Abstract
In this paper, a stochastic competitive model with distributed time delays and Lévy jumps is formulated. With or without a polluted environment, the model is denoted by (M) or (M0), respectively. The existence of positive solution, persistence in mean, and extinction of species for (M) and (M0) are both studied. The sufficient criteria of stability in distribution for model (M) is obtained. Finally, some numerical simulations are given to illustrate our theoretical results.
1. Introduction
The dynamics of the biological system has attracted many researchers and has no interruption in the past few decades. This includes the study of the persistence and extinction, stability in distribution of biological systems, optimal harvesting effects of renewable resources (for example, fish and plants), and so on. These studies have implications for the management of biological resources. The dynamics behaviors from the initial deterministic model to stochastic model have been extensively studied and a lot of nice results have been reported [1–4]. It has been verified that the growth rates of species are inevitably subject to white noise. And whether to consider the white noise is the difference between the stochastic model and the deterministic model. Following the method adopted in [4], we will model a stochastic system with white noise. For the biological system, usually there are three kinds of population relationship, i.e., predator-prey, mutualistic, and competitive scenarios, where the competitive scenario between populations is relatively popular [5, 6]. The general competitive model between two populations with white noise is as follows:where is the size of the i-th population at time t; represents the growth rate of i-th population; denotes the intraspecific competitive coefficients of ; and are positive and represent the competitive rates between and , respectively; stands for the standard Brownian motion defined on a complete probability space with a filtration satisfying the usual conditions (i.e., it is increasing and right continuous and contains all null sets); is the intensity of the white noise, .
However, the world economy is developing more and more rapidly and the economic development will inevitably destroy the ecological environment. With the increasing toxins and pollution into the ecological environment, the quality of human living environment is becoming worse and worse. Therefore, the study on the impact of toxin importation and environmental pollution on biological populations has become one of the most popular topics in the world [7–12], which is of vital significance to the development of sustainable economy and the protection of human’s only living environment. Based on model (1) and considering the environmental pollution factors, then we derive the following model:where , and are the concentrations of the toxicant in the organism of species and and environment at time t, respectively; denote the dose-response of species and to the organismal toxicant, respectively; and represent the absorbing and excretion rates of the toxicant from the environment respectively, is depuration rate of the toxicant, . denotes the loss rate of the toxicant because of volatilization; represents the exogenous rate of toxicant inputting into the environment and is always assumed to be bounded.
On the other hand, the behavior between predator and prey is often not always continuous. For example, in some cases, young predators cannot engage in predation; that is, young prey cannot be preyed on. These phenomena are called time delays. Similar phenomena include hibernation, pregnancy, and migration. In fact, time delays exist not only in biological systems, but in other domains as well. For example, R. Manivannan has studied a control system with probabilistic time-delay signals [13]. Therefore, time delays are very important to reveal the real world and should be taken into account in our system. Some scholars pointed out that discrete delays and continuous delays do not include each other, but the S-type distributed delays can be done [14, 15]. Therefore, taking S-type delays into account in above model is interesting. In addition, in nature there are some environmental perturbations such as earthquakes, epidemics, and hurricanes, which differ from white noise because of its sudden and destructive nature, so Lévy jumps are introduced to simulate them in mathematical modeling [16–21]. For example, Liu and Wang [18] studied the persistence and extinction of the two-species model with Lévy jumps. Liu and Bai [21] investigated the stability in distribution of a stochastic model with Lévy noises by Lyapunov functional approach.
Motivated by these, taking the S-type distributed time delays and Lévy noises into the above model, we get the following stochastic predator-prey model (M):with initial datawhere denotes the left limit of ; represents a compensated Poisson process, where is a Poisson counting measure, is the characteristic measure of on a measurable subset in with is the measure of ; is a bounded function with ; the term denotes the Lebesgue–Stieltjes integral, where denotes a nonnegative variation function defined on with satisfying The biological meanings of other parameters are the same as before. If the corresponding model is denoted by (M0), which means that the population is not contaminated.
We aim to study the dynamical behaviors of (M) and (M0) such as the extinction and persistence in mean for all species and explore the impacts on the dynamics of time delays and Lévy noise.
The article is structured as follows. For preliminaries, we give some notations and important lemmas in Section 2. In Section 3, we establish the sufficient criteria for the persistence in mean and nonpersistence of M and M0 and investigate the stable in distribution of (M). In Section 4, some numerical simulations are presented to verify our main results. Finally, conclusion and discussion are given to end this article in Section 5.
2. Preliminaries
For the simplicity, we first make the following notations:
Our later discussion are based on the following technological hypotheses.
Assumption 1. There exists a positive constant such that
Assumption 2. Suppose that , which means the internal competition is greater than the external competition (see [22]).
The following lemmas are necessary for our later proof.
Lemma 1. Let Assumption 1 hold, then for any given initial data , there exists a unique solution remaining in with probability 1.
Proof. It is obvious that the coefficients of model (M) are locally Lipschitz. By [5], model (M) has a unique local solution for any initial data and , where is the explosion time. It needs only to verify that Let be sufficiently large such that for each integer . Define the stopping timeClearly, is strictly increasing with . Let , then Then, we only need to prove If the statement is not true, then there exist and such that and an integer such thatDefinewhereChoose a constant and integer such thatFor model (M), by Ito’s formula, we getwhereBy basic inequality , we haveSubstituting (14) and (15) into , thenTherefore,Substituting (17) into and together with (11), there exists a constant such thatBy this result and according to the argument in [23], we havewhich leads to a contradiction, and hence, Therefore, The proof is completed.
