With the development of modernization, pollution is also being produced. Air pollution, water pollution, noise pollution and other pollution affect the stability of the ecosystem. At the same time, environmental pollution affects the survival of the natural population and human life     . For example, the use of chemical pesticides has effectively controlled the pest problem in agriculture, but it is also widely regarded as one of the problems that have a negative impact on the environment and food safety . The ecotoxicity produced by microplastics will be transferred and diffused to the entire aquatic environment, which affects the stability of the ecological environment . These examples show that uncontrolled input of toxicant affects the balance of the ecosystem, and even leads to the extinction of populations. Therefore, environmental pollution will inevitably attract people’s great attention. Research on the survival of populations in polluted environments has become a hot spot    .
Zhang and Tan  considered a stochastic predator-prey system in a polluted environment with impulsive toxicant input and impulsive perturbations. They obtained a set of sufficient conditions for extinction, weak persistence in the mean and global attraction to any positive solution of the system. Lv, Meng and Wang  investigated an impulsive stochastic chemostat model with nonlinear perturbation in a polluted environment. They showed that both stochastic and impulsive toxicant inputs have great effects on the survival and extinction of the microorganism. Liu, Du and Deng  established a stochastic modified Leslie-Gower Holling-type II predator-prey model with impulsive toxicant input. They got the threshold between persistence in the mean and extinction for each population; then they concluded that the white noise is harmful to the sustainable growth of species.
Ecosystems may suffer sudden and catastrophic environmental disturbances, such as earthquakes, tsunamis, volcanoes, hurricanes or epidemics, etc. To explain these phenomena, Bao et al.   considered a jump process into the stochastic Lotka-Volterra population systems and studied population dynamics of their systems at the first time. Zhao, Yuan and Zhang  established a stochastic competitive model with Lévy noise in an impulsive polluted environment. They showed that Lévy noise can significantly affect the persistence and extinction of each species. In this paper, we consider adding Lévy noise to the stochastic modified Leslie-Gower and Holling-type IV predator-prey system proposed by Xu et al. . Then we get
where and are the left limit of prey populations and predator populations respectively. represents the intrinsic growth rate of the population in a non-polluting environment. is the intensity of competition among individuals of . is the modified Leslie-Gower term, which states that the number of predators has fallen due to the shortage of the most important food. is Holling-type IV functional response, which refers to the change in the density of the prey that each predator is attached to per unit time. , and are positive constants, where indicates the maximum endurance of the environment without predators. N is a real-valued Poisson counting measure with characteristic measure on a measurable subset of with , . is the jump-diffusion coefficient.
According to the actual situation, we consider the impact of environmental pollution on the system (1). Let and be the concentration of toxicant in the prey organism and predator organism at time t respectively. Suppose that the growth rate is an affine function of , the parameter represents the dose response rate of the ith population to the concentration of the organismal toxicant:
Suppose denotes the concentration of toxicant in the environment at time t. is the organism’s net uptake rate of environmental toxicant. and represent the net ingestion rate and the depuration rate respectively. h represents the rate of toxin loss in the environment due to evaporation or other reasons. Assuming that external toxins affect the entire predator-prey system by impulsive toxicant input, let and q represent the period and the amount of impulsive toxicant input each time respectively. So we can get a stochastic modified Leslie-Gower Holling-type IV predator-prey model with Lévy noise in impulsive toxicant input environments:
Toxicants affect the system (3) by impulsive input, and the system also contains Lévy noise. There are few studies on the impact of this type of model on system dynamics, so it is of great significance. We first turn the system (3) into an impulseless system through approximate solving methods. Then we can use the ergodic method to prove the distribution stability of the system. We also get the extinction and persistence of the population by use of the comparison theorem and some inequality techniques.
The organization of this paper is as follows. In Section 2, we provide preparations for the proof and calculation of the system (3). Section 3 discusses the stability of the distribution of the impulseless system (6). Then in Section 4, the threshold between persistence in the mean and extinction for each species is established. We introduce some numerical simulations to support the theory in Section 5. The final section concludes this paper.
For the sake of convenience, we define the following notations:
Moreover, as a standing hypothesis throughout this paper, we assume that and N are independent. We also suppose that . In order to facilitate the search when using the formula later, we suppose
Assumption 1. There is a constant such that
Then we put forward some necessary lemmas to prepare for the main results later.
