Asymptotic behavior of a stochastic non-autonomous predator–prey model with impulsive perturbations

Commun Nonlinear Sci Numer Simulat 20 (2015) 965–974

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Asymptotic behavior of a stochastic non-autonomous predator–prey model with impulsive perturbations Ruihua Wu a,b, Xiaoling Zou a,⇑, Ke Wang a a b

Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, PR China College of Science, China University of Petroleum (East China), Qingdao 266555, PR China

a r t i c l e

i n f o

Article history: Received 21 February 2014 Received in revised form 14 June 2014 Accepted 14 June 2014 Available online 21 June 2014 Keywords: Predator–prey model Impulsive effects Persistence and extinction Stochastically ultimate boundedness

a b s t r a c t This paper is concerned with a stochastic non-autonomous Lotka–Volterra predator–prey model with impulsive effects. The asymptotic properties are examined. Sufficient conditions for persistence and extinction are obtained, our results demonstrate that the impulse has important effects on the persistence and extinction of the species. We also show that the solution is stochastically ultimate bounded under some conditions. Finally, several simulation figures are introduced to confirm our main results. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction The stochastic non-autonomous Lotka–Volterra predator–prey model with white noise is expressed by



dx1 ðtÞ ¼ x1 ðtÞ½r 1 ðtÞ  a11 ðtÞx1 ðtÞ  a12 ðtÞx2 ðtÞdt þ r1 ðtÞx1 ðtÞdB1 ðtÞ; dx2 ðtÞ ¼ x2 ðtÞ½r 2 ðtÞ þ a21 ðtÞx1 ðtÞ  a22 ðtÞx2 ðtÞdt þ r2 ðtÞx2 ðtÞdB2 ðtÞ:

ð1Þ

The stochastic processes x1 ðtÞ and x2 ðtÞ represent, respectively, the prey and predator populations. And r i ðtÞ denotes the intrinsic growth rate of the corresponding population at time t; a11 ðtÞ and a22 ðtÞ represent the density-dependent coefficients of the prey and the predator, respectively. The coefficient a12 ðtÞ is the capturing rate of the predator and a21 ðtÞ stands for the rate of conversion of nutrients into the reproduction of the predator. Both r1 ðtÞ and r2 ðtÞ are the coefficients of the effects of environmental stochastic perturbations on the prey and predator population, respectively. The standard Brownian motions Bi ðtÞ; i ¼ 1; 2, are independent each other. The functions ri ðtÞ; aij ðtÞ and ri ðtÞ (i; j ¼ 1; 2) are continuous bounded functions on Rþ ¼ ½0; þ1Þ. The stochastic model (1) has received great attention and many good results have been published, see [1–5] and the references cited therein. Liu and Wang analyzed the extinction and persistence of species of model (1) in [1]. In [2,3] Rudnicki investigated system (1) in the autonomous case, showed that the distributions of the solutions are absolutely continuous and proved that the densities can converge in L1 to an invariant density or can converge weakly to a singular measure. There are many other papers in the literature on models with white noise, the readers can refer to [6–12] and the references cited therein. ⇑ Corresponding author. Tel.: +86 06315687660. E-mail addresses: [email protected], [email protected] (R. Wu), [email protected] (X. Zou), [email protected] (K. Wang). http://dx.doi.org/10.1016/j.cnsns.2014.06.023 1007-5704/Ó 2014 Elsevier B.V. All rights reserved.

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However, in the real world, due to some natural and man-made factors, the growth of species often suffers from some discrete changes of relatively short time interval at some fixed times, such as drought, flooding, hunting, planting, etc. These phenomena cannot be considered continually, so in this case, system (1) cannot capture these phenomena. Introducing the impulsive effects into the model may be a reasonable way to accommodate such phenomena, see [13,14]. Lots of deterministic population dynamical systems with impulsive effects have been proposed and studied. Many results on dynamical behavior for such systems have been reported, see e.g. [15–18] and the references therein. Recently, authors of [19–22] considered the stability of stochastic differential equation (SDE) with impulsive effects. However, so far as we know, there are few papers published which study the impulsive stochastic population model, see [23–25]. By now, there are no results related to the stochastic predator–prey system with impulsive effects. Inspired by the above discussions, we propose the following stochastic Lotka–Volterra predator–prey model with impulsive effects:

