Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2016 / Article
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Qualitative Analysis on Differential, Fractional Differential, and Dynamic Equations and Related Topics

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Research Article | Open Access

Volume 2016 |Article ID 5749892 | https://doi.org/10.1155/2016/5749892

Xinzhu Meng, Xiaohong Wang, "Stochastic Predator-Prey System Subject to Lévy Jumps", Discrete Dynamics in Nature and Society, vol. 2016, Article ID 5749892, 13 pages, 2016. https://doi.org/10.1155/2016/5749892

Stochastic Predator-Prey System Subject to Lévy Jumps

Academic Editor: Elvan Akın
Received06 Jan 2016
Accepted13 Mar 2016
Published31 Mar 2016

Abstract

This paper investigates a new nonautonomous impulsive stochastic predator-prey system with the omnivorous predator. First, we show that the system has a unique global positive solution for any given initial positive value. Second, the extinction of the system under some appropriate conditions is explored. In addition, we obtain the sufficient conditions for almost sure permanence in mean and stochastic permanence of the system by using the theory of impulsive stochastic differential equations. Finally, we discuss the biological implications of the main results and show that the large noise can make the system go extinct. Simulations are also carried out to illustrate our theoretical analysis conclusions.

1. Introduction

Omnivory is considered as a common ecological phenomenon in the natural world. Omnivorous predator feeds on both animal prey and plant, so the intrinsic growth rate for predator should be positive. For example, the giant panda is omnivorous animal, since it can eat both meat and plant such as bamboo. With the development of the economy, pollution is becoming more and more serious. Thus pollution models have widely attracted the focus of the people [15]. A deterministic predator-prey system with omnivorous predator in an impulsive polluted environment takes the following form:where ,  ,  ,  ,  , ,  ,  ,  , and   are positive constants, and denote prey and omnivorous predator densities, and and denote the concentrations of the toxicant in the organism and in the environment, respectively. ,  , and are all positive bounded continuous functions on . and represent the prey intrinsic growth rate and the predator intrinsic growth rate, respectively, are the density-dependent coefficients of the prey and the predator, is the capturing rate of the predator, and are damage rates of the prey and predator by the toxicant, respectively, represents environmental toxicant uptake rate per unit mass organism, and are organismal net ingestion and depuration rates of toxicant, respectively, denotes the loss rate of toxicant from the environment itself by volatilization, and is the amount of pulsed input concentration of the toxicant at each .

System (1) is a deterministic model, where all parameters in the model are deterministic. However, there are some limitations in mathematical model from a biological viewpoint. Therefore, it is significant to study the effects of noises on population systems [69]. There are many kinds of environmental noises. First, we assume that the toxicant uptake rates are disturbed by white noise. If we still let represent the toxicant uptake rates, then and can be replaced bywhere are white noises and are the intensities of the white noises, which are bounded continuous functions on . Then we obtain the following stochastic system with impulsive toxicant input in a polluted environment: where are mutually independent standard Brownian motions defined on a complete probability space .

On the other hand, populations may be affected by sudden environmental fluctuations, such as severe weather, earthquakes, floods, and epidemics. Brownian motion cannot describe these phenomena better, so it is very important to introduce Lévy noise into the population system [10]. There are many researches about autonomous stochastic predator-prey system with Lévy jumps [11, 12].

Inspired by these, we focus on nonautonomous impulsive stochastic predator-prey system with white noises and Lévy jumpswhere and represent the left limit of and , respectively, is a Poisson counting measure with characteristic measure on a measurable bounded subset of with , and are independent of . The Poisson counting measure is represented by ;   are continuous functions on , which are assumed to be periodic with period . Other parameters are defined as in system (1).

The paper is arranged as follows. In Section 2, we prove that system (4) has a global positive solution. Section 3 shows the main result; in Section 3.1 we prove the extinction of system (4). We also examine almost sure permanence in mean and the stochastic permanence of the system in Sections 3.2 and 3.3. Finally we present some simulations and conclusions to close the paper in Section 4.