Lemma 1 . Consider the following subsystem of the system (3),
Subsystem (4) has a unique -periodic solution , which satisfies
Remark. and denote the concentrations of toxicant. According to their practical significance, we get must hold for all . From Lemma 1, it requires the following constraints :
In the following, we apply Lemma 1 to the system (3). Therefore, we only need to consider the following system
with initial value .
Lemma 2 . Suppose that , is a local martingale vanishing at time zero. Then
and is Meyer’s angle bracket process.
Lemma 3 . Suppose that population .
(i) If there exist some constants , , , , and such that, for all ,
(ii) If there exist some constants , , , , and such that, for all ,
Definition 1    . Let be a solution of system (6). Then
(a) the population is said to go to extinction if ;
(b) the population is said to be stable in mean if a.s., where K is a constant;
(c) the population is said to be stochastic strong persistence in mean if a.s.;
(d) the population is said to be stochastic weak persistence in mean if a.s..
Lemma 4. For any given initial value , system (6) has a unique global positive solution on and the solution will remain in almost surely.
Proof. To begin with, let us consider the following equations
with initial value . Clearly, the coefficient of system (6) satisfy the local Lipschitz condition, then there is a unique local solution on , where is the explosion time. According to Itô’s formula, is the unique positive local solution to system (6) with initial value . Now let us prove to show this solution is global. Consider the following four auxiliary equations
By the comparison theorem for stochastic differential equations , we have
where . According to Lemma 4.2 in , we have
Noting that and are existent on , then we obtain (Theorem 2.1 in  ).
3. Stability in Distribution
Lemma 5. Suppose that is the positive solution of system (6) with any initial value , then for any , there exists a positive constant K such that
Proof. The proofs are very standard. Detailed proofs can refer to   and hence are omitted here.
Lemma 6. If and , then system (6) is asymptotically stable in distribution, i.e., when , there is a unique probability measure such that the transition density of converges weakly to with any given initial value .
Proof. Let and be two solutions of system (6) with the same initial value and respectively. Then we define
According to the Itô’s formula with noise, we have
Note that , we have
According to the first equation of system (6), we get
Therefore, is continuously differentiable function. By Lemma 5, it gives that
where is a positive constant. Similarly, by the second equation of system (6), it can
where is a positive constant.
Based on Lemma 5, we can get that and are uniformly continuous function through (19) and (20). By the Barbalat’s conclusion of , we can observe that
Suppose that represents the transition probability density of the process and denotes the probability of with initial value . By Chebyshev’s inequality  and Lemma 5, the family of is tight. So we can obtained that a compact subset such that for any given .
Let be the probability measures on . For any given two measures , we define the metric
For any and , we can get
where , is a complementary set of . Because the family of is tight, so there exists a sufficiently large K such that for any given . By (21), there exists a such that
holds for , which yields .
It follows from the arbitrariness of that for , we have
Similarly, we get
holds for and .
That is to say, for any , the transition probability is Cauchy in with metric . So is Cauchy in with metric . In a word, there has a unique such that . From (21), we get
Using triangle inequalities, we have
This completes the proof of Lemma 6.
4. Extinction and Persistence
Lemma 7. If and , then
Proof. By Lemma 1, we have
Then for arbitrary , there is a T such that for ,
Set , therefore for ,
Consequently, for , by (10), we have
Substituting the above inequality into (18), we can get
Note that , under Assumptions 1,
Applying Lemma 2, we can obtain that
Then, it follows from and for arbitrary that
By substituting the above identities into (25) results in
And then let us prove . Applying Itô’s formula to (12) gives
That is to say
For arbitrary , there exists such that for ,
Let be sufficiently small such that , then applying (i) and (ii) in Lemma 3 to the above two inequalities respectively, we have
According the arbitrariness of gives that
Substituting this equation into (27), and then noting that , and , we can derive that . Thus by (14), we obtain
This completes the proof.
Next we will discuss the ecological dynamics of the system (3) or system (6).
Theorem 1. Consider system (6), we have the following valid statements
(i) If and , then both and are extinct, i.e.
(ii) If and , then is extinct and is stable in mean almost surely, i.e.
(iii) If and , then is stable in mean almost surely and is extinct, i.e.
(iv) If and then and are both stochastic strong persistence in mean almost surely.