Lemma 2. Let be a positive solution of (14) initial data . Then, for any , there exists a constant such that
Proof. We only prove . The rest is similar and omitted. Defining , by Ito’s formula, we havewhereLet thenIntegrating both sides of (21) from 0 to t and taking expectation leads toThen, . The proof is completed.
Lemma 3. (see [18]). Let , and satisfying (1)If there exist two constants and such that for all , then (i) and if ; and if .(2)If there exist two constants and such that for all then (i) if ; if .(3)If there exist two constants and such that for all
Then,
Lemma 4. Let be any positive solution of model (M), then(1)(2)For any positive constant ,
The proof of Lemma 4 is standard and is omitted (see, e.g., [24]).
3. Main Results
3.1. Persistence in Mean
Definition 1. (see [25]). The system is said to be persistence in mean if there are positive constants and such thatholds for any solution of model (M) with initial data
Theorem 1. Assume , then for any initial data , the solution of model (M0) has the following properties:
Proof. For , we computeFor model (M0), noticing and using Ito’s formula, we getwhereSinceBy Lemma 4, we getBy the same way, we can derive thatSubstituting (35) and (36) into (32) and (33), respectively, thenComputingwhere . From (37), we can easily get
By Lemma 4 again, for any , there is a such thatAccording to Lemma 3, thenThus, for any and sufficiently large , there isUsing (41) in (31), we getLemma 3 implies thatSimilarly, we havewhere Obviously,
From Lemma 4, for any , there is a such thatIt follows from Lemma 3 thatCombining (43) and (46) leads toSubstituting (47) into (30) and using Lemma 2, we getThe proof is completed.
Next, let us consider model (M).
Theorem 2. If hold, then for any initial , the solution of model (M) has the properties that
That is to say, model (M) will be persistence in mean.
Proof. Using Ito’s formula to compute , we haveThen,We can get from Lemma 3 thatFor any , there is sufficiently such thatSubstituting (54) into (50), we getTherefore,Similar computation leads toIn view of Lemmas 3 and 4, we haveFor any , there is a such thatSubstituting (59) into (51), we getand then we haveCombing (53), (56), (58), and (61) leads to the result. The proof is completed.
Remark 1. If the limit of and exist, that is, and , then Theorem 2 will be simplified as the following case.
If hold, then model (M) will have the properties that
3.2. Nonpersistence
Theorem 3. If holds, then for any initial data the solution of model (M) has the properties that
That is to say, model (M) is nonpersistent.
Proof. From (50), we getAccording to Lemma 3, we getWe can easily computeBy Lemma 4, for enough large and any , there isThen,By the assumptionand Lemma 3, we obtain from (68) thatSubstituting (70) into (50), thenFrom Lemma 3, we getand thus,The proof is finished.
Theorem 4. If holds, then for any initial data the solution of model (M) has the properties that
That is to say, model (M) is nonpersistent.
Proof. Under the assumption that , we can get from Lemma 3 and (64) thatSubstituting (75) into (51) readsThe assumption implies that Then, from (76) and Lemma 3, it is easy to verify thatThis completes the proof.
3.3. Stable in Distribution
Theorem 5. If all the assumptions hold, then model (M) is stable in distribution.
Proof. The proof of this result is divided into three steps.
Step 1. We first prove model (M) is globally attractive.
Let be any two solutions of (M) with initial data , respectively. Denote . We only need to verify thatDefine . By Ito’s formula, we getLetwhere
Using Ito’s formula to (80), we can easily compute thatAccording to (81), we getwhich meansConsequently,Furthermore, considering the continuity of and combining (M), we haveTherefore,It is not difficult to see that is differential. From Lemma 2, we getwhere is a positive constant. Therefore, is uniformly continuous. Similarly, we can also obtain that is uniformly continuous. By virtue of (84) and Barbalat’s conclusion in [26], we getStep 2. For any , it is denoted by the transition probability of the process , the probability of , and the space of all probability measures on . For any , definewhere Thanks to Lemma 2 and Chebyshev’s inequality, for any , the family is tight. That is, for any , there exists a compact subset such that for any .
We computewhere .Therefore, for sufficiently large and any , we can derive thatThat is to say, is Cauchy in with initial data .
Step 3. We prove
According to (92), for is Cauchy in , then there exists a unique such thatBy virtue of (78), we derive
. That is, specie will die out at some point.By the triangle inequality and together with (93) and (94), we haveThis completes the proof.
4. Numerical Simulations
In this section, some numerical simulations are given to verify our theoretical results. Take . It is easy to check that which means Assumption 2 holds.(1)Set and then it is easy to count that Theorem 1 implies that is, the species are both persistent in mean. Simulation also validates the result, see Figure 1.(2)Set By computation, we have By Theorem 2 and Remark 1, model (M) is persistent in mean, shown in Figure 2.(3)Set and then these parameters satisfy the conditions of Theorem 3. By Theorem 3, we know that is permanent, but will die out, see Figure 3.(4)Set Obviously, Theorem 4 shows that both and will die out, see Figure 4.
(a)
(b)
(a)
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5. Conclusions
The study of biological dynamics has been a popular topic in the field of biomathematics in recent years. With the development of economy, the environmental pollution is becoming more and more serious, which has become an important factor affecting the population relationship. Time delays are also important factors affecting the relationship. In this paper, we formulate a delayed predator-prey model with Lévy noise. Theorem 1 and 2 give the sufficient criteria of persistent in mean for cases (M) and (M0), respectively. Theorem 3 and 4 obtain the sufficient conditions of nonpersistence. Theorem 5 investigates the stable in distribution. Finally, numerical simulations are given to validate our conclusion.
In view of the complexity of the environments, other factors such as the telephone noise and impulsive input may bring important influence to the dynamics, which needs further research in the future.
Data Availability
No data were used to support the findings of this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (11861027).