Stochastic Periodic Solution andPersistence of aNonautonomous Impulsive System with Nonlinear Self-Interaction

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 periodicMarkovian process. Secondly, by comparison theory of the stochastic differential equation, we study the extinction and permanence in the mean of all species. -irdly, applying an important lemma, we investigate the stochastic persistence of this system. Finally, some numerical simulations are given to illustrate the main results.


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.
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][17][18]. For the determinate biological system, the existence of periodic solution is a very important dynamical behavior [10,[19][20][21][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 a 1 (t), a 2 (t), b 1 (t), b 2 (t), a(t), σ 1 (t), and σ 2 (t) are bounded, continuous, and periodic functions with period T. Assumption 2. e impulsive points satisfy 0 < t 1 < t 2 < · · ·, lim i⟶∞ t i � +∞, and there exists an integer q such that t k+q � t k + T and λ i,k+q � λ ik , i � 1, 2, k ∈ N. Assumption 3. By the biological meanings, we assume For the following stochastic differential equation (see [21]), with initial data x(t 0 ) � x 0 ∈ R n , we define the following differential operator: For any bounded and continuous function f(t), we use the following notations: e 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. e 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.

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.
x(t) satisfies the following integral equation: (7) for t ∈ [0, t 1 ] and satisfies 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 U l × R + (l > 0, U l � x: ‖x‖ ≤ l { }) and are T-periodic in t. Furthermore, there exists a T-periodic and once continuously differentiable Discrete Dynamics in Nature and Society differentiable with respect to x and satisfies the following conditions: and then there exists a solution of (9) which is T-periodic Markov process.
(a) If there exist t 0 > 0 and λ 0 > 0 such that, for all t > t 0 , then then To investigate (2), we consider the following nonimpulsive system: where Assume that the product equals unity if the number of factors is zero. Obviously, A 1 (t)and A 2 (t) are both T-periodic (for details, see [20]).

Remark 1.
e proof is similar to that of eorem 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 (x 1 (0), x 2 (0)) ∈ R 2 + , if a l 11 > b u 1 A u 1 and a l 22 > b u 2 A u 2 , then system (2) has a unique solution (x 1 (t), x 2 (t)) on t ≥ 0 and the solution remains in R + 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.
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 C 2 -function Discrete Dynamics in Nature and Society V(t, u 1 , u 2 ) and a closed set U ∈ R 2 + such that all conditions of Lemma 1 hold for (15). Define where w i (t) is a continuous differentiable function satisfying where . 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 V 1 (t, u 1 ) and V 2 (t, u 2 ), respectively, we obtain erefore, Denote us, For any small positive ε < 1, define a closed set where D ε is compact and its component Discuss LV as follows: and hence, LV < − 1.
Discrete Dynamics in Nature and Society erefore, To summarize, the conditions of Lemma 1 are all satisfied and the required assertion is directly derived. is 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]. eorem 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.

Extinction and Permanence in the Mean
that is, the solution of (2) is stochastically ultimately bounded.
Proof. By Lemma 3, it is only needed to study the equivalent system (15).

Remark 4.
e result of extinction for all species (case (i)) is in accordance with eorem 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.

Stochastic Persistence in Probability
Firstly, we give the following lemmas.
Proof. Using the property of expectation and Lemma 5, we have By stochastic integral inequality, for 0 < t 1 < t 2 and p > 2, we obtain □ Lemma 7 (see [25]). Let f be a nonnegative function defined on R + such that f is integrated and uniformly continuous, then lim t⟶∞ f(t) � 0.
Applying Itô's formula and computing the right derivative of function V(u 1 , u 2 ) along the solution of (15) yield where ζ i is the sup of u 1 and u 2 , i.e., u i ≤ ζ i (i � 1, 2). According to (H 2 ), there exist ρ > 0 and t 0 > 0 such that a 11 (t) − a 21 (t) > ρ and a 22 (t) − a 12 (t) > ρ for t ≥ t 0 ; hence, Integrating both sides of it from t 0 to t leads to (56) Hence, at is, (58) erefore, the global attractiveness of (15) is obtained by Lemmas 6 and 7.
On the other hand, and hence, E(u 1 ) is uniformly continuous. e uniform continuity of E(u 2 ) can be similarly obtained. According to (58) and Barbalat's conclusion [25], is completes the proof. Finally, we discuss the stochastic persistence in probability of (2). For the system where x(t) � (x 1 (t), x 2 (t), . . . , x m (t)), (B 1 (t), B 2 (t), . . . , B m (t)) is an m-dimensional standard Brownian motion. Suppose S ⊂ R n + is a set such that x(t) ∈ S for t ≥ 0. Denote the set of the states where the size of at least one species is less than or equal to η by S η � x ∈ S: x i ≤ η for some i . □ Definition 3 (see [26,27]). For any ε > 0, if there exists η > 0 such that lim sup t⟶∞ t (S η ) ≤ ε with X(0) ∈ S 0 a.s., then we say (61) is stochastically persistent.

Remark 5.
e definition of S η 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].

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

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. eorem 1 gives the sufficient conditions of the existence of periodic Markovian process. eorem 2 gives the conditions of extinction and permanence in the mean of both species. eorem 3 establishes the condition assuring the global attractivity. eorem 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.
ree 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.   Figure 4: e distribution and stochastic persistence for x 1 (t) and x 2 (t) of (2) with initial data x 1 (0) � x 2 (0) � 2. (a) e density graph of x 1 (t) and x 2 (t), (b) the time series graph of x 1 (t), (c) the time series graph of x 2 (t), and (d) the time series graph of z(t) � (x 2 1 (t) + x 2 2 (t)) 1/2 .

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