Analysis of a Stochastic CompetitiveModel with Distributed Time Delays and Jumps in a Polluted Environment

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. +e existence of positive solution, persistence in mean, and extinction of species for (M) and (M0) are both studied. +e sufficient criteria of stability in distribution for model (M) is obtained. Finally, some numerical simulations are given to illustrate our theoretical results.


Introduction
e dynamics of the biological system has attracted many researchers and has no interruption in the past few decades.
is 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. ese studies have implications for the management of biological resources. e dynamics behaviors from the initial deterministic model to stochastic model have been extensively studied and a lot of nice results have been reported [1][2][3][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]. e general competitive model between two populations with white noise is as follows: dy 1 (t) � y 1 (t) r 10 − a 11 y 1 (t) − a 12 y 2 (t) dt + σ 1 y 1 (t)dB 1 (t), dy 2 (t) � y 2 (t) r 20 − a 21 y 1 (t) − a 22 y 2 (t) dt + σ 2 y 2 (t)dB 2 (t), (1) where y i (t) is the size of the i-th population at time t; r i0 represents the growth rate of i-th population; a ii > 0 denotes the intraspecific competitive coefficients of y i ; a 12 and a 21 are positive and represent the competitive rates between y 1 and y 2 , respectively; B i (t) stands for the standard Brownian motion defined on a complete probability space (Ω, F, F { } t≥0 , P) with a filtration F { } t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous and F 0 contains all P null sets); σ 2 i is the intensity of the white noise, i � 1, 2.
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.
erefore, 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][8][9][10][11][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: dy 1 (t) � y 1 (t) r 10 − r 11 C 1 (t) − a 11 y 1 (t) − a 12 y 2 (t) dt + σ 1 y 1 (t)dB 1 (t), where C 1 (t), C 2 (t), and C E (t) are the concentrations of the toxicant in the organism of species y 1 and y 2 and environment at time t, respectively; r 11 and r 21 denote the doseresponse of species y 1 and y 2 to the organismal toxicant, respectively; k i and g i represent the absorbing and excretion rates of the toxicant from the environment respectively, −m i is depuration rate of the toxicant, i � 1, 2. −hC E (t) denotes the loss rate of the toxicant because of volatilization; u(t) 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. ese 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]. erefore, 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]. erefore, 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][17][18][19][20][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 data where y i (t − ) denotes the left limit of y i (t); Poisson process, where Γ is a Poisson counting measure, λ is the characteristic measure of Γ on a measurable subset Z in dμ i (θ) � 1, i � 1, 2. e biological meanings of other parameters are the same as before. If r 11 � r 21 � 0, the corresponding model is denoted by (M 0 ), which means that the population is not contaminated.
We aim to study the dynamical behaviors of (M) and (M 0 ) such as the extinction and persistence in mean for all species and explore the impacts on the dynamics of time delays and Lévy noise. e 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 M 0 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.

Preliminaries
For the simplicity, we first make the following notations: Δ � a 11 a 22 − a 12 a 21 , Our later discussion are based on the following technological hypotheses.
ere exists a positive constant L such that Assumption 2. Suppose that Δ > 0, which means the internal competition is greater than the external competition (see [22]). e following lemmas are necessary for our later proof.

Lemma 1. Let Assumption 1 hold, then for any given initial
Proof. It is obvious that the coefficients of model (M) are locally Lipschitz. By [5], model (M) has a unique local solution y(t) � (y 1 (t), y 2 (t)) ∈ R 2 + a.s. for any initial data where τ e > 0 is the explosion time. It needs only to verify that τ e � ∞ a.s. Let m 0 > 0 be sufficiently large such that (ξ 1 (0), Clearly, τ m is strictly increasing with m. Let τ ∞ � lim m⟶∞ τ m a.s., then τ ∞ ≤ τ m a.s. en, we only need to prove τ ∞ � ∞ a.s. If the statement is not true, then there exist T > 0 and 0 < ε < 1 such that P(τ ∞ < T) > ε and an integer m 1 > m 0 such that Discrete Dynamics in Nature and Society P τ m ≤ T > ε, for any m > m 1 .
where y � y 1 , y 2 , Choose a constant β > 0 and integer n > 0 such that a 21 For model (M), by Ito's formula, we get where By basic inequality a 2 + b 2 ≥ 2ab, we have 4 Discrete Dynamics in Nature and Society Substituting (14) and (15) into LV 1 (y 1 ), LV 2 (y 2 ), then erefore, Substituting (17) into dV(y) and together with (11), there exists a constant K > 0 such that By this result and according to the argument in [23], we have which leads to a contradiction, and hence, τ ∞ � ∞ a.s. erefore, τ e � ∞ a.s. e proof is completed.

(45)
It follows from Lemma 3 that Combining (43) , the solution (y 1 (t), y 2 (t)) of model (M) has the properties that at is to say, model (M) will be persistence in mean.
Substituting (70) into (50), then Discrete Dynamics in Nature and Society From Lemma 3, we get and thus, , the solution (y 1 (t), y 2 (t)) of model (M) has the properties that at is to say, model (M) is nonpersistent.

Conclusions
e 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. eorem 1 and 2 give the sufficient criteria of persistent in mean for cases (M) and (M 0 ), respectively. eorem 3 and 4 obtain the sufficient conditions of nonpersistence. eorem 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
e authors declare that they have no conflicts of interest. 14 Discrete Dynamics in Nature and Society