Abstract
This paper is concerned with a stochastic delay logistic model with jumps. Sufficient and necessary conditions for extinction are obtained as well as stochastic permanence. Numerical simulations are introduced to support the theoretical analysis results. The results show that the jump process can affect the properties of the population model significantly, which conforms to biological significance.
1. Introduction
Recently, Freedman and Wu [1] considered the following delay logistic model: where is the net birth rate, denotes the self-inhibition rate, represents the reproduction rate, and is the time-varying delay. There is an extensive literature concerned with the properties of system (1) and we here mention [2–4] among many others.
As we know, stochastic population models have recently been investigated by many authors (see, e.g., [5–10]). Particularly, May [5] has revealed that due to environmental noises, the growth rate should be stochastic. Suppose that the growth rate is perturbed by white noise (see, e.g., [9, 10]) where is the white noise, namely, is a Brownian motion defined on a complete probability space with a filtration satisfying the usual conditions; represents the intensity of the white noise. As a result, (1) becomes the following model:
On the other hand, the population may suffer from sudden environmental shocks, for example, massive diseases like avian influenza and SARS, earthquakes, hurricanes, epidemics, and so forth. Bao et al. [11, 12] and Liu and Wang [13, 14] incorporate a jump process into the underlying population system, which can describe these phenomena well and provide a more practical model. Particularly, the books by Applebaum [15] and Situ [16] are also good references in this area. Furthermore, some scholars have researched the theory about stochastic differential delay equations with jumps recently (see, e.g., [17–19]). However, as far as our knowledge is concerned, no articles on introducing a jump process into stochastic delay population has been introduced. Motivated by these, we will consider the stochastic delay logistic model with jumps: with the initial data , where denotes the family of all bounded, -measurable, -valued random variables and . Here, , is a real-valued Poisson counting measure with characteristic measure on a measurable subset of with , , is bounded function, and , . Furthermore, we assume that is independent of and is nonnegative constant.
Based on the fact that model (3) describes a population dynamics, it is very important to investigate the permanence and extinction. The main aims of this work are to investigate how jump process affect the permanence and extinction of model (3). Our results demonstrate that the jump process can change the permanence and extinction, which accords with biological significance. In addition, we establish the sufficient and necessary conditions for stochastic permanence and extinction of model (3).
For model (3) we always assume the following.(A1), , , , and are continuous bounded functions on with , , and , where .(A2)For each there exists such that where with .(A3)There exists a positive constant such that for .
For the simplicity, we define the following notations:
The rest of the paper is arranged as follows. In Section 2, we show that model (3) has a unique positive global solution. Afterward, sufficient and necessary conditions for extinction and stochastic permanence are established in Section 3. Section 4 mainly concentrates on introducing some figures to illustrate the main results. Finally, we close the paper with conclusions and remarks in Section 5.
2. Global Positive Solution
The classical existence and uniqueness result for solutions of a stochastic differential delay equation with jumps requires the coefficient functions to satisfy a local Lipschitz condition and a linear growth condition (see, e.g., [17–21]). Clearly, the coefficients of (3) satisfy the local Lipschitz condition, while they fail the linear growth condition. In this section, using the Lyapunov analysis method (mentioned in [12]), we will show that the jump processes can suppress the explosion and the solution of model (3) is positive and global. For later applications, let us cite the following lemma.
Lemma 1. The following inequalities hold:
Theorem 2. Let assumptions (A1)–(A3) hold. For any given initial value , (3) has a unique positive solution for any almost surely.
Proof. Since the coefficients of the equation are locally Lipschitz continuous, for any given initial value , there is a unique local solution on , where is the explosion time. To show that this solution is global, we need to show that a.s. Let be sufficiently large for . For each time integer , define the stopping time: where throughout this paper we set (as usually denotes the empty set). Clearly, is increasing as . Set ; hence a.s. If we can show that a.s., then a.s. and a.s. for all . In other words, to complete the proof all we need to show is that a.s. To show this statement, let us define a -function by . Let and be arbitrary. For , applying the Itô’s formula, we obtain where From the inequality for , , and assumptions (A1) and (A3), it is easy to see that is bounded, say by , in . We therefore obtain that Integrating both sides from 0 to , and then taking expectations, yields Letting , we obtain that . Note that for every , equals either or , and hence is not less than either or . Consequently, It then follows from (11) that where is the indicator function of . Letting gives . Since is arbitrary, we have , and so as required.
