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

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.


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 x(t) and y(t) are the densities of prey and predator at time t, respectively; a i (i � 1, 2) is the intrinsic growth rate of prey or predator; and b 1 is the density-dependent coefficient; ( ay(t)/b + x(t) ) represents Holling type II functional response, where a, b > 0 denote the capturing rate and half capturing saturation constant, respectively. Function ( b 2 y(t)/x(t) ) is the Leslie-Gower term, which measures the loss in the predator population due to rarity of its favorite food, where (a 2 x/b 2 ) is the carrying capacity. For more biological meanings of this model, see [3][4][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][8][9][10][11][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][14][15]. erefore, Jiang et al. [16] proposed the following nonautonomous stochastic model: where a i (t), b i (t), σ i (t)(i � 1, 2), a(t), and b(t) are all positive T-periodic functions; B 1 (t) and B 2 (t) are independent standard Brownian motions defined on the probability space (Ω, F, F t t ≥ 0 , P) with a filtration F t t ≥ 0 satisfying the usual conditions (i.e., it is right continuous, and F 0 contains all p-null set); and σ 2 i (i � 1, 2) denotes the density of white noise. Parameter n � 1 or n � 2 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. ese 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][18][19][20][21][22][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: Journal of Mathematics where all coefficients a 1 (t), a 2 (t), b 1 (t), b 2 (t), a(t), b(t), σ 1 (t), and σ 2 (t) are bounded, continuous, and periodic with period T. e impulsive points satisfy 0 < t 1 < t 2 < · · ·, lim k⟶∞ t k � +∞, and there exists an integer p such that t k+p � t k + T, λ i,k+p � λ ik , i � 1, 2. Furthermore, by the biological meanings, we assume ξ ik > − 1 for i � 1, 2. 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][25][26][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][29][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). e 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.

Preliminaries
For an n-dimensional stochastic differential equation [13], with initial value x(t 0 ) � x 0 ∈ R n , where B(t) is an n-dimensional standard Brownian motion. e differential operator L associated (6) is defined by For bounded and continuous function f(t), set To investigate the dynamics of (5), we consider the following nonimpulsive system: where It is easy to show that A 1 (t) and A 2 (t) both are periodic functions with period T (for details, see [27]). We assume the product equals unity if the number of factors is zero and ε 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 3 (see [20,22]). For the following impulsive stochastic differential equation with initial value x(0) � x 0 , a stochastic process . , x n (t)) T , t ∈ R + , is said to be a solution of the above system, if (14) and for all t ∈ [t k , t k+1 ], k ∈ N, x(t) obeys the following integral equation: Lemma 1 (see [13,27]). For the following Itô's differential equation if all the coefficients are T− periodic in t and satisfy the linear growing condition and the Lipschitz condition in every cylinder U l × R + for l > 0, where U l � x: ‖x‖ ≤ l { } and there exists a function v � v(t, x) which is twice continuously differentiable with respect to x and once continuously differentiable with respect to t in R n × R + , T is periodic in t and satisfies the following conditions: then there exists a solution of (16) which is T-periodic Markovian process.
is the solution of (5).

Remark 1.
e 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.
Remark 2. Lemma 3 implies that the existence of solution (x(t), y(t)) of (5) on t ≥ 0 is equivalent to the existence of the solution (X(t), Y(t)) of (10). e proof of the existence of (X(t), Y(t)) of (10) is similar with that of [23] and is omitted here.

