Modelling and analysis of a stochastic nonautonomous predator-prey model

In this paper, a new stochastic predator-prey model with impulsive perturbation and Crowley-Martin functional response is proposed. The dynamical properties of the model are systematically investigated. The existence and stochastically ultimate boundedness of a global positive solution are derived using the theory of impulsive stochastic differential equations. Some sufficient criteria are obtained to guarantee the extinction and a series of persistence in the mean of the system. Moreover, we provide conditions for the stochastic permanence and global attractivity of the model. Numerical simulations are performed to support our qualitative results.


Introduction
Predator-prey models are highly important in general and mathematical ecology [1]. In the past decades, many factors have been considered to describe the ecological predator-prey system more correctly and reasonably [2,3]. Notably, population models in the real world is inevitably influenced by numerous unpredictable environmental noise, and deterministic systems are fairly challenged in describing the fluctuation accurately [4,5]. Hence, an increasing number of researchers have paid attention to stochastic models and proposed various population models with stochastic perturbations, such as in [6][7][8][9][10]. Liu and Wang [10] introduced a stochastic non-autonomous predator-prey model for one species with white noise as follows: dx(t) = r(t)[x(t) − a(t)x(t)]dt + σ(t)x(t)dB(t). (1.1) The group analyzed the conditions for extinction and species persistence in Eq (1.1). On the basis of the theoretical and practical significance of this stochastic model, many results have been presented, particularly, in [11][12][13]. However, the influence of the functional response to systems has been rarely considered in previous stochastic population models. Generally, two types of functional response exist, that is, prey-and predator-dependent responses. The first functional response considers only the prey density, whereas the other accounts for both prey and predator densities [2]. When investigating biological phenomena, one must not ignore the predator's functional response to prey because of such response's effect on dynamical system properties [14][15][16][17][18]. Among many different forms of predator-dependent functional responses, the three classical ones include the Beddington-DeAngelis, Hassell-Varley and Crowley-Martin types. We let x 1 (t) and x 2 (t) denote the prey and predator population densities, respectively, at time t. Then, 1+a(t)x 1 (t)+b(t)x 2 (t)+a(t)b(t)x 1 (t)x 2 (t) becomes the Crowley-Martin functional response, where a(t), b(t) and ω(t) represent the effects of handling time, the magnitude of interference among predators, and capture rate, respectively. Interestingly, if a(t) = 0 and b(t) = 0, then the Crowley-Martin functional response becomes a linear mass-action functional response. If a(t) = 0 and b(t) > 0, the response represents a saturation response; if a(t) > 0 and b(t) = 0, then the response becomes a Michaelis-Menten functional response (or Holling type-II functional response) [1,19,20]. Given its importance and appeal, some scholars have studied stochastic predator-prey models incorporating Crowley-Martin functional response [21][22][23] and in this paper, we consider the Crowley-Martin functional response to embody interference among predators and provide insight into the dynamics of the predator-prey population model.
Meanwhile, the theory of impulsive differential equation was well developed recently, and impulsive differential equations were found as a more effective method for describing species and the ecological systems more realistically. Many important and peculiar results have been obtained regarding the dynamical behavior of these systems, including the permanence, extinction of positive solution and dynamical complexity. However, few studies have addressed the population dynamics of two species both with stochastic and impulsive perturbations, except in [24][25][26]. In [25], Zhang and Tan considered a stochastic autonomous predator-prey model in a polluted environment with impulsive perturbations and analyzed the extinction and persistence of the system. By contrast, the model proposed is autonomous, that is, the parameters are assumed as constants and independent of time. In [26], Wu, Zou, and Wang proposed a stochastic Lotka-Volterra model with impulsive perturbations. The asymptotic properties of the model were examined. However, their model was based on the prey-dependent functional response and did not consider the predator's functional response to prey. In addition, there are four approaches to introduce stochastic perturbations to the model as usual, through time Markov chain model, parameter perturbation, being proportional to the variables, and robusting the positive equilibria of deterministic models [27]. In this paper, we adopt the third approach to include stochastic effects to Eq (1.2). Inspired by the above discussion and [11,25,26], we consider the possible effects of impulsive and stochastic perturbations on the system and propose the following non-autonomous stochastic differen-tial equation: The parameters are defined as follows: r(t) and g(t) denote the intrinsic growth rate of the prey and predator population at time t; k(t) and h(t) are the density-dependent coefficients of prey and predator populations, respectively; f (t) represents the conversion rate of nutrients into the reproductive predator population; σ 2 i (t) (i = 1, 2) refers to the intensities of the white noises at time t;Ḃ 1 (t) anḋ B 2 (t) are standard white noises, in particular, B 1 (t), B 2 (t) are Brownian motions defined on a complete probability space (Ω, F , P) [6].
