Dynamics analysis of a stochastic non-autonomous one-predator–two-prey system with Beddington–DeAngelis functional response and impulsive perturbations

*Correspondence: mxz721106@sdust.edu.cn 1College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, P.R. China 2State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao, P.R. China Full list of author information is available at the end of the article Abstract

mechanism, which changes the prey density per unit time per predator as a function of prey or both prey and predator species. There are many kinds of famous functional response in the predator-prey system reported in the previous references, such as Holling types [19][20][21], Beddington-DeAngelis type [22][23][24][25], Michaelis-Menton type [26], Ivlev type [27], Hassell-Varley type [28], Crowley-Martin type functional response [29], which are suitable for different kinds of predator-prey systems, respectively. In 1975, Beddington [22] and DeAngelis [23] first introduced the Beddington-DeAngelis type predator-prey model taking the form ⎧ ⎨ ⎩ dx dt = r 1 xα 1 x 2 -c 1 xy a 1 +a 2 x+a 3 y , dy dt = r 2 yα 2 y 2 + c 2 xy a 1 +a 2 x+a 3 y , where x and y denote the population densities of prey and predator, respectively. The term c 1 x a 1 +a 2 x+a 3 y represents the Beddington-DeAngelis functional response, which turns into the Holling-II functional response if a 3 = 0 and linear functional response if both a 2 = 0 and a 3 = 0. That is to say, the B-D functional response is affected by both predator and prey. Therefore, the effect of mutual interference on the dynamics of population is worth studying.
On the other hand, the population systems in the real world are always inevitably influenced by all kinds of environmental noises which are an important component in an ecosystem. Usually, there are two types of environmental noises: white noise and color noise. White noise arises from a nearly continuous series of small or moderate perturbations that have small effects on the intrinsic growth rates of the species. Therefore, it is essential to reveal how the environmental noise disturbs the population systems. In recent years, many scholars have proposed and investigated stochastic models with white noise perturbations, please refer to [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46] and the references therein. For example, Ji et al. [30] considered a predator-prey model with modified Leslie-Gower and Holling type II schemes with stochastic perturbation and the condition for persistence and extinction of the system is established. Liu and Wang in [36] discussed a predator-prey system with Beddington-DeAngelis functional response with stochastic perturbation. They demonstrated that if the positive equilibrium of the deterministic system is globally stable, then the stochastic model will preserve this nice property provided the noise is sufficiently small.
However, periodic behavior often arises in implicit ways in various natural phenomena. For example, due to the seasonal variation, hunting, harvesting and so on, the birth rate, the mortality rate and other parameters in the population systems will not remain constant, but exhibit a more-or-less periodicity. Thus, it is natural to model the population by a periodic environment. Therefore, numerous authors have investigated the effect of seasonal variation and stochasticity (see [47][48][49]).
Furthermore, population growth in ecosystems is also affected by human activities, such as periodic harvesting or stocking for the species, which cannot be considered continuously. Stochastic systems that consider continuous phenomena are not suitable for these phenomena. Therefore, in this case, we should consider the effect of impulse in order to describe these phenomena more accurately. In recent decades, a variety of population dynamical systems with impulsive effects have been proposed and studied extensively (see [50][51][52][53][54][55][56]). For example, in [50] Liu and Wang concerned with an n-species stochastic nonautonomous Lotka-Volterra competitive system with impulsive effects. They obtained the sufficient conditions for stochastic permanence, extinction and global stability and investigated some dynamical properties. Zhang and Meng et al. [52] discussed a stochastic non-autonomous predator-prey system with impulsive effect. They concluded that the large stochastic disturbances can lead to the extinction of the population, and large impulse harvests can also result in the extinction of the population.
Taking all above influences into consideration, we focus on the stochastic nonautonomous one-predator-two-prey system with the Beddington-DeAngelis functional response and impulsive perturbations a 1 (t)+a 2 (t)x 1 (t)+a 3

(t)x 3 (t)
β 2 (t)x 1 (t)] dt + σ 2 (t)x 2 (t) dB 2 (t), dx 3 where x i (t) is the size of the ith population at time t, r i (t) represents the intrinsic growth rate of the ith population, α i (t) stands for the density-dependent coefficients of the ith population, β 1 (t) and β 2 (t) are the competitive coefficient of x 1 (t) and x 2 (t), respectively, c j (t) is the capturing rate of predator, e j (t) represents the rate of conversion of nutrients into the reproduction of predator, B i (t) (i = 1, 2, 3) is for independent standard Brownian motions defined on a complete probability space and σ i (t) is for the intensities of , e j (t), σ i (t) are positive, continuous and bounded functions defined on R + = (0, ∞), N denotes the set of positive integers, 0 < t 1 < t 2 < · · · , lim k→+∞ t k = +∞, i = 1, 2, 3, j = 1, 2, k ∈ N . We impose the following restriction on system (2) which is a reasonable way for giving biological meaning: h ik + 1 > 0, i = 1, 2, 3, k ∈ N . When h ik > 0, the impulsive effects represent releasing the specie, but if h ik < 0, the impulsive effects denote harvesting for the ith population.
The main goals of this paper are to investigate how impulsive perturbations and the white noises affect the permanence, persistence, extinction and global attractivity of system (2). The rest of the paper is organized as follows. In Sect. 2, we give some definition and prove the existence of a unique positive solution of the system. In Sect. 3, we will derive main theoretical results of this paper, such as sufficient conditions for the extinction, non-persistence in the mean, weak persistence, persistence in the mean and stochastic permanence of the system. Meanwhile the limit of the average in time of the sample path of the solution is estimated by two constants. In Sect. 4, the lower-growth rate and the upper-growth rate of the solutions are estimated. In Sect. 5, we investigate the global attractivity of the system. In Sect. 6, we give the con-clusions and several examples and numerical simulations to illustrate our theoretical results.

