Abstract and Applied Analysis

Volume 2014 (2014), Article ID 634871, 8 pages

http://dx.doi.org/10.1155/2014/634871

Research Article

Dynamics of a Stochastic Delayed Competitive Model with Impulsive Toxicant Input in Polluted Environments

Meiling Deng, Meng Liu, and Chuanzhi Bai

School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China

Received 1 March 2014; Revised 30 April 2014; Accepted 4 May 2014; Published 25 May 2014

Academic Editor: Rehana Naz

Copyright © 2014 Meiling Deng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In the world today, with the rapid development of modern agriculture and industry, a large quantity of pollutants enter into ecosystems one by one, which is a threat to the persistence of the exposed populations. This paper investigates a stochastic delayed competitive system with impulsive toxicant input in a polluted environment. Under a simple condition, sufficient and necessary conditions for stability in the mean and extinction of each species are established. Some recent works are improved and extended greatly. Some numerical simulations are also included to illustrate and support the findings.

1. Introduction

In this paper, we consider the following stochastic delay competitive model in polluted environments with impulsive toxicant input:

with initial data

where , , and is a continuous function on . All coefficients in model (1) are positive. and ; is the size of the th population, ; is the growth rate of the th population; is the response to the pollutant present in the organism of the th population; is the toxicant concentration in the organism; is the toxicant concentration in the environment; is the organism's net uptake of toxicant from the environment; is the egestion and depuration rates of the toxicant in the organism; is the toxicant loss from the environment itself; is the period of the impulsive effect about the exogenous input of toxicant; is the amount of toxicant input at every time. is a standard Brownian motion defined on a complete probability space ; is the intensity of the environmental noise.

Recently, population models with toxicant effect have received great attention; see, for example, [1–18]. Liu and Zhang [15] considered the following competitive model in polluted environments with impulsive toxicant input:

For model (3), the authors [15] proved the following.

Lemma 1 (see [15]). Define

Suppose that , , .

(A)If and , then and .(B)If and , then and .(C)If and , then , .

From the work of Liu and Zhang [15], some important and interesting questions arise naturally.

(Q1)In the real world, the growth of population is inevitably affected by random environmental fluctuations. May [19] have claimed that population systems should be stochastic. Therefore, what happens if system (3) is affected by environmental fluctuations?(Q2)Gopalsamy [20] have pointed out that, in order to be reality, time delays should not be ignored. Hence, what happens if system (3) incorporates with time delays?(Q3)Can we improve the results obtained in Lemma 1?

The aims of this paper are to investigate the above questions. Recall that stands for the growth rate. In practice, we often estimate it by an average value plus an error term. Generally, by the famous central limit theorem, the error term follows a normal distribution. Hence, for short correlation time, we can replace with (see, e.g., [21–29]), where is white noise and is the intensity of the noise. At the same time, incorporating with time delays, we get model (1). For model (1), we will show the following.

Theorem 2. Suppose that . Define

(I)If and , then both and go to extinction almost surely; that is,

(II)if and , then goes to extinction a.s. and is stable in the mean a.s.; that is,

(III)if and , then goes to extinction a.s. and is stable in the mean a.s.; that is,

(IV)if , ,

(A)if and , then goes to extinction a.s. and is stable in the mean a.s.

(B)if and , then goes to extinction a.s. and is stable in the mean a.s.

(C)if and , then both and are stable in the mean a.s.

Remark 3. It is useful to point out that if , , and , then and will not hold simultaneously.

Remark 4. In comparison with most of the existing results, our key contributions in this paper are as follows.

(i)To the best of our knowledge, this paper is the first attempt to consider stochastic delay competitive model in polluted environments.(ii)Our conditions are much weaker. For example, the authors [15] supposed and which are dropped in this paper.(iii)Our results improve some recent works. For example, Lemma 1 indicates that the superior limit is positive while Theorem 2 proves that the limit exists and establishes the explicit form of the limit.

2. Proof

For simplicity, define

Lemma 5. For any given initial data , there is a unique global positive solution to model (1) on a.s. and

Proof. The proof is a special case of Theorems 5.1 and 5.2 in Liu and Wang [25] and hence is omitted.

Lemma 6 (see [26]). Suppose that .

(I)If there exist and positive constants , such that

for , where are independent standard Brownian motions and are constants, , then we have the following: if , then a.s.; if , then a.s.(II)If there exist positive constants , , and such that

for , then a.s.

