Dynamical Analysis of a Class of Prey-Predator Model with Beddington-DeAngelis Functional Response , Stochastic Perturbation , and Impulsive Toxicant Input

A stochastic prey-predator system in a polluted environment with Beddington-DeAngelis functional response is proposed and analyzed. Firstly, for the system with white noise perturbation, by analyzing the limit system, the existence of boundary periodic solutions and positive periodic solutions is proved and the sufficient conditions for the existence of boundary periodic solutions and positive periodic solutions are derived. And then for the stochastic system, by introducing Markov regime switching, the sufficient conditions for extinction or persistence of such system are obtained. Furthermore, we proved that the system is ergodic and has a stationary distribution when the concentration of toxicant is a positive constant. Finally, two examples with numerical simulations are carried out in order to illustrate the theoretical results.


Introduction and Model Formulation
The Lotka-Volterra model [1][2][3] is a classical model in the study of biological mathematics, and the continuous Lotka-Volterra model which is modeled by ordinary differential equations and delay differential equations is widely used to characterize the dynamics of biological systems [4][5][6][7][8][9][10][11][12][13].The functional response functions are important in the population ecological models [14].In general, functional responses fall into two categories: one depends only on the density of the prey, such as Holling I-III [15][16][17]; the other depends on the density of both the prey and the predator, such as Beddington-DeAngelis type [18,19].Compared with the Holling II functional response, the Beddington-DeAngelis type functional response,  =  12 /( 2 +  +  12 ), has an additional term  in the denominator modeling mutual interference among predators.In other words, this type of functional response is affected by both predator and prey.Some biologists believe that if the predators compete with each other to obtain food, functional response should depend on the density of both the prey and the predator.Arditi et al. [20] and Jost et al. [21,22] used the actual observation data to verify this point.In particular, having collected observation data from 19 predator-prey communities, Skalski and Gilliam [23] found that predator-dependent functional responses were in agreement with the observation data, and in many instances, the Beddington-DeAngelis type looked better than the others.The Beddington-DeAngelis functional response has been widely used in the modeling of ecosystems in which there is mutual interference among predators [24,25].In [19], DeAngelis et al. have extensively investigated the dynamical properties of the following prey-predator system: where () and () represent the density of the prey and the predator, respectively. 1 is the intrinsic growth rate of the 2 Complexity prey,  12 ,  2 , and  12 are the consumption rate, the saturation constant, and the saturation constant for an alternative prey, respectively. 12 is the conversion rate of nutrients into the reproduction for the predator.The parameters   and   ( = 1, 2) are the nonpredatory loss rate and the interspecific competition rate.We refer the reader to [19] for more details.In many ecosystems, predators tend to be omnivorous, they have wide variety of food sources.For example, the giant panda is omnivorous animal, since it can eat both meat and plant such as bamboos.In the lake ecosystem, some fishes not only prey on aquatic invertebrates, but also feed on algae and other aquatic plants.Polis and Strong in [26] and McCann and Hastings in [27] studied omnivorous nature of animals in the food chain in 1996 and 1997, respectively.Based on the above literature, we established a kind of omnivorous model as follows: ẋ () =  () ( 1 −  1  () −  ()  +  () +  () ) , ẏ () =  () ( 2 +  ()  +  () +  () −  2  ()) , where  2 represents the growth rate of  due to omnivorous nature and   ( = 1, 2) denote the density-dependent coefficient of the prey and the predator, respectively., , , , and  are the consumption rate, the saturation constant, the predator interference, the saturation constant for an alternative prey, and the conversion rate, respectively.All parameters are positive in system (2).It is well known that the biological population is inevitably affected by environment perturbation while the stochastic population model is more in line with the actual situation.Recently, various models based on stochastic differential equations (SDEs) have extensively been paid the attention of the researchers (see, e.g., [28][29][30][31][32][33][34][35][36][37]).Parameter perturbation induced by white noise is an important and common form to describe the effect of stochasticity (see, e.g., [37][38][39][40][41][42][43][44][45][46][47][48]).In this paper, we consider the white noise perturbation for the intrinsic growth rates of the prey and predator; that is,  1 →  1 +  1 Ḃ 1 () and  2 →  2 +  2 Ḃ 2 (), where  1 (),  2 () are mutually independent Brownian motions and  1 ,  2 denote the intensities of the white noise.On the other hand, it can be seen from the recent literature that the environmental pollution has an important effect on the population systems [49][50][51][52][53][54][55][56][57][58][59][60].In 1983, Hallam et al. [61,62] studied the influence of environmental pollution on the population and established a relationship model between environmental toxins and population.Subsequently, Hallam et al. [63,64] studied the persistence and extinction of population in polluted environment.The mathematical model established by Hallam et al. considered only the toxins in the organism to cause a decrease in the birth rate or an individual death, which is reasonable in the case of lower concentration of the toxicant in the environment.When pollution is serious, the emission of pollutants may directly lead to the death of the species; see [65][66][67][68][69].The authors in [68] added the environmental toxic term directly to the model; this is reasonable in the heavily polluted environment.For example, in a lake ecosystem, the discharge of large amounts of industrial waste water may directly lead to the death of fish, aquatic invertebrates, and so on.Therefore, we assume that the emission of pollutants to the environment is impulsive and directly affects the survival of the species in such an environment, so we get the following system: where  1 (),  2 () are positive, nonconstant, and continuous functions of period ,   () stands for the concentration of the toxicant in the environment, ℎ denotes the loss rate of toxicant at time ,  is the impulsive input period and  is the impulsive input amount, and  1 and  2 represent the dose-response of the prey and predator to the environmental toxicant, respectively.Furthermore, the prey-predator model may be perturbed by telegraph noise which is distinguished by factors such as rain falls and nutrition and can be represented by switching among two or more regimes of environment [40,60,[70][71][72][73][74][75][76][77][78][79][80].For example, population growth rates in different seasons are not the same.The intraspecific competition coefficient varies according to the changes in nutrition and food resources.Generally, the switching between different regimes is memoryless and the waiting time for the next switch is exponentially distributed [81,82].Therefore, it can be described by a continuous-time Markov chain () taking values in a finite state space S = {1, 2, . . ., }.Taking into account the influences of white noise and telegraph noise, we propose the following stochastic differential system with impulsive toxicant input: For any  ∈ S,   (),   (),   (),   () ( = 1, 2), (), (), (), (), and () are all positive constants.In model ( 4), the population is inevitably affected by severe stochastic interference such as drought; the parameter switches one state () =  into another state () =  and it will switch into the next regime until the next major environmental change.The rest of this paper is organized as follows.In Section 2, we provide preliminaries which are used in the following sections.In Section 3, we show that system (3) admits a nontrivial positive -periodic solution by constructing Lyapunov function.In Section 4, we explore the sufficient conditions for extinction and permanence in mean of system (4).Finally, some examples with numerical simulations have been given to illustrate our theoretical results.