According to biological meanings, ρ 1k > −1, ρ 2k > −1. Moreover, we assume that there are some positive constants m, M, m and M that satisfy 0 < m ≤ The remaining portion of this paper is arranged as follows. In the next section, some preliminaries are introduced. We analyze the impulsive stochastic differential model and obtain the existence, uniqueness and stochastically ultimate boundness of the positive solution in Section 3. In Section 4, sufficient conditions for extinction and a set of persistence in the mean, including non-persistence, weak persistence, and strong persistence in the mean, are presented. Additionally, we provide conditions to guarantee the stochastic permanence of the system. In Section 5, the global attractiveness of Eq (1.2) is studied. Finally, some numerical simulations, which verify our theoretical results, are given in Section 6. We compare the results of stochastic models under positive or negative impulsive perturbations with those without such disturbances, as well as, the figures with different stochastic perturbations and same impulse. By doing so, we clearly show that the impulsive and stochastic perturbations are of great importance to species permanence and extinction.
(b) For each t k , x(t) obeys the equivalent integral Eq (2.1) for almost every t ∈ R + \ t k and satisfies the impulsive conditions at t = t k a.s..
for all t ≥ T , where β i is a constant, 1 ≤ i ≤ n, then, x * ≥ λ/λ 0 a.s. Lemma 2.2. ( [30]) Let f be a non-negative function defined on R + such that f is integrate and is uniformly continuous. Then lim t→+∞ f (t) = 0.

Properties of the solution
In this section, the existence, uniqueness, and stochastically ultimate boundedness of the global positive solution are obtained.
Firstly, we denote then by virtue of Lemma 2.1 in [30], the following lemma can be obtained. Lemma 3.1. For the stochastic equations without impulses is a solution of Eq (1.2) with initial value (x 1 (0), x 2 (0)) = (y 1 (0), y 2 (0)). The proof can be given easily as in [31], but such approach is not applied herein. Theorem 3.1 For any given value (x 1 (0), x 2 (0)) = X 0 ∈ R 2 + , a unique solution (x 1 (t), x 2 (t)) exists for Equation (1.2) on t ≥ 0 and the solution will remain in R 2 + with probability one. The proof of Theorem 3.1 is standard and is presented in Appendix A. Theorem 3.2 The solutions of Eq (1.2) are stochastically ultimately bounded for any initial value X 0 = (x 1 (0), x 2 (0)) ∈ R 2 + . The proof of Theorem 3.2 is presented in Appendix A.

Long time behavior of Eq (1.2)
In this section, sufficient conditions for extinction and a series of persistence in the mean, such as non-persistence, weak persistence and strong persistence in the mean, are established. Furthermore, we obtain conditions to guarantee the stochastic permanence of the system. Before giving the main theorems, we introduce a lemma essential to our proofs.
The proof of Lemma 4.1 is given in Appendix A.