Proof. According to Itô’s formula, we can obtain that
In other words, we have
Now let us prove (i). It follows from (24) and (28), for sufficiently large t, that
where is sufficiently small such that . Noting that and and, hence . In the same way, if , and according to (29), .
(ii) Since , thus (i) implies . By (24) and (29), for sufficiently large t, we have
If , then there exist arbitrarily sufficiently small and , for all , by making use of (i) and (ii) in Lemma 3 to (30) and (31) respectively, we have
According to the arbitrariness of , the above inequality gives
(iii) Since , thus (i) implies , By (24) and (28), for sufficiently large t, we have
If , then there exist arbitrarily sufficiently small and , for all , by making use of (i) and (ii) in Lemma 3 to (32) and (33) respectively, we have
According to the arbitrariness of , the above inequality gives
(iv) From (29), we get
Then, we have
noting that , and and (23), for sufficiently large t, we can derive that
Moreover, follows from (24) and (29), for sufficiently large t, we have
then there exist arbitrarily sufficiently small and , for all , by making use of (ii) in Lemma 3, we get
According to the given condition , we get
Integrating the both side of (35), then by (34) and for sufficiently large t, we have
There exist arbitrarily sufficiently small and , for all , by making use of (i) and (ii) in Lemma 3 to (33) and (36) respectively, we get
5. Numerical Simulations
In this section, we apply Split-step Backward Euler method    to prove our theoretical results.
(1) We assume the parameters , , , , , , , , , , , , , , , , , , , , then , .
We observe that two species will go to extinction from Figure 1, and the result of (i) in Theorem 1 are shown.
(2) Let , , then other conditions remain unchanged, we have , .
We observe that will go to extinction, and will be stable in mean from Figure 2, where
and then the result of (ii) in Theorem 1 are shown.
(3) We set , , then other conditions remain unchanged, we have , .
From Figure 3, we get that will be stable in mean and will be extinct, and then the result of (iii) in Theorem 1 are shown, where
(4) We set , , then other conditions remain unchanged, we have , , then , .
From Figure 4, we get that both and will be stochastic strong
Figure 1. This figure is time series graph of and . We choose , , step size , initial value , , , , and , , .
Figure 2. This figure is time series of and with , , step size , initial value , , , , and , , .
Figure 3. This figure is time series of and with , , step size , initial value , , , , and , , .
Figure 4. The above figure is time series graph of and . The under figure is the 2D phase diagram of and . We choose , , step size , initial value , , , , and , , .
persistent in mean. The 2D phase diagram of and mean that two species are in a predator-prey relationship. The result of (iv) in Theorem 1 are shown, where
6. Discussion and Conclusions
In this paper, we add Lévy noise to the stochastic modified Leslie-Gower and Holling-type IV predator-prey model, and assume that the toxicants are added in periodic pulses in the model. We show that the model has a unique global solution and study the stability in distribution of solutions. We get the thresholds to determine extinction and persistent in mean of two species; thus sufficient and necessary conditions are established for the extinction and persistent in mean of two species.
From the Theorem 1 and the numerical simulation results in Figures 1-4, we can see that Lévy noise has a strong effect on the system (3). At the same time, through the expression of the thresholds and changing the parameter value multiple times, it shows that the line shape in the numerical simulation is undulating, because white noise can reflect that the model is affected by the environment. We also know that the value of and f will affect the survival dynamics of the species from (iv) in Theorem 1. The expression of also reflects that the toxicants and population’s own performance also more or less affects the survival dynamics of the species.
Indeed, when the population encounters sudden environmental disturbances, such as tsunamis, earthquakes, etc., the survival environment of the population is threatened. Ecological stability is bound to be affected because they can’t adapt to this sudden environmental fluctuation in a short time. Lévy jump has a great impact on the survival of species. With the rapid development of modern industrial technology, pollution has been increased as well. Impulsive toxicant will inevitably have a certain impact to species’ living environment and their own growth.
This article has practical significance for the survival analysis of a stochastic modified Leslie-Gower and Holling-type IV predator-prey model with Lévy noise in impulsive toxicant input environments. But considering that some more complex systems will be more in line with the actual situation, for example, during the rainy season and the dry season, the growth rate and mortality rate of the species are different, so we can consider adding the regime switching to the system (3). In the next research work, we can try to consider the influence of continuous-time Markov chain on the system.
The authors would like to thank the referees for their useful suggestions which have significantly improved the paper.
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