Preliminaries
Throughout this paper, let (Ω, F, F ≥0 , P) be a complete probability space with a filtration F ≥0 satisfying the usual conditions,   () ( = 1, 2) is one-dimensional Brownian motion on this space, and () is a right-continuous Markov chain and independent of the Brownian motion   ().The state space of this Markov chain is S = {1, 2, . . ., }.Suppose that the generator matrix of () is Γ = (  ) 1≤,≤ , where   stands for the transition rate from state  to  and satisfies the following conditions: here,   ≥ 0 if  ̸ = , while   = − ∑  ̸ =   , ,  = 1, 2, . . ., .As a standing hypothesis, we assume that the Markov chain () is irreducible, which means that system (4) can switch from one regime to another.Under this assumption, the Markov chain has a unique stationary distribution  = ( 1 ,  2 , . . .,   ) which is the solution of the system of linear equations Γ = 0 subject to ∑  =1   = 1 and   > 0 for all  ∈ S. Hence, for any vector  = ((1), . . ., ())  , we have that lim Let us consider the following stochastic differential equation with Markov conversion.

Complexity
Next, we consider the following stochastic differential equation: Lemma 3 (see [78]).Suppose that the coefficients of ( 12) are -periodic in  and there exists a function (, ) ∈  2 which is -periodic in , and (, ) satisfies the following conditions: Then there exists a solution for (12) which is a -periodic Markov process.
Furthermore, we introduce some results from [80,83] in Lemmas 4 and 5, which will be used in next section.
holds for all  ≥  and constants   ( = 1, 2, . . ., ), then (ii) if there are three positive constants , , and  0 such that holds for any  ≥ , then Finally, we give some basic properties of the following subsystem of system (3), Lemma 5 (see [83]).System ( 17) has a unique -periodic solution  *  () which is globally asymptotically stable.Here For convenience and simplicity, define α = min ∈S   , α = max ∈S   , and

Existence of Periodic Solutions of System (3)
In this section, we devote our attention to the investigation of the existence of periodic solutions of system (3).From Lemma 5, we know that system (17) has a globally asymptotically stable periodic solution  *  (); therefore, the limit system of (3) is where  1 (),  2 (), and  *  () are all positive and continuous functions of period .Now, we discuss the existence of periodic solutions of system (18).Define Then, we have the following theorem about periodic solutions of system (18).Theorem 6.If ℎ 1 < 0 and ℎ 2 > 0, there exists a prey extinction periodic solution (0,  * ()) of system (18).
Proof.From the first equation of system (18), it is easy to see Applying Itô's formula and then integrating from 0 to , we obtain ln where () = ∫  0  1 ()d 1 () is local martingale.From strong law of large numbers for martingales (see [84]), we have It then follows from ( 21) by dividing  on both sides and letting  → ∞ that lim sup namely, () tends to zero exponentially almost surely.Since lim →∞ () = 0, a.s., from the second equation of system (18), its limit system is According to Theorem 4.2 in [85], when  2 −  2  *  () > 0 and (24) has a unique positive -periodic solution  * ().
The proof of this theorem is completed.
In order to investigate the existence of a nontrivial positive -periodic solution for system (18), first of all, we assume following conditions hold.
Proof.Obviously, the coefficients of system (18) are continuous bounded positive periodic functions in .Now, we show that conditions (i) and (ii) of Lemma 3 hold.Define a nonnegative  2 -function where where   = (1/, ) × (1/, ).Therefore, condition (i) of Lemma 3 holds.Next, we will prove that condition (ii) of Lemma 3 also holds.Applying Itô's formula, one has Therefore, Define a bounded closed set where 0 <  < 1 is a sufficiently small number such that and  3 ,  4 are quantities to be determined in the rest of the proof.Denote (29), it implies that where In fact, from condition ( 3 ), one can get that is to say,  1 < 0.