8 dx1 ðtÞ ¼ x1 ðtÞ½r1 ðtÞ  a11 ðtÞx1 ðtÞ  a12 ðtÞx2 ðtÞdt þ r1 ðtÞx1 ðtÞdB1 ðtÞ; t – t k ; k 2 N; > > > < dx ðtÞ ¼ x ðtÞ½r ðtÞ þ a ðtÞx ðtÞ  a ðtÞx ðtÞdt þ r ðtÞx ðtÞdB ðtÞ; t – t ; k 2 N; 2 2 2 21 1 22 2 2 2 2 k > x1 ðt þk Þ  x1 ðt k Þ ¼ b1k x1 ðt k Þ; k 2 N; > > : x2 ðt þk Þ  x2 ðt k Þ ¼ b2k x2 ðt k Þ; k 2 N;

ð2Þ

where N denotes the set of positive integers, 0 < t1 < t2 <    ; limk!þ1 tk ¼ þ1. For biological meanings, we impose the following restriction on bik :

1 þ bik > 0;

i ¼ 1; 2; k 2 N:

For example, if bik > 0, the impulsive effects may denote planting of the species, while bik < 0 may represent harvesting. The main aims of this paper are to investigate how impulses affect the asymptotic behaviors of system (2). The rest of this paper is organized as follows. In Section 2, we show that the solution of system (2) is global and positive. In Section 3, we study the conditions under which it is observed extinction or persistence of the species modeled by system (2). The stochastically ultimate boundedness is examined in Section 4. Finally, we present several numerical simulations to illustrate out results and the effects of impulse. 2. Global positive solutions For convenience, we adopt the following notations. If f ðtÞ is a continuous bounded function on Rþ , define f u ¼ Rt Q Q supt2Rþ f ðtÞ; f l ¼ inf t2Rþ f ðtÞ. f ðtÞ ¼ t 1 0 f ðsÞds; f  ¼ lim supt!þ1 f ðtÞ; f ¼ lim inf t!þ1 f ðtÞ. And i¼1;2 yðiÞ denotes i¼1;2 yðiÞ ¼ l l l yð1Þyð2Þ. Throughout this paper, we assume that aii > 0; aij P 0; r 2 > 0; i – j; i; j ¼ 1; 2. Moreover, we always assume that a product equals unity if the number of factors is zero. Before we discuss the asymptotic properties of solutions to (2), we should first guarantee the existence of global positive solutions. First, we give the definition of solutions to impulsive stochastic differential equations (ISDE). Definition 1 [23]. For a given ISDE:



dXðtÞ ¼ Fðt; XðtÞÞdt þ Gðt; XðtÞÞdBðtÞ; Xðt þk Þ  Xðt k Þ ¼ Bk Xðt k Þ;

t – t k ; k 2 N;

k 2 N;

ð3Þ

with the initial condition Xð0Þ. A stochastic process XðtÞ ¼ ðX 1 ðtÞ; X 2 ðtÞ; . . . ; X n ðtÞÞT is said to be a solution of (3) on Rþ , if the following conditions are satisfied: (a) XðtÞ is Ft -adapted and is continuous on ð0; t1 Þ and each interval ðt k ; tkþ1 Þ; k 2 N; Fðt; XðtÞÞ 2 L1 ðRþ ; Rn Þ; Gðt; XðtÞÞ 2 L2 ðRþ ; Rn Þ, where Lk ðRþ ; Rn Þ is all Rn -valued measurable Ft -adapted processes f ðtÞ satisfying RT jf ðtÞjk dt < 1 almost surely for every T > 0. 0    þ (b) For each tk ; k 2 N; Xðtþ k Þ ¼ limt!t k XðtÞ and Xðt k Þ ¼ limt!t k XðtÞ exist and Xðt k Þ ¼ Xðt k Þ with probability one. ðcÞ XðtÞ obeys the equivalent integral equation of (3) for almost every t 2 Rþ n tk and satisfies the impulsive conditions at each t ¼ t k ; k 2 N, with probability one. We are now in position to prove the existence of the positive solution to stochastic system (2) with impulsive effects, the following theorem states this. Theorem 1. For any given initial value ðx10 ; x20 Þ 2 R2þ ¼ fðx; yÞ 2 R2 jx > 0; y > 0g, the system (2) has a unique solution xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞÞ on t P 0 and the solution will remain in R2þ a.s. (almost surely). Proof. We follow [23]. Consider the following SDE without impulse:

R. Wu et al. / Commun Nonlinear Sci Numer Simulat 20 (2015) 965–974

8 > > > > > < > > > > > :

"

Y

dy1 ðtÞ ¼ y1 ðtÞ r1 ðtÞ  a11 ðtÞ " dy2 ðtÞ ¼ y2 ðtÞ r 2 ðtÞ þ a21

ð1 þ b1k Þy1 ðtÞ  a12 ðtÞ

#

Y

ð1 þ b2k Þy2 ðtÞ dt þ r1 ðtÞy1 ðtÞdB1 ðtÞ;

0
0
Y

Y

ð1 þ b1k Þy1 ðtÞ  a22 ðtÞ

0
967

#

ð4Þ

ð1 þ b2k Þy2 ðtÞ dt þ r2 ðtÞy2 ðtÞdB2 ðtÞ;

0
with initial value ðy10 ; y20 Þ ¼ ðx10 ; x20 Þ. According to the classic theory of SDE without impulse, Eq. (4) has a unique global Q positive solution yðtÞ ¼ ðy1 ðtÞ; y2 ðtÞÞ, see [26]. Let xi ðtÞ ¼ 0
"

Y

dx1 ðtÞ ¼ d

# ð1 þ b1k Þy1 ðtÞ ¼

0
¼

Y 0
"

Y

ð1 þ b1k Þdy1 ðtÞ

0
# 2 X Y Y ð1 þ b1k Þy1 ðtÞ r 1 ðtÞ  a1j ðtÞ ð1 þ bjk Þyj ðtÞ dt þ r1 ðtÞ ð1 þ b1k Þy1 ðtÞdB1 ðtÞ
j¼1

"

0
# 2 X a1j ðtÞxj ðtÞ dt þ r1 ðtÞx1 ðtÞdB1 ðtÞ: ¼ x1 ðtÞ r 1 ðtÞ  j¼1

Similarly, we have

dx2 ðtÞ ¼ x2 ðtÞ½r 2 ðtÞ þ a21 ðtÞx1 ðtÞ  a22 ðtÞx2 ðtÞdt þ r2 ðtÞx2 ðtÞdB2 ðtÞ: And for every k 2 N and t k 2 Rþ ,

xi ðt þk Þ ¼ limþ xi ðtÞ ¼ limþ t!t

k

t!t

k

Y

ð1 þ bij Þyi ðtÞ ¼

0
Y

ð1 þ bij Þyi ðtþk Þ ¼ ð1 þ bik Þ

0
Y

ð1 þ bij Þyi ðtk Þ ¼ ð1 þ bik Þxi ðtk Þ:

0
Moreover,

xi ðt k Þ ¼ lim xi ðtÞ ¼ lim t!t k

t!t k

This completes the proof.

Y 0
ð1 þ bij Þyi ðtÞ ¼

Y

ð1 þ bij Þyi ðtk Þ ¼

0
Y

ð1 þ bij Þyi ðt k Þ ¼ xi ðt k Þ:

0
h

3. Persistence and extinction Theorem 1 demonstrates that the solutions to model (2) will remain in the positive cone R2þ . This nice positivity allows us to further examine how the solutions vary on R2þ in more detail. First, we give several definitions, then try to explore sufficient criteria for them. Definition 2 [7]. Let xðtÞ be a solution to system (2). 1. 2. 3. 4.

If If If If

limt!1 xðtÞ ¼ 0 a.s., then species xðtÞ is said to be extinct. limt!1 xðtÞ ¼ 0 a.s., then species xðtÞ is said to be non-persistent in the mean. x > 0 a.s., then species xðtÞ is said to be weakly persistent in the mean. x > 0 a.s., then species xðtÞ is said to be weakly persistent.

