Abstract
In this paper, we explore an impulsive stochastic infected predator-prey system with Lévy jumps and delays. The main aim of this paper is to investigate the effects of time delays and impulse stochastic interference on dynamics of the predator-prey model. First, we prove some properties of the subsystem of the system. Second, in view of comparison theorem and limit superior theory, we obtain the sufficient conditions for the extinction of this system. Furthermore, persistence in mean of the system is also investigated by using the theory of impulsive stochastic differential equations (ISDE) and delay differential equations (DDE). Finally, we carry out some simulations to verify our main results and explain the biological implications.
1. Introduction
With the development of the economy, environmental pollution is caused by various industries and other activities of human, which has been one of the most important social problems in the world today. Many species have gone extinct due to the toxicant in the environment. Therefore, controlling the environmental pollution has been the important topics around the world. There are many researchers which have investigated the pollution models in recent years [1–3]. In addition, a lot of animal populations suffer from infectious disease, so some scholars investigated the predator-prey systems with diseases [4–8]. For example, a deterministic predator-prey model with infected predator in an impulsive polluted environment is described by the following equation:where , , , , , is the density of the prey at time , and and represent the density of susceptible predator and infected predator at time , respectively. and stand for the concentrations of the toxicant in the organism and the environment at time , respectively. are dose-response parameter to the toxicant. stands for the intrinsic growth rate of the prey. denote the death rates of and , respectively. is the infection rate. are density-dependent coefficients, and and represent predation rate and ingestion rate, respectively. are environmental toxicant uptake rates, denote organismal net depuration rates, represent organismal net ingestion rates, is the loss rate of toxicant from environment, and denotes the amount of pulsed input concentration of the toxicant at every time. The above parameters are all positive constants. Next, we propose a new mathematical model by taking more factors into account based on model (1).
In the natural world, time delay often occurs in almost every situation. Thus it is significant to take time delay into consideration [9–14]. As we know, deterministic model is not enough to describe the species activities. Sometimes, the species activities may be disturbed by environmental noises. May [15] revealed that the birth rates, death rates, carrying capacities, competition coefficients, and other parameters involved in the system should exhibit random fluctuation to a greater or lesser extent. Hence some parameters should be stochastic [16–26]. First, we assume that the intrinsic growth rate and the death rates of species are disturbed by white noise, then and can be replaced bywhere are standard Brownian motions and are the intensities of . are mutually independent defined on a complete probability space .
Furthermore, populations may suffer from sudden environmental fluctuations, such as floods and earthquakes, which cannot be described by Brownian motions. To explain these phenomena, introducing a jump process into the underlying population dynamics is one of the important methods. Thus, there are many scholars introduce Lévy jumps into the population system [27–31]. Taking all above influences into consideration, we focus on the infected stochastic predator-prey system with Lévy jumps and delays in a polluted environmentwhere , , and stand for the left limits of , , and , respectively. denotes a Poisson counting measure with characteristic measure which defines a measurable bounded subset of with and , and is bounded and continuous with respect to and is -measurable, and (see [27–30]). Moreover, are independent of , are death rates of species, and represent the time delay. Other parameters are defined as system (1).
The rest of this paper is arranged as follows. Section 2 introduces some lemmas which will be used in our main results. In Section 3, we show the main results. We examine the extinction of system (3) in Section 3.1; in Section 3.2 we also prove the permanence in mean of this system. Finally, we present some simulations and conclusions in Section 4.
2. Preliminary Results
Throughout the paper, we assume that , , , and are continuous at and is left continuous at and . Moreover, let be a complete probability space with a filtration satisfying the common conditions (i.e., it is increasing and right continuous while contains all -null sets).
For the sake of convenience, we introduce some notions and some lemmas which will be used for the main results. We definewhere is a bounded continuous function on .
Then we show some basic properties of the subsystem of system (3)
Lemma 1 (see [3]). System (5) has a unique positive -periodic solution which is globally asymptotical stable. Furthermore, if and , then and for all , where for , , and .
It can be obtained from a simple calculation that Since is a periodic function, then we getThus we have
From Lemma 1, we know that the long time dynamical behaviors of system (3) can be replaced by the dynamical behaviors of the following limiting system:
Now we give an assumption which will be used in the following proof.
Assumption 2. There exist constants such that
Lemma 3. For any given initial value , there is a unique solution of (10) on and the solution will remain in with probability 1.
Proof. This proof is the same as Theorem in [11] by defining where Thus, we omit it here.
The stochastic comparison theorem and limit superior and limit inferior theory are given as follows.
Lemma 4 (see [27]). Suppose that .
(i) If there exist three positive constants , , and such thatfor all , where and are constants, then(ii) If there exist three positive constants , , and such thatfor all , then
First, we explore the following auxiliary system:
Lemma 5. For system (17), let be the solution of this system with initial value .