Preliminary
Let (Ω, F, {F } t≥0 , P) be a complete probability space with a filtration {F t } t≥0 satisfying the common conditions (i.e. it is increasing and right continuous while F 0 contains all Pnull sets). Let B(t) = (B 1 (t), B 2 (t), B 3 (t)) T be an n-dimensional Brownian motion defined on this probability space. Let R 3 We define the norm as |x| = For the constants 2. x(t) is said to be non-persistent in the mean if lim t→+∞ t 0 x(s) ds t = 0.
4. x(t) is said to be persistent in the mean if lim inf t→+∞ t 0 x(s) ds t > 0.
5. x(t) is said to be stochastically permanent if for every ε ∈ (0, 1) there are two constants β > 0, δ > 0 such that Now we give an assumption which will be used in the following proof.

Extinction and persistence
In this section we will derive sufficient conditions for the extinction, non-persistence in the mean, weak persistence, persistence in the mean and stochastic permanence of the solutions of system (2).
Proof Applying Itô's formula to the first equation of system (3), we could find that In the same way, combining with the last two equations of system (3) we have d ln y 2 (t) = r 2 (t) - which leads to Integrating both sides of inequalities (4), (5) and (6) on the interval [0, t], one can easily see that where Making use of the strong law of large numbers for local martingales (see [58]) results in On the other hand, it follows from (7) that 0<t k <t In other words, we can compute that Taking superior limit on both sides of (8) and noting that lim t→+∞ This completes the proof.
Proof According to the definition of the limit, for arbitrary fixed i > 0, there exists a constant T 0 > 0, for every t > T 0 , such that Substituting above inequalities into (8) yields Taking exponent on both sides of (9), we can show that Integrating inequality (10) from T 0 to t yields Taking logarithm of both sides of inequality (11), we can derive that Taking superior limit on (12) elicits that lim sup Then it follows from L'Hospital's rule that lim sup This completes the proof of this theorem.
On the other hand, for ∀ω ∈ S, we have lim t→+∞ x i (t, ω) = 0. Thus it follows from the boundedness of α i (t) that Substituting these inequalities into (13) and making use of lim t→+∞ This completes the proof.
Remark 2 Theorems 3.1-3.3 have an interesting biological interpretation. Observe that the extinction and persistence of species x i (t) only depend on δ * i . If δ * i > 0, the population x i (t) is weakly persistent. If δ * i < 0, the population x i (t) goes to extinction.
Proof Applying Itô's formula to the first equation of system (3), we can observe that Applying general calculations to (14), it is easy to verify that It then follows from Theorem 3.2 that lim sup Since lim t→+∞ Substituting the above inequalities into (15), we get, for t > T 1 , where In the similar way, we can conclude that, for any i , there exists some T i > 0 such that where and Let T * = min 1≤i≤3 T i > 0, then from (16) and (17), we can easily see that Denote Taking the exponent on both sides of (18), we can obtain Integrating inequality (19) from T * to t yields Taking logarithm of both sides of inequality (20), it can be verified straightforwardly that Taking superior limit on both sides of (21), we obtain Then it follows from L'Hospital's rule that This completes the proof of this theorem.

.
It is easy to see that g(y 1 ) has a unique maximum y * 1 = Therefore, which yields On the other hand, by applying Itô's formula and the last two equations of system (3) then making some estimations, we can easily see that Then, similar to the above discussions, we can also derive that where Combining (29) and (30), we can conclude that Multiplying e -t on both sides of (31) and taking the superior limit yield This leads to Then, for any ε > 0, let δ = Θ ε , it follows from Chebyshev's inequality that According to Definition 2.1, it follows from (27) and (33) that system (2) is stochastically permanent.
Therefore, system (2) has the property lim sup

Asymptotic properties
In this section we will discuss the asymptotic properties of the solution of system (2).