It follows from the definitions that extinction implies non-persistence in the mean; weak persistence in the mean indicates weak persistence. But generally, the reverses are not true. We will discuss them one by one. First, we give a lemma which give an estimation of the trajectory. Lemma 1. The solution xðtÞ of the predator–prey system (2) obeys

lim sup t!þ1

R0
a:s: i ¼ 1; 2:

Proof. The proof is similar to [26], here we omit it. h We are now in position to present our main results. For the prey population x1 , we have the following results.

ð5Þ

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Theorem 2. Rt P  (I) If h1 < 0, then the population x1 will be extinct, where h1 ðtÞ ¼ 1t ½ 0 0; lim supt!1 0




(I) If h2 ða11 ðtÞÞ þ ða21 ðtÞÞ h1 < 0 and h2 < 0, then the predator x2 will go extinct. P    (II) If h2 ða11 ðtÞÞ þ h1 ða21 ðtÞÞ ¼ 0; h2 < 0 and lim supt!1 0 0 and lim supt!1 0 0; h1 6 0 and lim supt!1 0
ln½x1 ðtÞ=x10  M1 ðtÞ ¼ h1 ðtÞ  a11 ðtÞx1 ðtÞ  a12 ðtÞx2 ðtÞ þ t t

ð6Þ

and

ln½x2 ðtÞ=x20  M2 ðtÞ ¼ h2 ðtÞ þ a21 ðtÞx1 ðtÞ  a22 ðtÞx2 ðtÞ þ ; t t

ð7Þ

Rt

where M i ðtÞ ¼ 0 ri ðsÞdBi ðsÞ; i ¼ 1; 2. Then M i ðtÞ is a local martingale whose quadratic variation is hM i ; M i iðtÞ ¼ u 6 ðr2i Þ t. By the strong law of large numbers [27], we have

lim

t!1

M i ðtÞ ¼ 0; t

a:s: i ¼ 1; 2:

Rt 0

r2i ðsÞds ð8Þ

Proof of Theorem 2. Case (I). Passing to the superior limit as t ! 1 in (6) and using (8), we follow that 





½t 1 ln x1 ðtÞ 6 h1  a11 ðtÞx1 ðtÞ  a12 ðtÞx2 ðtÞ 6 h1 < 0 a:s: which leads to limt!1 x1 ðtÞ ¼ 0 a.s. Case (II). By the property of limit and (8), for sufficiently small e > 0, there exists a T > 0 such that  ðln x10 Þ=t 6 e=3; h1 ðtÞ < h1 þ e=3 and M 1 ðtÞ=t 6 e=3 for all t > T. Substituting above inequalities into (6) we find

ln x1 ðtÞ e e e  6 þ h1 þ  a11 ðtÞx1 ðtÞ þ 6 e  al11 x1 ðtÞ: t 3 3 3 Using Lemma 3.2 in [28] leads to x1  6 e=al11 a.s. By the arbitrariness of e we conclude our result. P  Case (III). By the condition lim supt!1 0
au11 x1  þ au12 x2  P h1 > 0 a:s:

ð9Þ

Therefore x1 > 0 a.s. In fact, for 8x 2 fx1 ðt; xÞ ¼ 0g, it follows from (9) that x2 ðt; xÞ > 0. On the other hand, taking the superior limit for (7) and using x1  ¼ 0 a.s. result in 











½t 1 ln x2 ðt; xÞ 6 h2  al22 x2 ðt; xÞ 6 h2 < 0: That is to say limt!1 x2 ðt; xÞ ¼ 0 which is a contraction. So we have x1  > 0 a.s.  Proof of Theorem 3. Case (I). If h1 6 0, it follows from Theorem 2 that x1  ¼ 0 a.s. By the property of superior limit, for  sufficiently small e > 0, there exists a T > 0 such that h2 ðtÞ < h2 þ e, for all t > T. Let e > 0 be sufficiently small such that  h2 þ e < 0. By (7), we have

 1    t ln x2 ðtÞ 6 h2 þ e þ au21 x1  ¼ h2 þ e < 0 a:s: 

That is to say limt!1 x2 ðt Þ ¼ 0 a.s. If h1 > 0, it follows from (6) and (8) that

ln x1 ðtÞ e e e  6 þ h1 þ  ½ða11 ðtÞÞ  ex1 ðtÞ þ : t 3 3 3

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By Lemma 3.2 in [28], we obtain 

x1  6 h1 =ða11 ðtÞÞ

a:s:

ð10Þ

Substituting the above inequality into (7), we see that 









½t1 ln x2 ðtÞ 6 h2 þ a21 x1  6 h2 þ ðða21 ðtÞÞ þ eÞx1  6 h2 þ ðða21 ðtÞÞ þ eÞ





h1 h ða11 ðtÞÞ þ ðða21 ðtÞÞ þ eÞh1 ¼ 2 ða11 ðtÞÞ ða11 ðtÞÞ

< 0 a:s: Then our result is obtained.  Case (II). In case (I) we have already shown that if h1 6 0, then limt!1 x2 ðtÞ ¼ 0 a.s. In other words, x2  ¼ 0 a.s. Now P  suppose that h1 > 0, if x2  > 0 a.s. by Lemma 1 and assumption lim supt!1 0




0 ¼ ½t1 ln x2 ðtÞ 6 h2 þ a21 ðtÞx1  6 h2 þ ða21 ðtÞÞ x1  On the other hand, for arbitrary

ðln x20 Þ=t < e=4;

a:s:

e > 0, there exists a T > 0 such that 

a21 ðtÞx1 < ða21 ðtÞÞ x1  þ e=4; M 2 ðtÞ=t < e=4 a:s:

h2 ðtÞ < h2 þ e=4;

Substituting above inequalities into (7), we find 

t1 ln x2 ðtÞ 6 h2 þ ða21 ðtÞÞ x1  þ e  ða22 ðtÞÞ x2 ðtÞ: 



By Lemma 3.2 in [28], x2  6 ½h2 þ ða21 ðtÞÞ x1  þ e=ða22 ðtÞÞ , which indicates that x2  6 ½h2 þ ða21 ðtÞÞ x1  =ða22 ðtÞÞ . Substituting (10) into the above inequality yields 

x2  6



h2 ða11 ðtÞÞ þ h1 ða21 ðtÞÞ ¼ 0 a:s: ða11 ðtÞÞ ða22 ðtÞÞ

Then contraction rises. Consequently, x2  ¼ 0 a.s. Case (III). Multiplying (6) and (7) by a21 ðtÞ and a11 ðtÞ, respectively, we find

a21 ðtÞt 1 ln½x1 ðtÞ=x10  þ a11 ðtÞt 1 ln½x2 ðtÞ=x20  ¼ a21 ðtÞh1 ðtÞ þ a11 ðtÞh2 ðtÞ  ½a11 ðtÞa22 ðtÞ þ a21 ðtÞa12 ðtÞx2 ðtÞ Z t Z t a21 ðsÞr1 ðsÞdB1 ðsÞ=t þ a11 ðsÞr2 ðsÞdB2 ðsÞ=t: þ 0

ð11Þ

0

Using the strong law of large numbers for local martingales leads to

Rt lim

t!1

0

a21 ðsÞr1 ðsÞdB1 ðsÞ ¼ 0; t

Rt lim

t!1

0

a11 ðsÞr2 ðsÞdB2 ðsÞ ¼ 0; a:s: t 



Taking the superior limit for (11) and noting that ½t1 ln x1 ðtÞ 6 0; ½t 1 ln x2 ðtÞ 6 0 a.s. result in

½au11 au22 þ au21 au12 x2  P a21 ðtÞh1 ðtÞ þ a11 ðtÞh2 ðtÞ > 0 a:s: Then the desired assertion is required.  Case (IV). By Lemma 1, ½t1 ln x2 ðtÞ 6 0 a.s. If x2 > 0 a.s. is not true, then PðEÞ > 0, where E ¼ fx2 ¼ 0g. Note that, for 8x 2 E, we have limt!1 x2 ðt; xÞ ¼ 0. Theorem 2 indicates that h1 6 0 implies x1  ¼ 0 a.s. Substituting the above equality and    Eq. (8) into (7), we follow that ½t1 ln x2 ðt; xÞ ¼ h2 > 0 a.s. Then Pð½t 1 ln x2 ðtÞ > 0Þ > 0 is obtained which is a contradiction. This completes the proof. h

Remark 1. If the impulse bik ¼ 0, then lnð1 þ bik Þ ¼ 0. In this case, our results coincide with Theorem 3 in [1] without impulsive effects. Therefore, our results generalize the conclusions of [1]. Remark 2. If there exist positive constants m; M > 0 such that the impulse satisfies:

m6

Y

ð1 þ bik Þ 6 M;

8t > 0; i ¼ 1; 2:

ð12Þ

0
Then, comparing our results with the conclusions in [1], we observe that the impulse has no nature impact on the extinction and persistence of the population. However, if the impulse does not satisfy (12), it can change the properties of population significantly, see Examples 2 and 3.