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 T-periodic Markovian process.
Proof. According to the equivalent property and existence of solutions (Lemmas 3 and 4), we only need to prove that, under (H 1 ), the solution of system (10) is a periodic Markovian process. Lemma 1 shows that it suffices to find a C 2 -function V(t, X, Y) and a closed set U ∈ R 2 + such that all conditions of Lemma 1 hold. Define where q and c are two constants defined later and μ 1 (t) and μ 2 (t) are the positive continuous function such that It is not difficult to verify that where is T-periodic and satisfies the first condition of Lemma 1. Applying the Itô's formula to V 1 (t, X) and V 2 (t, Y), then Journal of Mathematics where ζ and η are defined as above, and Journal of Mathematics Choose any small positive ε 1 and ε 2 such that Define an open subset as follows: Obviously, D ε 1,2 is compact and its component It is easy to verify that Φ(X, Y) ⟶ − ∞ when X ⟶ 0 + , or Y ⟶ 0 + , or X ⟶ ∞, or Y ⟶ ∞. erefore, LV(X, Y) ≤ − 1 holds for any (X, Y) ∈ R 2 + /D ε , which means the second condition of Lemma 1 holds. Using Lemma 1, then the existence of periodic solution of (10) is obtained. is completes the proof. □ Remark 3. For system (5), if there is no impulsive effect, i.e., ξ ik � 0(i � 1, 2), by eorem 1, we can obtain the sufficient conditions assuring the existence of T-periodic solution, which is in accordance with eorem 3 in Reference [16]. And if n � 1, i.e., the case of Holling type II functional response, eorem 1 yields the same result as eorem 3.3 in reference [23]. It is in this sense that we improved or generalized the main results in [16,23].

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 equation let x(t) be a solution with any given initial data Remark 4. e existence of T-periodic solution can be obtained by eorem 1. e presentation of x(t) and the global attractivity are referred to [21].

Lemma 6.
For the solution (x(t), y(t)) of system (5), we have that is, the solution of (5) is stochastically ultimately bounded.
Proof. From the first equation of (5), we have Consider the following comparison system: By Lemma 5, the solution u(t) of (37) is positive and T-periodic, right continuous, and globally attractive. erefore, u(t) has maximum value and minimum value. Define u * � max t⟶+∞ u(t), then by comparison theorem for stochastic equation, for any sufficiently small positive ε > 0, we have Journal of Mathematics On the contrary, using x(t) ≤ β 1 , we can obtain from (5) that In the same manner, we have where v(t) is the solution of the following stochastic comparison system: and v * � max t⟶+∞ v(t). erefore, limsup t⟶+∞ P y(t) < β 2 > 1 − ε. is completes the proof.
Proof. Make use of Lemma 3, we only need to prove these conclusions hold for (10) for some constants m 1 ′ , m i ′ , M i ′ (i � 1, 2). For system (10), by applying the Itô's formula to lnX and lnY, we have (i) If ζ > 0, then integrating both sides of (43) from 0 to t yields ln 8
To summarize, the above conclusions hold for system (10). Using Lemma 3, the required assertion is directly obtained. e proof is completed. □ Remark 5. For system (5), if the impulsive is absent, then eorem 2 implies the sufficient conditions of the extinction of species x(t) or y(t), which is accordance with eorem 2 in [16].
□ On the contrary, by the given condition, there exist two constants τ 0 and c such that for all t ≥ τ 0 and l > 0, we have e previous proof shows that there exists a positive integer k 6 > t 0 /T such that 〈Y〉 * ≤ κ for any t > k 6 T + τ * and ω ∈ Ω 3 . Due to T * − T * ⟶ ∞, then there exists a positive integer n 0 > 0 such that τ * > k 6 T + τ * for any n ≥ n 0 , and hence for any t ∈ [k 6 T + τ * , τ * ] and ω ∈ Ω 3 . Using (65) again, we have E ln( X( t n , ] n ) )I Ω 3 ≥ E ln X( k 6 T + τ * ), ] n * * * which leads to Obviously it implies a contradiction, and our claim is obtained. erefore In a similar way, we can derive that Finally, making use of Lemma 3 yielding the required assertion. e proof is completed.

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]. eorem 1 gives the sufficient conditions of the existence of periodic Markovian process. eorem 2 represents the extinction and permanence in the mean of predator and prey species. eorem 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
e authors declare that there are no conflicts of interest regarding the publication of this paper.

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