Survival and ergodicity of a stochastic Holling-III predator–prey model with Markovian switching in an impulsive polluted environment

Based on the effects of white noise and colored noise, we propose a stochastic Holling-III predator–prey model in an impulsive polluted environment. Firstly, we prove an existence and uniqueness theorem of the presented model. Secondly, we establish sufficient criteria of extinction, nonpersistence in mean, and weak persistence in mean for both prey and predator species. Thirdly, with the aid of Lyapunov functions, we prove that this system is ergodic and has a unique stationary distribution under certain conditions. Finally, we verify the theoretical results by performing some numerical simulations.


Introduction
Environmental pollution from various industries attracts more and more demographers and ecologists because it has seriously threatened the survival of humans and other exposed living organisms [2,26,38]. In the early years, the survival analysis of populations in the polluted environment was carried out by establishing deterministic models. For example, Hallam et al. [3,[6][7][8] used some deterministic models to show the effects of toxicant on populations, He and Wang [10,11] analyzed the dynamics of two single-species models in polluted environments, and so on. The deterministic models are not suitable for modeling the ubiquitous noise-driven systems, so stochastic models in the polluted environment are frequently used to explore the dynamics behavior of species [27]. Generally, there are two main types of environmental noise, white noise and colored noise. For white noise-driven models, Gard [5] investigated a stochastic single-species model to explain the influence of the toxicant on organisms and compared it to the corresponding deterministic model. Liu and Wang [20] studied the dynamics of stochastic single-species population models with and without pollution.
Wei et al. [36,37] further investigated the stochastic single-species population model in a polluted environment. In addition, Lv et al. [23] introduced a new impulsive stochastic chemostat model. On the other hand, to illustrate the switching between two or more environmental regimes, it is meaningful to consider the influence of colored noise on the population [4,32]. Liu and Wang [21] proposed a stochastic single-specie model under regime switching. Moreover, a stochastic two-species model is presented in [44]. For the dynamics behaviors of the populations with environmental toxins and noises, we also refer the readers to [22,34,39,43] and references therein.
Many models mentioned are based on the Lotka-Volterra model with linear functional responses. However, these models ignore some important natural phenomena (compared with the models with nonlinear functional responses), especially for the predator-prey case [9,28]. To this end, Holling [12,13] proposed the most widely used functional responses and classified them into three basic types (denoted types I, II, and III). As a typical nonlinear functional response, Holling type III [29] has a powerful role in describing the predation behavior of vertebrates, for example, some predators learn more special skills for hunting or prey handling. Recently, some important models with Holling type III has been discussed. For instance, Huang et al. [14] established a prey-predator model with Holling-III response function and a prey refuge to show that the refuge has a steadying influence on prey-predator interactions. Su et al. [33] indicated that both periodically varying environment and stochastically released natural enemies have a great impact on the survival of the species by using the predator-prey system with generalized Holling-III type functional response. Wu and Li [40] combined Holling-III type with Hassell-Varley type functional responses to demonstrate the permanence and global attractivity of a discrete predator-prey system. More recently, Sengupta et al. [30] analyzed the dynamics of the deterministic and stochastic models with Holling-III response function, respectively.
To the best of our knowledge, there are rare results on the effects of pollution inputs and noise fluctuations for the dynamics behavior of Holling-III predator-prey systems.
Undoubtedly, the theoretical analysis of Holling type-III is more challenging than other functional responses because its nonlinear form is more complicated. In this paper, we are devoted to two main goals: • to analyze the long-time behavior of stochastic Holling-III predator-prey systems with regime switching in an impulsive polluted environment, and • to investigate the effects of pollution inputs and noise fluctuations on the dynamics of predators and preys.
In particular, when the considered model of this paper reduces to that of [30], our conditions in Theorem 3.2 are more convenient to verify the weak persistence in the mean of predator species in comparison with [30,Theorem 4.4].
This paper is organized as follows. In Sect. 2, we begin to state our model and prepare some preliminaries including the existence of a unique positive solution. In Sect. 3, we obtain some sufficient conditions of the extinction and weak persistence in mean for two species. In Sect. 4, we investigate the existence and uniqueness of stationary distribution.
In Sect. 5, we present numerical simulations confirming our theoretical results. Finally, in the last section, we give a brief conclusion.

The model and preliminaries
We begin this section by stating the stochastic Holling-III predator-prey model with Markovian switching in an impulsive polluted environment step by step and prepare some preliminaries including the existence and uniqueness theorem.