Extinction and persistence of the system
The results about persistence in the mean and extinction of the prey and predator populations are presented in Theorems 4.1.1 and 4.1.2. Theorem 4.1.1 For the prey population x 1 of Eq (1.2), we have (a) If r 1 < 0, then the prey population x 1 is extinct with probability 1, where r 1 (t) = r(t) − 0.5σ 2 1 (t), (b) If r 1 = 0, then the prey population x 1 is non-persistent in the mean with probability 1. (c) If r 1 > 0 and r 2 < 0, then the prey population x 1 is weakly persistent in the mean with probability 1, where r 2 = lim sup then the prey population x 1 is strongly persistent in the mean with probability 1, Proof. (a) According to Itô's formula and Eq (3.1), the function can be expressed as Taking integral on both sides of Eq (4.1) results in Using the strong law of large numbers for martingales, we show that lim sup Therefore, Making use of Eq (4.2) and the superior limit as t → ∞ in Eq (4.4), we get lnx 1 According to the definition of superior limit and Eq (4.2), for an arbitrary ε > 0, there is a Substituting the above inequalities into the first equation of (4.4), we easily show that By virtue of Lemma 2.1, we obtain x 1 (t) * ≤ ε k * . In accordance with the arbitrariness of ε, we achieve the result.
(c). By virtue of Eq (4.2), superior limit and Lemma 4.1, we show that Thus, x 1 (t) * > 0 a.s. By reduction to absurdity, we can assume that for any υ ∈ { x 1 (t, υ) * = 0}, by Equation (4.6), we obtain x 2 (t, υ) * > 0. Meanwhile, using the superior limit for the second equation of (4.3) and x 1 (t, v) * = 0 leads to Therefore, lim t→∞ x 2 (t, υ) = 0. This expression is a contradiction. Then, According to the definition of superior limit, interior limit and Eq (4.2), for the above-mentioned positive constant ε, there is a T > 0 satisfying 1 Using Lemma 2.1 and the arbitrariness of ε, we have that (e). Passing to the first equation of (4.3), we yield Thus, x 1 (t) * ≤ r 1 k l M x , which is obtained by a similar process in the proof of conclusion (2) and is omitted.
Let (x 1 (t),x 2 (t)) be the solution of the following comparison equation with initial value (x 1 (0), x 2 (0)) ∈ R 2 + , then we hold the following theorem. (a) if k * r 2 + f * r 1 < 0, then the predator population x 2 is extinct with probability 1 ; (b) if k * r 2 + f * r 1 = 0, then the predator population x 2 is non-persistent in the mean with probability 1; (c) if r 2 + fx 1 1+ax 1 +bx 2 +abx 1x2 * > 0, then the predator population x 2 is weakly persistent in the mean with probability 1; (d) if r 2 + f a * > 0, then the predator population x 2 (t) has a superior bound in time average, that is, Proof. (a). Case I. If r 1 ≤ 0, then by virtue of Theorem 4.1.1, we obtain x 1 (t) * = 0. According to the definition of superior limit, for an arbitrary Case II. If r 1 > 0, by Eq (4.3), for the above constant ε > 0, there is a T 1 > 0 such that lnx 1 (4.10) Substituting the above inequality into the second equation of (4.3) and using the arbitrariness of ε, we yield we only need to show that if r 1 > 0, then x 2 (t) * = 0 is also valid. Otherwise, x 2 (t) * > 0, and by Lemma 4.1, we obtain that [ lnx 2 (t) t ] * = 0. From Eq (4.11), we note that Meanwhile, for any constant ε > 0, there is a T > 0 satisfying 1 By the second equation of (4.3), Then, making use of Lemma 2.1, we achieve . By virtue of Eq (4.10) and the arbitrariness of ε, we obtain x 2 (t) * ≤ k * r 2 + f * r 1 h * k * = 0. This is a contradiction. Therefore, x 2 (t) * = 0 a.s.