3. Permanence and Extinction for Model (3)
Theorem 3. Let assumptions (A1)–(A3) hold. If and ; then the population represented by (3) goes to extinction a.s.
Proof. Now applying Itô’s formula to (3) leads to Then we have where . The quadratic variation of is . By virtue of the exponential martingale inequality, for any positive constants , , and , we have Choose , , and . Then it follows that Making use of the Borel-Cantelli lemma yields that, for almost all , there is a random integer such that, for , That is to say, , for all , a.s. Substituting this inequality into (15), we can obtain that for all , a.s. In other words, we have shown that for , a.s., Define, for , . Under assumption (A3), , by the strong law of large numbers for local martingales (see, e.g., [22]); we then obtain Taking superior limit on both sides of (20) and then making use of (21) yield . That is to say, if , one can see that a.s.
Definition 4 (see, e.g., Bao et al. [11]). Population size is said to be stochastic permanence if, for arbitrary , there are constants and such that and .
Theorem 5. Let assumptions (A1)–(A3) hold. If and , then the population modeled by (3) will be stochastic permanence.
Proof. First, we prove that for arbitrary , there is constant such that .
Let ; we compute
where is a positive constant and
Making use of (6) in Lemma 1, we obtain . In view of the inequality above, and , we have that is bounded in ; namely, . Therefore
Once again by the Itô’s formula we have
We hence derive that
This implies immediately that . Now, for any and , then by Chebyshev’s inequality,
Hence . This implies .
Next, we claim that for arbitrary , there is constant such that .
Obviously,
Hence if , we can find a sufficiently small such that
Define for . Then, by Itô’s formula (see, e.g., [23, Theorem 2.5]),
Define , where satifies (29). In view of Itô’s formula,
Now, let be sufficiently small satisfying
Define . By virtue of Itô’s formula,
for . Note that is upper bounded in ; namely, . Consequently,
for sufficiently large . Integrating both sides of the above inequality and then taking expectations give . So for any , set . Then the desired assertion follows from the Chebyshev’s inequality [24–26]. This completes the whole proof.
Remark 6. Obviously, if assumptions (A1)–(A3) hold, , , exists, and , then Theorems 3 and 5 establish the sufficient and necessary conditions for stochastic permanence and extinction of model (3).
Remark 7. In line with in Theorem 3 and in Theorem 5, where (see Lemma 2.2 in [12]), we found that the jump process exists considerable level of detriment to permanence and leads to the extinction of the population, which conforms to biological significance.
Remark 8. If , , , and which means the jump process degenerates to zero, then our result about extinction and permanence coincide with the ones in paper [27]. Moreover, in view of Theorem 5, we found that time delay has no impact on the permanence.
4. Examples and Numerical Simulations
In this section, we will use the Euler scheme (see, e.g., [28]) to illustrate the analytical findings.
Here, we choose , , , , , , , , , and step size . The only difference between conditions of Figures 1(a) and 1(b) is that the representations of are different. In Figure 1(a), we choose , and then . In view of Theorem 3, population will go to extinction. In Figure 1(b), we consider , and then . Making use of Theorem 5, the population will be stochastic permanence. By the numerical simulations, we can find that the jump process can affect the properties of the population model significantly.
(a)
(b)
5. Conclusions and Remarks
In this paper, we investigate the permanence and extinction of a stochastic delay logistic model with jumps. Sufficient and necessary conditions for extinction are established as well as stochastic permanence.
Besides, some interesting topics deserve further consideration. One may propose some more realistic but complex models, such as introducing the colored noise into the model [29]. Another significant problem is devoted to multidimensional stochastic delay model with jumps, and these investigations are in progress.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This paper is supported by the National Natural Science Foundation of China (11271101), the NNSF of Shandong Province in China (ZR2010AQ021), and the Scientific Research Foundation of Harbin Institute of Technology at Weihai (HIT (WH) 201319).