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Remark 3. From the results, we can see that the negative impulse and white noise can make the values of h1 ðtÞ and h2 ðtÞ less than zero. In other words, the negative impulse and white noise can contribute to the extinction of the prey and predator, see Example 3. Rt P Remark 4. For the prey, if b1k > 0 is big enough such that 0 0 ðr1 ðsÞ  0:5r21 ðsÞÞds, i.e. h1 ðtÞ > 0, then the prey will not be extinct. This implies that the positive impulse is advantageous for the prey and it can resist the effect of the white noise, see Example 2. 



Remark 5. Let us consider the case: h2 > 0 and h1  0. This result reveals that even if the prey species is extinction, due to the effect of the impulse, the predator may be weak persistent. This situation is not going to happen in the absence of the impulse.

4. Stochastically ultimate boundedness In the previous section, we consider how the solutions of (2) vary on R2þ extensively. In this section, we continue to examine another important asymptotic property: stochastically ultimate boundedness which means that the solution will be ultimately bounded with the large probability. We first give its definition. Definition 3 ([8]). A solution xðtÞ to Eq. (2) is said to be stochastically ultimate bounded if, 8 2 ð0; 1Þ; 9H ¼ H > 0 such that

lim supP½jxðtÞj > H <  t!þ1

for any initial value xð0Þ 2 R2þ . We first examine the pth moment boundedness. Theorem 4. Let xðtÞ be a solution to (2) with initial value xð0Þ 2 R2þ , if al22 > au21 , then for all p > 0, there exist constants L1 ðpÞ and L2 ðpÞ such that xðtÞ obeys

lim supE½xp1 ðtÞ 6 L1 ðpÞ and lim supE½xp2 ðtÞ 6 L2 ðpÞ: t!1

t!1

Proof. Applying Itô’s formula and then taking expectations follow that

  1 p1 2 þ r1 ðsÞ þ r1 ðsÞ  a11 ðsÞx1 ðsÞ  a12 ðsÞx2 ðsÞ ds p 2 0   Z t Z t 1 u 6 xp10 þ pE es xp1 ðsÞ þ r u1 þ 0:5pðr21 Þ  al11 x1 ðsÞ ds 6 xp10 þ E es L1 ðpÞds ¼ xp10 þ L1 ðpÞðet  1Þ; p 0 0

E½et xp1 ðtÞ ¼ xp10 þ pE

Z

t

es xp1 ðsÞ

where u pþ1

L1 ðpÞ ¼

½1 þ pr u1 þ 0:5p2 ðr21 Þ  ðp þ 1Þpþ1 ðal11 Þ

p

:

Taking the superior limit, we find

lim supE½xp1 ðtÞ 6 L1 ðpÞ:

ð13Þ

t!1

Similarly,

  1 p1 2  r2 ðsÞ þ r2 ðsÞ þ a21 ðsÞx1 ðsÞ  a22 ðsÞx2 ðsÞ ds p 2 0   Z t 1 u 6 xp20 þ pE es xp2 ðsÞ þ 0:5pðr22 Þ þ au21 x1 ðsÞ  al22 x2 ðsÞ ds p 0    Z t  1 u p p s 6 x20 þ p e Ex2 ðsÞ þ 0:5pðr22 Þ  al22 x2 ðsÞ þ au21 Ex1 ðsÞxp2 ðsÞ ds p 0   Z t  Z t Z t 1 pau21 u 6 xp20 þ p es Exp2 ðsÞ þ 0:5pðr22 Þ  al22 x2 ðsÞ ds þ pau21 es E½xpþ1 es E½xpþ1 2 ðsÞds þ 1 ðsÞds p pþ1 0 0 0   Z t  Z t 1 pau21 u ¼ xp20 þ p es Exp2 ðsÞ þ 0:5pðr22 Þ  al22 x2 ðsÞ þ au21 x2 ðsÞ ds þ es E½xpþ1 1 ðsÞds: p p þ1 0 0