(c). In the following, we show that x 2 (t) * > 0 a.s.. By reduction to absurdity, for arbitrary ε 1 > 0 and initial value (x 1 (0), x 2 (0)) ∈ R 2 + , there is a solution (x 1 (t),x 2 (t)) of Eq (1.2) satisfying P{ x 2 (t) * < ε 1 } > 0. Let ε 1 be sufficiently small that From the second equation of (4.3), it can be shown that Herein,x 1 (t) ≤x 1 (t),x 2 (t) ≤x 2 (t), a.s. for t ∈ [0, +∞). Note that is a positive function on R + . By virtue of Itô's formula and Eq (4.9), we achieve the following expression: Moreover, integrating the above inequality from 0 to t and dividing by t on both sides of the above Then we achieve By substituting the above inequality into Eq (4.12) and taking the superior limit of the inequality, we obtain Equation (4.14) contradicts Lemma 4.1, therefore x 2 (t) * > 0 a.s. The proof is hence completed. (d). By the second equation of (4.3), we obtain the following equation Moreover, from the definition of superior limit and Eq (4.2), for the given positive number ε, there is a In accordance with Lemma 2.1 and the arbitrariness of ε, we easily achieve The desired result is obtained. (e). If x * 2 > 0 a.s is false, let Ω = {x * 2 = 0}, then P(Ω) > 0. For an arbitrary ν ∈ Ω, we have lim t→∞ x 2 (t, ν) = 0. From the second equation of (4.3) and by virtue of Eq (4.2), we show that [ lnx 2 (t,ν) t ] * = r 2 > 0 a.s. Then we follow that P{[ lnx 2 (t,ν) t ] * > 0} > 0, which contradicts with Lemma 4.1. The result is then concluded. Remark 1. By the proof of Theorem 4.1.2, we observe that if r 1 < 0, then k * r 2 + f * r 1 < 0. Thus, if the prey species is extinct, then the predator species will also be extinct. This notion is consistent with the reality. Moreover, if r 1 > 0 and k * r 2 + f * r 1 < 0, then even if the prey population is persistent, the predators end in extinction because of an excessively large diffusion coefficient σ 2 2 . Remark 2. According to conclusion (5) of Theorem 4.1.2, with the effect of impulsive perturbations despite the regression of the prey population to extinction, the predator may remain weakly persistent.
Let U(y 1 , y 2 ) = 1 V(y 1 , y 2 ) , according to the Itô's formula, we obtain Under the condition of this theorem, a positive constant θ can be chosen to satisfy Eq (4.16). By the Itô formula, Then we choose p > 0 to be sufficiently small such that the term satisfies Eq (4.17). We define W(X) = e pt (1 + U(X)) θ and consequently achieve LW(X) = pe pt (1 + U(X)) θ + e pt L(1 + U(X)) θ = e pt (1 + U(X)) θ−2 p(1 + U(X)) 2 − θU 2 (X)y 1 r(t) − k(t) By Eq (4.17), a positive constant S satisfying LW(X) ≤ S e pt is easily noted. Consequently, E[e pt (1+ U(X)) θ ] ≤ (1 + U(0)) θ + S (e pt −1) Notably, (y 1 + y 2 ) θ ≤ 2 θ (y 2 1 + y 2 2 ) θ 2 = 2 θ |X| θ , where X = (y 1 , y 2 ) ∈ R 2 + . Then, we obtain where m = min{m, m}. Therefore, for arbitrary ε > 0, we let δ = (ε \ M 0 ) 1 θ in accordance with Chebyshev's inequality, thereby yielding In the following relations, we prove that for any ε > 0, there exists χ > 0 satisfying P * {|X 1 (t)| ≤ χ} ≥ 1 − ε. Define V 4 (X) = y q 1 + y q 2 , herein 0 < q < 1 and X = (y 1 , y 2 ) ∈ R 2 + , then by virtue of Itô's formula, we obtain the expression Let n 0 be a sufficiently large constant, such that y 1 (0), y 2 (0) remain within the internal [ 1 n 0 , n 0 ]. For each integer n ≥ n 0 , we define the stopping time t n = inf{t ≥ 0 : y 1 (t) (1/n, n) or y 2 (t) (1/n, n)}. Obviously, t n increases as n → +∞. Using Itô's formula again for exp{t}V 4 (X) and accounting for the expectations on both sides, we show that Then, we achieve lim sup max{M, M}. Therefore, at any given ε > 0, we let χ = M(K 1 +K 2 ) 1/q ε 1/q , by virtue of the Chebyshev inequality, we easily show that Consequently, P * {|X 1 (t)| ≤ χ} ≥ 1 − ε. Theorem 4.2.1 is proven. Remark 3. From the conditions of Theorem 4.2.1, we find that although the stochastic disturbance greatly influence the dynamical property of the system, the bounded impulsive perturbations do not affect the stochastic permanence of the model. Remark 4. We should point out that the definition of stochastically permanent which requires that all species have positive upper bounds and at least one species has a positive lower bound, cannot demonstrate the permanence of all species. It has some limitations and deficiency. If there is only one species having a positive lower bound and all the other species go extinction, the system is still permanent. A new definition of stochastic permanence [31] may be more appropriate.