The model
First, let us introduce the deterministic Holling-III predator-prey model [41] with the initial value (x(0), y(0)) ∈ R 2 + , where x(t) and y(t) denote the population densities of prey and predator species at time t, respectively, a 1 and a 2 stand for the intrinsic growth rate of prey population and the death rate of predator population, respectively. Both b 1 and b 2 represent intraspecific coefficients of competition. The nonlinear function αx 2 1+βx 2 is the Holling-III functional response, where α stands for the predation rate of predators on prey populations, β denotes the handling time of predators for each prey that is consumed, and k denotes the conversion rate concerning the number of newborn predators for each captured prey. All parameters of the model are positive.
Taking into account the impact of environmental pollution on species [3,[6][7][8], most of the existing studies assume that the exogenous input of toxicant is continuous. However, the actual situation is that the toxin is released in regular pulses; for example, the factories drain sewage into rivers on a regular basis. Therefore we focus on the case of toxic exogenous pulse input and then obtain the following model: where χ(t) = χ(t + )χ(t) with χ = x, y, C 1 , C 2 , C E , C 1 (t), C 2 (t), and C E (t) stand for the concentrations of toxicant in the organism of the prey, predator, and environment at time t, respectively, r 1 and r 2 denote the dose-response of the prey and predator to the toxicant, respectively, e i is the uptake rate of toxicant from environment, g i and m i indicate the excretion and depuration rates of toxicant, respectively, h is the loss rate of toxicant, and u and ρ represent the toxicant input amount and the period of the exogenous toxicant input, respectively. Here we assume that the environmental capacity is large enough so that the effects of toxins excreted by the organism into the environment have negligible influences on the concentration of environmental toxins. Now we further take the white noise into consideration. Following the approach used in [15,17,19], the parameters a 1 and -a 2 of system (2.2) are perturbed with where the dot denotes the time formal derivative, B 1 (t) and B 2 (t) are mutually independent one-dimensional standard Brownian motions defined on the complete probability space ( , F, {F t } t≥0 , P), and σ 2 i (i = 1, 2) are the intensities of the white noises. Thus system (2.2) becomes the stochastic Holling-III predator-prey model in the impulsive polluted environment It is worth pointing out that there are many phenomena that cannot be modeled by Brownian motion-driven stochastic differential equations (SDEs) [45]; for example, when the growth environment of some species changes significantly, their birth and death rates will be much different [1,46]. In general, the switching among different environments is memoryless, and the waiting time of the next switch obeys an exponential distribution. Hence these random changes can be described by a continuous-time Markov chain ξ (t), t > 0, taking values in a finite-state space S = {1, 2, . . . , N} with the generator = (γ ij ) N×N given by where all functions a i , b i , r i , σ i (i = 1, 2) and α, β, k are R + -valued. In addition, we assume that ξ (t) is irreducible and independent of Brownian motions B i (t) (i = 1, 2). In fact, the Markov chain ξ (t) has a unique stationary distribution π = (π 1 , π 2 , . . . , π N ) ∈ R 1×N , which can be obtained by solving the linear equation π = 0 subject to N j=1 π j = 1 and π j > 0, j ∈ S. As a result, for any vector θ = (θ (1), . . . , θ (N)) T , lim t→∞ 1 t t 0 θ (ξ (s)) ds = i∈S π i θ (i). In reality, environmental noise has little effect on the toxin concentration of the organism, so we assume that the parameters e i , g i , m i , and h are independent of noises.
(2) for any ε > 0 and sufficiently large t, Because C 1 (t), C 2 (t), and C E (t) can be obtained only from (2.5), system (2.4) reduces to the subsystem with initial values Subsystem (2.8) can be written as the SDE We end this section with the following existence and uniqueness theorem.
Proof Since both the drift and diffusion coefficients of equation (2.8) satisfy the local Lipschitz condition, there is a unique local solution ( where τ e denotes the explosion time (see [25]). To verify that the solution is global, we need to prove that τ e = ∞ a.s. Let m 0 > 1 be sufficiently large such that Thus we only need to prove that τ ∞ = ∞ a.s. If this were not true, then there would be constants T > 0 and ∈ (0, 1) such that P{τ ∞ ≤ T} > and an integer m 1 ≥ m 0 such that Define the C 2 -function V: R 2 + × S → R + as follows: According to the definition of the operator L (see (2.11)) and the vertex formula of quadratic functions, we have where H is a finite positive constant. Then the generalized Itô's formula [25] yields which implies where τ m ∧ T = min{τ m , T}. On the other hand, set m = {τ m ≤ T} for m ≥ m 1 , so P( m ) ≥ by (2.12). Note that for all ω ∈ m , at least one of x(τ m , ω) and y(τ m , ω) equals either m or 1/m. Then Reviewing (2.13), we can claim that

Extinction and persistence
This section aims to investigate the extinction, nonpersistence in mean, and weak persistence in mean of both prey x(t) and predator y(t) separately.