E½et xp2 ðtÞ ¼ xp20 þ pE

Z

t

es xp2 ðsÞ

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L1 ðpÞ and T, for t > T, we have E½xp1 ðtÞ 6 f L1 ðpÞ. Therefore, for t > T By (13), there are positive constants f

E½et xp2 ðtÞ 6 xp20 þ

Z

t

es L3 ðpÞds þ

0

pau21 f L1 ðp þ 1Þ pþ1

Z 0

t

es ds ¼ xp20 þ ½L3 ðpÞ þ

pau21 f L1 ðp þ 1Þðet  1Þ; pþ1

where u pþ1

L3 ðpÞ ¼

½1 þ 0:5p2 ðr21 Þ 

ðp þ 1Þpþ1 ðal22  au22 Þ

p

:

Taking the superior limit results in

lim supE½xp2 ðtÞ 6 L3 ðpÞ þ t!1

This completes the proof.

pau21 f L1 ðp þ 1Þ :¼ L2 ðpÞ: pþ1

h

Theorem 5. Let conditions of Theorem 4 be fulfilled, then the solution of system (2) is stochastically ultimate bounded. Proof. For x 2 R2þ and p > 0, we have p

2ð12Þ^0 jxjp 6 xp1 þ xp2 : By Theorem 4, we see that p

p

lim supEjxðtÞjp 6 1=2ð12Þ^0 lim supE½xp1 ðtÞ þ xp2 ðtÞ 6 1=2ð12Þ^0 ðL1 ðpÞ þ L2 ðpÞÞ :¼ LðpÞ: t!1

For any

t!1

 2 ð0; 1Þ, let H ¼ ðLðpÞ=Þ1=p . Then by Chebyshev’s inequality

PfjxðtÞj > Hg 6

EjxðtÞjp : Hp

Hence

lim supPfjxðtÞj > Hg 6  t!1

as required. This completes the proof.

h

5. Numerical simulations In this section, we will use the Milsteins Method (see [29]) to illustrate our results and the impulsive effects on the population. Example 1. For model (2), set the initial value ðx10 ; x20 Þ ¼ ð0:5; 0:5Þ, with the following choice of parameters r1 ðtÞ ¼ 0:2þ 0:01 sin t; r2 ðtÞ ¼ 0:1 þ 0:05 sin t; a11 ðtÞ ¼ a22 ðtÞ ¼ 0:1 þ 0:01 sin t; a12 ðtÞ ¼ a21 ðtÞ ¼ 0:22 þ 0:02 sin t; r21 ðtÞ=2 ¼ r22 ðtÞ=2 ¼ 0:21 1 þ0:02 sin t; b1k ¼ b2k ¼ e k2  1; tk ¼ k. 







By computation, we have h1 ¼ p2 =6  0:01 < 0; h2 ¼ p2 =6  0:31 < 0, therefore h2 ða11 ðtÞÞ þ ða21 ðtÞÞ h1 < 0. By Theorems 2 and 3, the prey and predator will go to extinction, Fig. 1 illustrates this. Now we give an example to demonstrate the effects of positive impulses on the system. Example 2. For model (2), choose the initial value ðx10 ; x20 Þ ¼ ð0:5; 0:5Þ, and the parameters are chosen as follows: r 1 ðtÞ ¼ 0:2 þ 0:01 sin t; r2 ðtÞ ¼ 0:1 þ 0:05 sin t; a11 ðtÞ ¼ a22 ðtÞ ¼ 0:1 þ 0:01 sin t; a12 ðtÞ ¼ a21 ðtÞ ¼ 0:22 þ 0:02 sin t; r21 ðtÞ=2 ¼ r22 ðtÞ=2 ¼ 0:21 þ 0:02 sin t. By the results of [1], both species will go to extinction, Fig. 2 shows this. This figure together with Fig. 1 can illustrate two points: (a) We can easily see that the impulse bik of Example 1 satisfies the condition (12). We follow from our results that this kind of impulse has no nature impact on the model as the conclusion of Remark 2 and Figs. 2 and 3 confirm this; (b) Comparing Fig. 1 with Fig. 2 we see that the negative impulse can accelerate the extinction of species. 1 Now let b1k ¼ e0:01  1; b2k ¼ ek2  1 be positive impulses, then the result changes greatly. By computation, we have   h1 ¼ 0; h2 ¼ 1:33 > 0. By Theorems 2 and 3, x1 ðtÞ is non-persistent in the mean and x2 ðtÞ will be weakly persistent in the mean. Fig. 3 illustrates this. Figs. 2 and 3 reveal that the positive impulses are advantageous for the ecosystem. In the following, we present an example to investigate the effects of negative impulses on the species.