Extinction and Persistence in Mean of System (4)
In this section, we investigate the long-term dynamic behaviors of the prey-predator system (4) with white noise and telegraph noise in a polluted environment and then discuss the extinction and average persistence of prey and predator.According to Lemma 5, the periodic solution  *  () of the toxicant input is globally asymptotically stable, so the limit system of ( 4 In order to obtain the threshold conditions of persistence and extinction of system (44), we assume that Theorem 8. Given initial value ((0), (0), (0)) ∈ R 2 + × S for system (44) By the ergodic theory of the Markov chain and the strong law of large number, we have lim sup (iii) By the condition ( 1 ), one can get lim →∞ () = 0 a.s., so that the limit system of the second equation of system (44) This completes the proof of the theorem.
Remark 9.If   () ( = 1, 2) remains unchanged and   () ( = 1, 2) or  min increases so that  1 < 0 or  2 < 0, then condition ( 1 ) or ( 2 ) is established.That is to say, if the intrinsic growth rate and the predation intensity are relatively fixation, the increase of white noise intensity or pollutant concentration will lead to the extinction of the biological population.
Next, we will discuss the persistence of system (44).Applying Itô's formula to the first equation of system (44) when  is large enough, we derive ln  () where  is a sufficiently small positive number.In view of ( 4 ) and Lemma 4, we deduce lim inf That is, the prey population of system (44) will be persistence in mean under condition ( 4 ).Furthermore, the persistent property of the predator species of system (44) can be investigated as follows.
From the first equation of system ( 44   Obviously, () ≤ (), () ≤ ().Using Lemma 2, if  5 > 0, system (61) is ergodic and there exists a unique stationary distribution   , such that Applying Itô's formula to (60) and then integrating from 0 to , we get ln  () − ln  (0 From the first equation of system ( 44  ) d By Lemma 4, if In summary, one gets the following.
Theorem 10.Given initial value ((0), (0), (0)) ∈ R 2 + × S for system (44), then (i) if ( 4 ) is established, the prey population will be persistent in mean, (ii) if conditions ( 5 ) and ( 6 ) are satisfied, the predator population will be persistent in mean.Remark 11. (i) It can be seen from  4 , in the case where the intrinsic growth rate and the predation intensity are relatively constant, only by reducing the intensity of white noise or pollutant concentration, so that ( 4 ) can be established to ensure the lasting survival of the prey population.
(ii) Obviously,  4 <  5 ; if the prey population is persistent, the predator population is persistent as long as the white noise interference intensity or the toxin concentration is small enough, such that ( 6 ) is established.As can be seen from condition ( 6 ), the omnivorous nature of  contributes to its permanence.
In system (4), if the concentration of the toxicant in the environment remains unchanged, that is,   () =  is a positive constant, then the system can be converted into the following system: Lemma 12 (see [77]).System (7) for all  ∈ R  , with some constant  ∈ (0, 1] for all  ∈ R  .(iii) There exists a bounded open set D ⊂ R  with a smooth boundary satisfying that, for each  ∈ S, there is a twice continuously differentiable nonnegative function (⋅, ⋅) : D  → R and that for some  > 0, L(, ) ≤ −, for any (, ) ∈ D  × S.
Moreover, the Markov process ((), ()) has a unique ergodic stationary distribution (⋅, ⋅).Hence, for any Borel measurable function (⋅, ⋅) : Recently, the ergodicity and stationary distribution have been explored by many authors.In the following, we give sufficient conditions for the existence of stationary distribution of system (68) and prove the following theorem by showing that system (68) satisfies the three conditions in Lemma 12.
Using Lemma 12, we obtain the conclusion that ((), (), ()) is ergodic and positive recurrent; that is, system (68) is positive recurrent and has a unique stationary distribution.This completes the proof of Theorem 13.

Conclusions and Numerical Simulations
In this article, we discussed the dynamics of stochastic preypredator models with Beddington-DeAngelis functional response in polluted environment.Firstly, for system (3), there are the following properties: (1) If ℎ 1 < 0 and ℎ 2 > 0, the limit system of (3) has a prey extinction periodic solution (0,  * ()).

The Extinction and Persistence of System
It is easy to see from Figures 3 and 4 that the increase of the intensity of white noise can result in the extinction of prey and predator.
Figures 5 and 6 show that the increase of the amount of toxicant can also result in the extinction of the prey and predator.