Theorem 4.1 If Assumption 2.1 holds and any solution x(t) = (x 1 (t), x 2 (t), x 3 (t)) T of system
(2) has the property that and, moreover, 2r -(σ ) 2 > 0, then Proof It follows from Itô's formula and combining with inequality (4), (5) and (6) that d e t ln y i (t) = e t ln y i (t) dt + e t d ln y i (t) Integrating above inequality (34) where N i (t) = t 0 e t σ i (s) dB i (s) is the exponential martingale, whose quadratic variation is Thus, it follows from the exponential martingale inequality (see [57]) that By virtue of the Borel-Cantelli lemma, for almost all ω ∈ Ω, there exists k 0 (ω) such that, for every k ≥ k 0 (ω), for 0 ≤ t ≤ kγ . Substituting inequality (36) into (35) and making some estimations yield If we denote f (y i ) = ln y i + δ u i + (σ u i ) 2 2 .
But this holds for any that satisfies 2r > (2 + 1)(σ ) 2 , we therefore have It then follows that This completes the proof of this theorem.
Remark 4 Theorem 4.1 shows that, for any > 0, there exists a random variable T > 0 such that t -1 2r-(σ ) 2 + ≤ |x(t)| ≤ t 1+ for t ≥ T almost surely. That is to say, the solution will not decay faster than t -1 2r-(σ ) 2 + and will not grow faster than t 1+ with probability one. We are now in the position to estimate the limit of the average in time of the sample paths of solutions.

Lemma 5.3 (see [61]) Let f be a non-negative function defined on t ≥ 0 such that f is integrable on t ≥ 0 and is uniformly continuous on t ≥ 0.
Then lim t→+∞ f (t) = 0.

Theorem 5.1 If Assumption 2.1 holds and
then system (2) is globally attractive.
Integrating both sides gives Am 1 y 1 (t)y 1 (t) + Bm 2 y 2 (t)y 2 (t) + Cm 3 y 3 (t)y 3 (t) ds. Therefore Making use of V (t) ≥ 0 and (44) results in Consequently, by Lemmas 5.2 and 5.3, one can observe that Then This completes the proof.

Conclusion and numerical simulations
In this paper, a stochastic non-autonomous one-predator-two-prey system with Beddington-DeAngelis functional response and impulsive perturbations is proposed and investigated. First, we obtain some sufficient conditions for extinction, non-persistence in the mean, weak persistence, persistence in the mean and stochastic permanence of the solution, and we verify some asymptotic behaviors of the solutions of system (2), such as the limit of the average in time, the lower-growth rate, the upper-growth rate and global attractivity. Now we summarize the key results as follows: (I): (2) is extinct.
(2) If δ * i = 0, then the ith species (i = 1, 2, 3) in system (2) is non-persistent in the mean. (3) If δ * i > 0, then the ith species (i = 1, 2, 3) in system (2) is weakly persistent. (4) If θ i * > 0, then the ith species (i = 1, 2, 3) in system (2) is persistent in the mean.  (2) is globally attractive. By our results, we can analyze that the smaller stochastic perturbations cannot affect the stochastic permanence and extinction of the population. However, if the stochastic perturbations are larger, the stochastic permanence of the populations will be extinct. Similarly, the small impulsive perturbations have a little influence on the stochastic permanence and extinction of the populations. However, if the impulsive perturbations are large, the stochastic permanence and extinction of the populations could be changed.
We will give some numerical experiments to verify our analytical results by using the Milstein method (see [62]) by supplementing impulsive perturbations into it. We choose the same initial value (x 1 (0), x 2 (0), x 3 (0)) = (0.5, 0.5, 0.5) and the same parameters in the following numerical examples.
The parameters are as follows: At first, we will discuss the effects of different stochastic perturbations to system (2) under the same impulse interference in following Examples 1-6.
Let h 1k = h 2k = h 3k = e -0.2 -1, it is easy to verify that which means the Assumption 2.1 holds. In system (2) without stochastic perturbations, we can see that the prey and predator populations are all permanent (see Fig. 1).
Example 5 Let σ 2 1 (t) = σ 2 2 (t) = 0.1+0.04 sin t, σ 2 3 (t) = 2.1624+0.04 sin t. Then we get δ * 3 = 0. According to Theorem 3.2, we can see that the population x 3 (t) is non-persistent in the mean (see Fig. 6(a),(c)). x i (s) ds a.s. i = 1, 2, 3, then we have 0.466 ≤ x * 1 (t) ≤ 5, 0.2784 ≤ x * 2 (t) ≤ 3.6897, 0.7174 ≤ x * 3 (t) ≤ 2.2636. In Fig. 7(a), we can see that the persistence in the mean of system (2). In Fig. 7(b), it is clear to see that the curve of x * i (t) gradually transcend the line x * i (t) and stays between the x * i (t) and x * i (t) lines of the same color, which verify the conclusion. Finally, we give Example 7 to discuss the effect of the impulsive perturbations on system (2), according to the choice of parameters in Example 1.
Example 7 Let h 1k = h 2k = e -1.2 -1, h 3k = e -0.6 -1. In Fig. 8, one can easily see that all of the species in system (2) become extinct gradually. This means suitable impulsive control strategy might be useful for the permanence of the system while arbitrary impulsive perturbations might lead to the extinction of system (2). Therefore, through the numerical simulations given in Examples 1-6, we can see that the large stochastic disturbance is disadvantageous for the persistence of the population. However, the small stochastic perturbation is little effects on the permanence and extinction of the population. By Fig. 8, we can see that the small impulsive perturbations cannot affect the stochastic permanence and extinction of the prey and predator populations. But the large impulsive perturbations can lead to population extinction.