Nonlinear Dynamics in Applied Sciences Systems: Advances and PerspectivesView this Special Issue
Analysis of Stochastic Delay Predator-Prey System with Impulsive Toxicant Input in Polluted Environments
A stochastic delay predator-prey model in a polluted environment with impulsive toxicant input is proposed and studied. The thresholds between stability in time average and extinction of each population are obtained. Some recent results are extended and improved greatly. Several simulation figures are introduced to support the conclusions.
Environmental pollution by industries, agriculture, and other human activities is one of the most important socio-ecological problems in the world today. Due to toxins in the environment, lots of species have gone extinct, and many are on the verge of extinction. Thus, controlling the environmental pollution and the conservation of biodiversity are the major focus areas of all the countries around the world. This motivates scholars to study the effects of toxins on populations and to find out a theoretical persistence-extinction threshold.
Recently, a lot of population models in a polluted environment have been proposed and investigated; here, we may mention, among many others, [1–23]. Particularly, Yang et al.  pointed out that in many cases toxicants should be emitted in regular pulses, for example, the use of pesticides and the pollution by heavy metals (see, e.g., ). Thus, they proposed the following two-species Lotka-Volterra predator-prey system in a polluted environment with impulsive toxicant input: where all the parameters are positive constants and, and : the size of prey population and the predator population, respectively; : the intrinsic growth rate of the th population without toxicant; : the th population response to the pollutant present in the organism; : the concentration of toxicant in the th organism; : the concentration of toxicant in the environment; : the organism’s net uptake of toxicant from the environment; : the egestion and depuration rates of the toxicant in the th organism; : the toxicant loss from the environment itself by volatilization and so on; : the period of the impulsive effect about the exogenous input of toxicant; : the toxicant input amount at every time.
Yang et al.  showed that in the following Lemma holds.
Lemma 1. For system (1), define (a)If , then , .(b)If and , then and goes to extinction.(c)If , then ,.
Some interesting and important problems arise naturally. In the real world, the growth of species depends on various environmental factors, such as temperature, humidity and parasites and so forth. Therefore population models should be stochastic rather than deterministic (May ). Thus, what happens if model (1) is subject to stochastic noises? In addition, time delays occur in almost every situation. Kuang  has pointed out that ignoring time delays means ignoring reality. Therefore, what happens if model (1) takes time delays into account? Can we improve the results given in Lemma 1?
The aim of this paper is to study the above problems. Suppose that stochastic noises mainly affect the growth rates, with (see, e.g., [27–39]), where is a white noise and is the intensity of the noise. Moreover, taking time delays into account, we obtain the following model: with initial condition where , , is continuous on . Our main result is the following theorem.
Theorem 2. For system (3), define (i)If , then both and go to extinction almost surely (a.s.); that is, .(ii)If and , then goes to extinction and is stable in time average a.s.; that is, (iii)If , then both and are stable in time average a.s.
Remark 3. By comparing Lemma 1 with our Theorem 2, we can see that on the one hand, if and , then , , , and our stochastic delay system (3) becomes model (1); on the other hand, our results in Theorem 2 improve that in Lemma 1. Lemma 1 shows that the superior limit is positive, while Theorem 2 reveals that the limit exists and gives the explicit form of the limit. The contribution of this paper is therefore clear.
For the sake of simplicity, we introduce some notations:
Lemma 4. For any given initial value , there is a unique global positive solution to the first two equations of system (3) a.s.
Proof. The proof is similar to Hung  by defining and hence is omitted.
To begin with, let us consider the following subsystem of (3):
Lemma 6 (see ). Suppose that . (I) If there exist and positive constants , such that for , where are independent standard Brownian motions and are constants, , then one has the following: if , then a.s.; if , then . (II) If there exist positive constants , and such that for , then a.s.
Now, let us consider the following auxiliary system: with initial value .
Lemma 7. If , then the solution of system (15) obeys
Proof. By Lemma 5, Then, for all, there exists such that An application of Itô’s formula to (15) yields That is to say, we have shown that When (18) is used in (20), we can see that for , Let be sufficiently small such that . Making use of (I) and (II) in Lemma 6 to (22) and (23), respectively, we have It then follows from the arbitrariness of that Substituting (17) and (25) into (20) and noting that , one can derive that Employing (20) and (21) in the expression yields In view of (25), we get By (17), (26), (27), and (28), for all, there exists such that, for , If , then we can choose sufficiently small such that . Then, by (29) and (I) in Lemma 6, we obtain . If , then we can choose sufficiently small such that . An application of (I) and (II) in Lemma 6 to (29) and (30), respectively, makes one observe that Therefore, using the arbitrariness of results in This completes the proof.
