section, by use of the numerical method [28], we give some simulations.

For system (2), with no special mention, we always take , .

An easy computation shows holds. Theorem 1 implies that the solution of (2) is a periodic Markovian process. Using the Milstein method and writing MATLAB code, we get the simulation result of periodic solution of (2), see Figure 1. Under , Figures 1(a) and 1(b) are the time series graphs of and for impulsive and stochastic case, respectively. Figures 1(c)–1(f) are the phase graphs of the periodic solution of deterministic system with , impulsive system with stochastic system with , and stochastic impulsive system with and as before, respectively. If that is, the stochastic disturbance is large, then the periodic process does not appear, illustrated in Figure 1(g). If then there exists no periodic process, see Figure 1(h). Figures 1(g) and 1(h) show that impulse and stochastic perturbation affects the existence of stochastic periodic solution.

Next, we simulate the extinction and permanence in the mean of solution of (2). Let or then . Hence, it follows from Theorem 2 that and are both extinct, illustrated in Figures 2(a) and 2(b). Let then . By Theorem 2 again, is extinct and is permanent in the mean, illustrated in Figure 2(c). Let then . Using Theorem 2 yields that is permanent in the mean and is extinct, illustrated in Figure 2(d). Let then . Similarly, Theorem 2 implies and are both permanent in the mean, see Figures 2(e) and 2(f). Figure 2 shows that the impulsive disturbance and stochastic disturbance bring some important influence to the dynamics of all species.

(a)

(b)

(c)

(d)

(e)

(f)

Finally, we simulate the global attractivity and stochastic persistence in probability of solution of (2). By computation, all parameters meet the conditions of Theorem 4; therefore, the solution of (2) is globally attractive (see Figures 3(a) and 3(b)) and the distribution of all species are asymptotic stability (see Figure 4(a)). The stochastic persistence is simulated in Figures 4(b)–4(d), which are the time series graphs of , , and , respectively.

(a)

(b)

(a)

(b)

(c)

(d)

7. Conclusions and Discussions

In this paper, for the stochastic competitive system with impulsive and nonlinear term of self-interaction proposed in [15], we further study such dynamics as the existence of periodic solution, the extinction and permanence in the mean, the global attractivity of solutions, and stochastic persistence of this system. Theorem 1 gives the sufficient conditions of the existence of periodic Markovian process. Theorem 2 gives the conditions of extinction and permanence in the mean of both species. Theorem 3 establishes the condition assuring the global attractivity. Theorem 4 establishes the condition of stochastic persistence in probability of this system. Finally, simulations (Figures 1–4) are given to verify the obtained results. Our main results are new and different from [15], which is presented by giving Remarks 3, Remark 4, and Remark 5 in detail.

Three or more species often coexist in the real world, and time delays often appear in biological system, then how to deal with the effects of time delays on the stochastic bahaviors of three-species biological system is very interesting to be further investigated. On the other hand, regime switching is another common random perturbation, e.g., stochastic hybrid phytoplankton-zooplankton model with toxin-producing phytoplankton, stochastic tumor-immune model with regime switching, and impulsive perturbations. 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 they have no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

WK carried out all studies and drafted the manuscript. YS conceived the study, performed the simulation, and helped to draft the manuscript.

Acknowledgments

This work was supported by the Natural Science Foundation of China (11861027).

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Copyright © 2020 Weili Kong 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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