(i) If , then (ii) If , when , then(iii) If , , when , then(iv) If , when , then
Proof. Applying Itô’s formula to system (17) leads toIntegrating both sides of (22), we haveThen we can obtainCase (i). From Lemma 4 and (24), if , then we have Obviously, we findCase (ii). By Lemma 4, it is derived from (24) and conditions thatSince then we obtainUsing Assumption 2 and the strong law of large numbers for local martingales, one hasThen, substituting (31) into (24) yieldsSincethen we have that, for any , there exists such thatCombining (25) with (36) yieldsFrom Lemma 4 and conditions, we have Obviously, we have Case (iii). By (25), we obtainIt can infer from (37), (40), and Lemma 4 that Since is an arbitrary number, then we get Combining this equality with (25) leads toSimilar to (34) and (35), we have that, for any , there exists such thatNote that , so we have Case (iv). By (26), we getWhen , from (44) and (46), using Lemma 4 results inSince is an arbitrary number, then we can obtain Combining this equality with (26) leads toThis completes the proof of Lemma 1.
3. Main Results
3.1. Extinction
Now we are going to show our main results. By Lemma 5, we can get the extinction of system (10).
Theorem 6. For system (10), let be the solution of this system with initial value .
(i) If , then (ii) If , when , then(iii) If , , when , then
Proof. By stochastic comparison theorem, we haveBy (34), we deriveIn the same way, we can verify thatIt follows from (33), (43), and (49) thatApplying Itô’s formula to system (10) leads toThen we can obtainComputing (60) × + (61) × leads towhere is given in Appendix.
Case (i). By Lemma 5, we know that , if . From (53), we have Obviously, we get Case (ii). By Lemma 5, we know that , if and . From (53), we have Then we obtain By using (60) and Lemma 4, we can prove Case (iii). Similarly, by Lemma 5, we get that , if , , and . By (53), we obtain Combining Lemma 4 with (56) and (63), we get From (60), we haveThen we infer from Lemma 4 that By (61), we get From Lemma 4, we obtain By (60), we have In view of Lemma 4, we can see that Consequently, This completes the proof of Theorem 6.
3.2. Permanence in Mean
In this section, we prove the permanence in mean of system (10).
Theorem 7. Let be the solution of system (10) with initial value . If and , when , then
Proof. Computing (62) × + (63) × , we obtain where is given in Appendix.
When , by Lemma 4, we have So Computing (63) × (62) × , we have where is given in Appendix.
When , by Lemma 4, we have From (71), when , we obtain By (61), we get then from Lemma 4 we prove that By (60), we have and then from Lemma 4 we get It can be inferred from (62) thatand then from Lemma 4 we have Consequently, This completes the proof of Theorem 7.
4. Conclusion and Simulations
This paper explores the dynamics of a stochastic predator-prey model with time delays in the polluted environment. We show some properties of the subsystem of the predator-prey system. Then by using comparison theorem and limit superior theory, we obtain the sufficient conditions for the extinction of this system. From Theorem 6, we know that if environmental noises are large enough, the species will be extinct (see Figure 1) and if the amount of pulsed input concentration of the toxicant is large enough or the pulse period is small enough, the species will also be extinct (see Figures 2 and 3). In addition, Theorem 7 indicates that the permanence of populations has high correlation with the intensity of environmental noises (see Figure 2(a)). The main results show that time delays can lead to extinction of this system. Therefore, we realize that the environmental noises, the toxicant input, and delays are harmful to the permanence of the populations. Now we show some simulations to verify our main results.
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
Choose some parameters as follows , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and Based on the above parameters, we give simulations to explain the biological implications.
In Figure 1, we keep and . In Figure 1(a), choose , , and , there is no Lévy noises, and we can see that the species are permanent. In Figure 1(b), if , , and , the intensities of Lévy noises are large and then we get , which means that the productiveness of the prey is less than its death loss rate; from Theorem 6, the species go extinction. In Figure 1(c), if , , and , then and , from Theorem 6 we have that is permanent and and are extinct; furthermore, we have . In Figure 1(d), if , , and , then , , and ; from Theorem 6 we obtain that and are permanent and is extinct and we get and .
In Figure 2, we keep , , , and . In Figure 2(a), we choose , and then , , and . From Theorem 7, we have that , , and are permanent and we obtain that , and . In Figure 2(b), if , then and , and from Theorem 6, we get that and are permanent and is extinct and and . In Figure 2(c), if , then , is permanent, and are extinct by Theorem 6, and . In Figure 2(d), we choose , and then ; from Theorem 6 the species are extinct.
In Figure 3, we keep , , , and . In Figure 3(a), we choose and then , , and . From Theorem 7, we have that , , and are permanent and we obtain that , , and . In Figure 3(b), if , then and , and from Theorem 6, we get that and are permanent and is extinct and and . In Figure 3(c), if , then , is permanent, and are extinct by Theorem 6, and . In Figure 3(d), we choose and then ; from Theorem 6 the species are extinct.
From Figures 1–3, we can obtain the following conclusions:(1)Large stochastic disturbance can cause the populations to go to extinction; that is, the persistent population of a deterministic system can become extinct due to the white noise stochastic disturbance.(2)Large impulsive input concentration of the toxicant or small impulsive period of the exogenous input of toxicant can cause the populations to go to extinction.
Therefore, the above numerical simulations illustrate the performance of the theoretical results, and the biological results show that the white noise stochastic disturbance and impulsive toxicant input are disadvantage for the permanence of system.
Appendix
For the sake of convenience, we define
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (11371230 and 11561004), the SDUST Research Fund (2014TDJH102), Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, Shandong Province, the Open Foundation of the Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, Gannan Normal University, China, and Shandong Provincial Natural Science Foundation, China (ZR2015AQ001).