We are now in the position to prove our main results.
Proof of Theorem 2. Applying Itô’s formula to (3) leads to
It follows from (17) and (33) that for sufficiently large . Since , then we can choose sufficiently small such that . Then, by (I) in Lemma 6, When (36) is used in (34), one can see that for sufficiently large , where obeys . In view of Lemma 6 again, ,.
By the stochastic comparison theorem , one can observe that Note that and ; it then follows from Lemma 7 that ,. Making use of (38) gives ,. Thus, for all, there exists such that, for , Substituting the above inequalities into (33) and then using (18), we obtain Let be sufficiently small such that , and then, applying (I) and (II) in Lemma 6 to (40) and (41), respectively, one can see that An application of the arbitrariness of gives
Clearly, implies , and then, by Lemma 7, Thus, similar to the proof of (28), we get Therefore, by (26), (28), and (38), we can observe that Employing (33) and (34) in the expression yields When (18), (46) and (47), are used in (48), one can obtain for sufficiently large , where obeys . It then follows from in Lemma 6 that By virtue of the arbitrariness of , we can see that Consequently, for every , there is such that Substituting the above inequality into (33) and then using (18) and (47), one can see that for sufficiently large . Since , and then, by Lemma 6 and the arbitrariness of , one can observe that When this inequality, (18) and (47), are used in (34), we can see that for sufficiently large . Then, it follows from Lemma 6 and the arbitrariness of that Substituting the above inequality and (18) into (33), we get for sufficiently large . By in Lemma 6 and the arbitrariness of again, we obtain Then, the required assertion follows from (51), (54), (56), and (58).
3. Numerical Simulations
Let us use the famous Milstein method (see, e.g., ) to illustrate the analytical results.
To begin with, we choose , , , , , , , , , , , , , and . Then, By in Lemma 1, the solution of model (1) obeys However, when the white noises are taken into account, the properties of the system may be changed greatly. In Figure 1, we let the coefficients be same with the above. The only difference between conditions of Figures 1(a), 1(b), and 1(c) is that the value of is different. In Figure 1(a), we choose . Therefore, Then, by in Theorem 2, both and are extinctive. Figure 1(a) confirms these. In Figure 1(b), we choose . That is to say and . It then follows from in Theorem 2 that is extinctive and is stable in time average: See Figure 1(b). In Figure 1(c), we choose . Then, . In view of in Theorem 2, we can obtain that both and are stable in time average: Figure 1(c) confirms these.
In Figure 2, we choose , , , , , , , , , , , , , , and . The only difference between conditions of Figures 1(c) and 2 is that the value of is different. In Figure 2, we choose . Then, . It follows from in Theorem 2 that both and are extinctive. Figure 2 confirms these. By comparing Figure 1(c) with Figure 2, one can see that the impulsive period plays a key role in determining the stability in time average and the extinction of the species.
4. Conclusions and Future Directions
This paper is concerned with stochastic delay predator-prey model in a polluted environment with impulsive toxicant input. For each species, the threshold between stability in time average and extinction is established. Some recent results are improved and extended. Our Theorem 2 reveals some interesting and important results. (A)Firstly, time delay is harmless for stability in time average and extinction of the stochastic system (3).(B)The white noise and can change the properties of the system greatly.(C)The impulsive period plays an important role in determining the stability in time average and the extinction of the species.
Some interesting questions deserve further investigations. One may consider some more realistic but more complex systems, for example, stochastic delay model with Markov switching (see, e.g., [30, 32, 39]). It is also interesting to investigate what happens if is stochastic.
The author thanks the editor and reviewer for these valuable and important comments. This research is supported by NSFC of China (nos. 11301207, 11171081, 11301112 and 11171056), Natural Science Foundation of Jiangsu Province (No. BK20130411) and Natural Science Research Project of Ordinary Universities in Jiangsu Province (no. 13KJB110002).
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