For convenience, we define
Proof Applying generalized Itô's formula to (2.8) yields where M i (t) = Firstly, it follows from equation (3.1) that Using (3.3) and the ergodicity of ξ (t), we obtain which implies that lim t→∞ x(t) = 0 a.s., so the prey x(t) is extinct, and (1) is proved. Secondly, for given > 0 small, there exists a constant T > 0 such that for all t > T, (3.5) Inserting (3.5) into (3.1) leads to In particular, when A 1 -B 1 = 0, it follows from (3.6) and the arbitrariness of that x(t) * = 0 a.s., which states that the prey x(t) is nonpersistent in mean. So (2) is proved. Thirdly, taking the upper limits of both sides of (3.1) shows Recalling that x(t) * < ∞ a.s. (see Remark 2.1), we get that the left side of (3.7) is nonpositive. Theň which shows that y(t) * = 0, a contradiction, so x(t) * > 0 a.s., that is, the prey x(t) is weakly persistent in mean. Hereto, all conclusions of the theorem are proved.

Stationary distribution
In this section, we prove that the solution of model (2.8) has a unique stationary distribution under certain conditions. To facilitate its proof, we will first prove a useful lemma.

Lemma 4.1 ([31, 45]) Suppose that all of the following conditions hold:
(1) γ ij > 0 for all i = j; (2) For all k ∈ S, the symmetric D(·, k) admits a constant ∈ (0, 1] such that (3) There exists a bounded open set D ⊂ R n with a regular boundary satisfying that for all k ∈ S, there exists a twice continuously differentiable function V (·, k) : D c → R + such that for some ς > 0, Then the solution (z(t), ξ (t)) of (2.10) is ergodic and positive recurrent, that is, it has a unique stationary distribution.

Numerical simulations
In this section, we perform some numerical simulations to verify the theoretical results established in the previous sections.
Example 5.1 Consider the following stochastic Holling-III predator-prey model with Markovian switching in an impulsive polluted environment: 2/7. Thus Theorems 3.1 and 3.2 state that both x(t) and y(t) are weakly persistent in mean, which is illustrated in Fig. 4(a). On the other hand, if δ =ˇb 1 2β = 2/7, then λ = 611/7556 > 0. Hence Theorem 4.1 indicates that model (5.1) has a unique stationary distribution, which is confirmed by Figs. 4(c) and 4(d).

Conclusions and future work
In this paper, we explore a stochastic Holling-III predator-prey system and regime switching in an impulsive polluted environment.
The major contributions of this work are: • We obtain sufficient conditions for the extinction, nonpersistence in mean, and weak persistence in mean. More specifically, for the prey x(t), A 1 -B 1 is the threshold of the extinction and weak persistence in mean. That is, if A 1 -B 1 < 0, then x(t) is extinct; if A 1 -B 1 > 0, then x(t) is weakly persistent in mean. For the predator y(t), if A 2 -B 2 < 0, then y(t) is extinct; if there exists a constant δ ∈ (0,ˆb 1 β ) such that A 3 > 0, then y(t) is weakly persistent in mean. • From Theorems 3.1 and 3.2 we can see that both intensities σ i (i = 1, 2) of white noise and the distribution π of Markov chain ξ (t) are related to the values of A 1 and A 2 , which will change the survival of x(t) and y(t). To be specific, the greater the values of σ i (i = 1, 2), the greater the risk of extinction of x(t) and y(t); see Figs. 1 and 2(a). Also, the distribution π of Markov chain ξ (t) plays a significant role in the survival of x(t) and y(t); see Figs. 1 and 2(b). In addition, the smaller the value of impulsive input period ρ, the greater the risk of extinction of x(t) and y(t); see Figs. 1 and 3. • Finally, we discuss the positive recurrence and ergodicity of the stochastic model, namely, there exists a unique stationary distribution under some conditions by constructing Lyapunov functions. Nowadays, environmental pollution has become a concern of people around the world. And environmental noise makes a huge difference to the biological systems in real life. Besides Holling type, we can also consider the stochastic models with other meaningful functional responses under regime switching, such as Beddington-DeAngelis type and Watt type. Furthermore, we will try to collect the real data to validate our theoretical results and explain biological significance.