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x1 x2

1.2

1

0.8

0.6

0.4

0.2

0

0

5

10

15 Time t

20

 12

Fig. 1. b1k ¼ b2k ¼ e

k

25

30

 1; tk ¼ k.

0.8

x1 x2

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

5

10

15

20

Time t

25

30

35

40

45

Fig. 2. b1k ¼ b2k ¼ 0.

1.4

x1 x2

1.2

1

0.8

0.6

0.4

0.2

0

0

5

10

15

20

25 Time t

30

35

40

45

50

1

Fig. 3. b1k ¼ e0:01  1; b2k ¼ ek2  1; tk ¼ k.

Example 3. For model (2), choose the initial value ðx10 ; x20 Þ ¼ ð0:5; 0:5Þ with the following choice of parameters r 1 ðtÞ ¼ 0:2 þ 0:01 sin t; r2 ðtÞ ¼ 0:1 þ 0:05 sin t; a11 ðtÞ ¼ a22 ðtÞ ¼ 0:1 þ 0:01 sin t; a12 ðtÞ ¼ a21 ðtÞ ¼ 0:22 þ 0:02 sin t; r21 ðtÞ=2 ¼ 0:1þ :020 sin t; r22 ðtÞ=2 ¼ 0:09 þ 0:02 sin t.

R. Wu et al. / Commun Nonlinear Sci Numer Simulat 20 (2015) 965–974 1.4

973

x1 x2

1.2

1

0.8

0.6

0.4

0.2

0

0

5

10

15

20

25 Time t

30

35

40

45

50

Fig. 4. b1k ¼ b2k ¼ 0.

4

x1 x2

3.5 3 2.5 2 1.5 1 0.5 0

0

10

20

30

40

50 Time t

60

70

80

90

100

Fig. 5. b1k ¼ e1  1; b2k ¼ e0:02  1; t k ¼ k.

By the results of [1], both prey population x1 ðtÞ and predator population x2 ðtÞ are weakly persistent in the mean. Fig. 4 confirms this. Now let b1k ¼ e1  1; b2k ¼ e0:02  1 be negative impulses, then the result changes greatly. By computation, we have   h1 ¼ 0:9 < 0; h2 ¼ 0:21 < 0. By Theorems 2 and 3, both x1 ðtÞ and x2 ðtÞ will go to extinction. Fig. 5 illustrates this. Comparing Fig. 4 with Fig. 5, we conclude that the negative impulses are disadvantageous for the ecosystem. 6. Conclusions and further discussions In this paper, we consider a stochastic predator–prey model with impulsive perturbations and examine the asymptotic properties of the positive solutions. Our contributions are as follows: (1) We consider the effects of the white noise and impulsive perturbations on the model at the same time. (2) The effects of the impulses on the population are considered in detail, see Remarks 2–5, examples and figures. Our results reveal that the impulse has great impacts on the model. (3) We present several examples and simulation figures, with or without impulse, to illustrate the influences of the impulsive perturbations on the population. (4) The pth moment boundedness and stochastically ultimate boundedness are also considered. Some interesting topics deserve further investigation. One may investigate the stochastic model with impulsive perturbation under regime switching. Further, one may consider n-species population system, and talk about the dynamics. Acknowledgments The authors thank the editor and referees for their very important and helpful comments and suggestions. They also thank the National Natural Science Foundation of P.R. China (Nos. 11301112, 11171081, 11171056, 11301207), Project

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