Dynamical Analysis of a Stochastic Predator-Prey System with Lévy Noise and Impulsive Toxicant Input

This paper established a modified Leslie-Gower and Holling-type IV stochastic predator-prey model with Levy noise and impulsive toxicant input. We study the stability in distribution of solutions by inequality techniques and ergodic method. By comparison method and Ito’s formula, we obtain the sufficient conditions for the survival of each species. Some numerical simulations are introduced to show the theoretical results.


Introduction
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 [1] [2] [3] [4] [5]. 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 [5]. The ecotoxicity produced by microplastics will be transferred and diffused to the entire aquatic environment, which affects the stability of the ecological environment [6]. 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 popula-tions in polluted environments has become a hot spot [7] [8] [9] [10].
Zhang and Tan [11] 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 [12] 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 [13] 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. [14] [15] 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 [16] 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. [17]. Then we get According to the actual situation, we consider the impact of environmental pollution on the system (1). Let Suppose that the growth rate i a is an affine function of ( ) i C t [13], the parameter 1 i a represents the dose response rate of the ith population to the concentration of the organismal toxicant: 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 prepara-Journal of Applied Mathematics and Physics tions 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.

Preliminaries
For the sake of convenience, we define the following notations: Moreover, as a standing hypothesis throughout this paper, we assume that ( ) ( ) 1 2 , B t B t and N are independent. We also suppose that ( ) . In order to facilitate the search when using the formula later, we suppose Then we put forward some necessary lemmas to prepare for the main results later.

M M t is Meyer's angle bracket process.
Lemma 3 [21]. Suppose that population ( ) [ ] ∈ be a solution of system (6). Then  Proof. To begin with, let us consider the following equations Clearly, the coefficient of system (6) satisfy the local Lipschitz condition, then there is a unique local solution is the unique positive local solution to sys- By the comparison theorem for stochastic differential equations [24], we have Noting that are existent on 0 t ≥ , then we obtain e τ = +∞ (Theorem 2.1 in [15]).

Stability in Distribution
is the positive solution of system (6) with Proof. The proofs are very standard. Detailed proofs can refer to [15] [25] and hence are omitted here. (6) is asymptotically stable in distribution, i.e., when t → +∞ , there is a unique probability measure ( ) ν ⋅ such that the transition density ( ) , , converges weakly to ( ) ν ⋅ with any given initial value Proof. Let Then we define According to the Itô's formula with noise, we have   ; ; According to the first equation of system (6), we get is continuously differentiable function. By Lemma 5,it gives that  P t ξ  denotes the probability of ( ) , z t ξ ∈  with initial value ( ) 2 R ξ θ + ∈ . By Chebyshev's inequality [27] and Lemma 5, the family of ( )            :

Numerical Simulations
In this section, we apply Split-step Backward Euler method [28] [29] [30] to prove our theoretical results.
(1) We assume the parameters 1 0.8 a = , 2 0.6 a = , 11 We observe that two species will go to extinction from Figure 1, and the result of (i) in Theorem 1 are shown. We observe that ( ) x t will go to extinction, and ( ) y t will be stable in mean from

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 i β 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 i β 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 1 2 , , , n c θ θ and f will affect the survival dynamics of the species from (iv) in Theorem 1. The expression of i β 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.