2. Notations and Global Positive Solution

For the purpose of convenience, we introduce some notions and some lemmas which will be used for our main results. We throughout this paper assume that ,  , and are continuous at , and is left continuous at and and let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is increasing and right continuous while contains all -null sets). Further assume that is a scalar Brownian motion defined on the complete probability space .

We denote as the positive cone in ; that is, . If is a bounded continuous function on , we define Now we give some basic properties of the following subsystem of systems (1) and (4):

Lemma 1 (see [3]). System (6) has a unique positive -periodic solution which is globally asymptotically stable. Moreover, for all if , wherefor and .

Lemma 2. For any positive solution of system (1) or (4) with initial value , one has

Proof. Through a simple calculation, we can get Moreover, since is a periodic function, we have Hence one can observe that

Then we show an assumption which will be used in the following proof.

Assumption 3. There exists a bounded continuous function such that

Theorem 4. For any given initial value , there is a unique solution of (4) and the solution will remain in with probability 1; that is, for all almost surely.

Proof. The coefficient of (4) is locally Lipschitz continuous, and so, for any given initial value , there is a unique local solution for , where is the explosion time. To demonstrate that this solution is global, we need to show that   a.s. Let be sufficiently large for and . For each integer , define the stopping timewhere we set ( denotes the empty set). Obviously, is increasing as . Set ; thus   a.s., so we just need to demonstrate that . If is not true, then there exist two constants and such that . Thus there is an integer such that for all .
Define a function as follows: Let be arbitrary. Applying Itô’s formula and Assumption 3 leads to where where SoIntegrating (18) from to and taking the expectations for both sides result inLet for ; we have . ThenIt is inferred from (19) and (20) that where is the indicator function of . Let ; we have that is a contradiction; then we have .
This completes the proof of Theorem 4.

3. Main Results

3.1. Extinction

For convenience, we prepare the following lemma.

Lemma 5. Suppose that and let Assumption 3 hold.
(I) If there exist two positive constants and such that for all , where ,  , and are constants, then (II) If there exist two positive constants and such that for all , then provided that .

Lemma 5 is proved in Appendix by using a similar method in [13].

Now, we will prove the extinction of system (4).

Define

Theorem 6. Ifthen

Proof. By Assumption 3 and the strong law of large numbers for local martingales, one has DefineApplying Itô’s formula yieldsIntegrating both sides of (32) and (33) from 0 to , respectively, we haveFrom (34), we can obtain thatTaking the limit superior results inFrom (27) we can know that Thus we haveApplying (35) leads toTaking the limit superior, together with (28) and (38), we can see that Therefore, This completes the proof of Theorem 6.

For simplicity, we define

Theorem 7. If and , then

Proof. According to (34), we can obtain that It follows from Lemma 5 that By (35), we have and then Since , then we know that which combined with (34) and the property of the limit superior shows that, for any , there exists a random number for such that By Lemma 5, we have thatThis completes the proof of Theorem 7.

3.2. Permanence in Mean

In this section, we need to show the permanence in mean. Note that implies that productiveness of the prey is less than its death loss rate; then the prey can go extinct. Naturally, we assume in the rest of this paper.

Define

Theorem 8. Ifthen

Proof. Define such that , where satisfiesIn [9], we know that It can be inferred from comparison theorem thatFrom (34) and (35), we know thatBy   ×  (57) +   ×  (58), we obtain thatWhen is used in (59), applying Lemma 5 and (56) leads toAccording to (57) and (60), we can haveFrom the conditions of Theorem 8, we know that and applying Lemma 5 results inBy (58) and (62), we obtain thatFrom the conditions of Theorem 8, we see that and applying Lemma 5 leads toBy (58) and (64), we can haveAccording to the conditions of Theorem 8, we see that and applying Lemma 5 yieldsThis completes the proof of Theorem 8.

3.3. Stochastic Permanence

For the sake of proving stochastic permanence, we should examine the th moment boundedness firstly.

Define

Theorem 9. If there exists a constant such thatthen

Proof. DefineApplying Itô’s formula and (68) yieldswhere