The proof of Theorem 5.1 is given in Appendix A.

Numerical simulations
In this section, some numerical simulations and examples are given to illustrate and augment our theoretical findings of Eq (1.2) by means of the Milstein method mentioned in Higham [35]. Moreover, the effects of impulsive and stochastic perturbations on population dynamics are discussed. Example 1.
We set r(t) = 0.2 + 0.01 sin t, a(t) = 0.1 + 0.04 sin t, b(t) = 0.5 + 0.05 sin t, k − 1 and τ k = k, then it can be obtained that r 1 = −0.1 < 0. By Theorems 4.1.1 and 4.1.2, both prey and predator populations ( x 1 and x 2 , respectively) regress to extinction, which is also further confirmed by Figure 1. = 0.4 + 0.02 sin t, ρ 1k = ρ 2k = e (−1) k+1 1 k − 1, and r(t) = 0.4 + 0.01 sin t. The other parameters are the same as that in example 1, then r 1 = 0.1 > 0, r 2 = −0.6 < 0, and k * r 2 + f * r 1 = −0.116 < 0. In Figure 2, although the prey population x 1 is weakly persistent in the mean, the predator population x 2 end in extinction because of the effects of the white noises, which are of great importance in maintaining the coexistence of populations.
We let Both the prey and predator populations ( x 1 and x 2 , respectively) are weakly persistent in the mean of Figure 3.
Both prey population x 1 and predator population x 2 go to extinction.    In the following instance, the effects of negative and positive impulses on the species are investigated. We let ρ 1k = e −1.92 − 1, ρ 2k = e −0.02 − 1, and τ k = k. The other parameters are the same as those in Example 3, then we obtain that r 1 = −0.03 < 0 and r 2 = −0.36 < 0. On the basis of Theorems 4.1.1 and 4.1.2, both the prey and predator populations end in extinction, which is further confirmed by Figure 6(b). By comparing Figure 6(a) with Figure 6(b), we observe that the negative impulses do not benefit species coexistence. Moreover, considering the impulsive perturbations are (a) ρ 1k = ρ 2k = 0, (b) ρ 1k = e 1 k 2 − 1, ρ 2k = e 0.8 − 1, (c) ρ 1k = ρ 2k = e 0.8 − 1 and the other parameters are the same as those in Example 1. Hence, the system can be altered from extinction to persistence with the effects of positive impulsive perturbations ( Figure 7(a)-7(c)). Herein, persistence can be divided into two cases as follows: first, the predator population x 2 (t) is weakly persistent and the prey population x 1 (t) pro-      ceeds to extinction and second, both of the populations are persistent. Therefore, positive impulses are advantageous for the coexistence of ecosystems. Moreover, comparing figures 3 and 6, figures 1 and 7, we can derive that if the impulsive perturbations are unbounded, some properties may be changed significantly.

Conclusions
In this paper, we propose a stochastic non-autonomous predator-prey system with impulsive perturbations and investigate the qualitative dynamic properties of the model. Under some sufficient conditions, we present the extinction and a series of persistence in the mean of the system, including non-persistence, weak persistence and strong persistence in the mean. Furthermore, we obtain the global attractivity of the model. From the assumptions of Theorems 4.1.1, 4.1.2 and 4.2.1, we demonstrate that the stochastic and impulsive disturbances greatly influence the extinction and persistence of the system. Positive impulses are advantageous for the coexistence of ecosystem, whereas negative impulses are not beneficial for species coexistence. Moreover, the results show that the bounded impulsive perturbations do not affect all the properties, such as the stochastic permanence of the model. However, if the impulsive perturbations are unbounded, some properties may be changed significantly.
Some interesting topics require further investigations. If we also consider the effects of time delays and telephone noise [36][37][38] on Eq (1.2) to propose a more realistic model, then how will the properties change? We leave it for future investigation.