To begin with, let us consider the following subsystem of (1):

Lemma 7 (see [12, 13]). System (16) has a unique positive -periodic solution and for each solution of (16), and as . Moreover, and for all if and , , where

for and . In addition,

Proof of Theorem 2. It follows from Lemma 7 that

That is to say, for all , there is such that

Applying Itô's formula to (1) leads to

In other words, we have shown that

(I) Assume that and . In view of (22),

By a.s., (19) and we have that

Therefore, , a.s. Similarly, it follows from (9) that if , then , a.s.

(II) Assume that and . Since , then, by (I), , a.s. Hence, for arbitrary , there exists such that for

When the above inequalities and (20) are used in (22), we can obtain that, for ,

Note that ; we can let be sufficiently small such that . Applying (i) and (ii) in Lemma 6 to (26) and (27), respectively, one can see that

It therefore follows from the arbitrariness of that

The proof of (III) can be obtained similarly and hence is omitted.

Now, we are in the position to prove (IV). Assume that and . For , consider the following stochastic equation:

By the classical stochastic comparison theorem [30], one can see that

Note that , ; an argument, identical to the argument used in the proof of (II), shows that

Consequently,

This, together with (31), implies that

On the other hand, computing (9) − (22) gives

In view of (13) and (34), for arbitrary , there exits such that, for ,

When the above inequalities and (20) are used in (35), we can see that for

Similarly, computing (22) − (9) gives

Hence, for sufficiently large ,

(A) Assume that and . Since , then we can let be sufficiently small such that . Applying (i) in Lemma 6 to (37) results in , a.s. The proof of a.s. is the same as that in (II) and hence is omitted. The proof of (B) is similar to that of (A) and hence is omitted.

(C) Assume that and . Notice that ; then, by (37) and Lemma 6,

It then follows from the arbitrariness of that

Similarly, by (39), Lemma 6, and the arbitrariness of , we have

Let be sufficiently small satisfying . Substituting (34), (41), and (20) into (22) yields

for sufficiently large . By (ii) in Lemma 6 and the arbitrariness of , one can observe that

Similarly, when (34) and (42) and (20) are used in (9), we can see that , a.s. This, together with (41), (42), and (44), indicates that

3. Numerical Simulations

In this section, using the classical Milstein method (see, e.g., [31]), we work out some numerical figures to support the analytical results. In Figure 1, we choose , , , , , , , , , , , and . Then, , , ,, , and . The only difference between conditions of Figures 1(a), 1(b), 1(c), 1(d), 1(e), and 1(f) is that the values of and are different.

(a)In Figure 1(a), we choose , . Then, , . By virtue of (I) in Theorem 2, both and are extinct; see Figure 1(a).(b)In Figure 1(b), we set , . Then, , . In view of (II) in Theorem 2, is extinct and . Figure 1(b) confirms this.(c)In Figure 1(c), we let , . Then, , . It follows from (III) in Theorem 2 that is extinct and

See Figure 1(c).(d)In Figure 1(d), we set , . Then, , , and . According to (A) in Theorem 2, is extinct and

Figure 1(d) confirms this.(e)In Figure 1(e), we choose , . Then, , , , and . By (B) in Theorem 2, is extinct and

See Figure 1(e).(f)In Figure 1(f), we let , . Then, and . It follows from (C) in Theorem 2 that

See Figure 1(f).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1: Solutions of system (1) for , , , , , , , , , , , , , and step size . (a) is with ; (b) is with , ; (c) is with , ; (d) is with , ; (e) is with , ; (f) is with , .

4. Conclusions and Future Directions

This paper investigates a stochastic delay competitive model in a polluted environment with impulsive toxicant input. For each population, the critical value between stability in the mean and extinction is obtained. Some recent works are extended and improved. Our Theorem 2 has some important and interesting interpretation.

(1)Time delay is harmless for stability in the mean and extinction of the stochastic model (1).(2)White noises can change the dynamics of the population model greatly.

Some interesting problems deserve further study. One can consider some more realistic systems, for example, stochastic delayed population model with the Markov switching (see, e.g., [22, 23, 29]). It is also interesting to extend Theorem 2 to -species case.

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the reviewers for their valuable comments. The authors also thank Dr. H. Qiu and Dr. C. Zhang for helping them to improve the English exposition. This research is supported by National Natural Science Foundation of PR China (nos. 11301207, 11171081, 11271364, and 11301112), Natural Science Foundation of Jiangsu Province (nos. BK2011407 and BK20130411), and Natural Science Research Project of Ordinary Universities in Jiangsu Province (no. 13KJB110002).

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