Long-Time Behaviours of a Stochastic Predator-Prey System with Holling-Type II Functional Response and Regime Switching

Considering the impacts of white noise, Holling-type II functional response, and regime switching, we formulate a stochastic predator-prey model in this paper. By constructing some suitable functionals, we establish the sufficient criteria of the stationary distribution and stochastic permanence. By numerical simulations, we illustrate the results and analyze the influence of regime switching on the dynamics.


Introduction
Functional responses are very important in the predatorprey system, which is the amount of prey catch per predator per unit of time and has significant effect on the dynamical properties. Usually there are two kinds of functional response: prey dependent (such as Holling II and Holling IV, see [1][2][3]) and predator dependent (such as Hassell-Varley, Beddington-DeAngelis, and Crowley-Martin, see [4,5]). Recently, a number of researchers have devoted their efforts to the predator-prey system with functional response and obtained some nice results [1][2][3][4][5][6][7]. For the ecological system, the growth rate of population is inevitably affected by environmental white noise, which almost exists everywhere in real world [8][9][10]. May reveals that due to stochastic fluctuations in environmental conditions, all the natural parameters exhibit a certain amount of random perturbations, and hence, random disturbance is introduced in many mathematical models to reveal the effect of white noise [10][11][12][13][14][15]. Besides the white noise, the growth of species also suffers from fluctuating environments such as hurricanes and earthquakes, which is described by colorful noise in mathematical modelling [16][17][18]. e colorful noise may take several values and switch among different regimes of environments. e switching is memoryless, and the waiting time for the next switching follows an exponential distribution. at is, in mathematical sense, it is a Markovian process. Actually, when the environments fluctuate frequently, colorful noise may bring great influence to population dynamics and even change the permanence and extinction of species, so the impacts of colorful noise on population dynamics have attracted many researchers, see, e.g., [19][20][21][22]. Motivated by above discussion, in this article, we formulate a stochastic model with Holling-type II functional response and colorful noise. By stochastic analysis, we aim to study the stability in distribution and stochastic permanence of the system. e rest of this paper is structured as follows. Section 2 begins with our model and some notations. Section 3 is devoted to the stability in distribution of the above system. Section 4 focuses on the stochastic permanence. Some examples are given to illustrate our main results in Section 5. Finally, a brief conclusion and discussion are given to end the paper in Section 6.
where r 1 > 0 and − r 2 < 0 represent the birth rate of prey and death rate of predator, respectively; b 1 and b 2 are intraspecific competition rate between species; c 1 > 0 is the capture rate, and c 2 > 0 is the conversion rate of food; σ 2 i (i � 1, 2) denotes the density of white noise; (y(t)/1 + x(t)) is the Holling-type II functional response. B 1 (t) and B 2 (t) are independent standard Brownian motions defined on the probability space (Ω, F, F t t ≥ 0 , P) with a filtration F t t ≥ 0 satisfying the usual conditions (i.e., it is right continuous and F 0 contains all p-null set). In view of the impact of regime switching (colorful noise) analyzed before, system (1) turns to the following: where ϵ > 0, χ ij is the transition rate from the ith stage to the jth stage and χ ij ≥ 0 if i ≠ j while χ ii � − i≠j χ ij . It is often assumed that every sample of α(t) is a right continuous step function and irreducible with a finite simple jumps in any finite subinterval of R + � [0, ∞). It obeys a unique stationary distribution π � (π 1 , π 2 , . . . , π N ) satisfying πχ � 0 and N k�1 π k � 1, π k > 0, ∀k ∈ S. e detailed switching mechanism of the hybrid system is referred to [19,23].
For the later discuss, we introduce some notations about the Itô's integral for stochastic differential equations with Markovian switching [19,22]. Let . Define the operator LV as follows: e generalized Itô's formula is defined as Lemma 1 (see [21]). If the following conditions hold.
regular boundary (i.e., smooth) such that, for any k ∈ S, there exists a nonnegative function V(·, k): D C ⟶ R satisfying V(·, k) is twice continuously differentiable and for some ϵ > 0, en, (5) is ergodic and positive recurrent; that is, there exists a unique stationary density μ(·, ·), for any Borel measurable function f(·, ·): About the existence and uniqueness of positive solutions and the moment boundedness of (2), we have the following two lemmas. e proofs of them are very standard and are omitted here. Readers may refer to [3,21].

Stationary Distribution
In this section, we discuss the stationary distribution of (2).

Theorem 1.
For any initial value (x(0), y(0), α(0)) ∈ R 2 + × S and any k ∈ S, the solution (x(t), y(t), α(t)) of (2) is ergodic and has a unique stationary distribution in R 2 + × S if the following condition holds: Proof. According to the equivalent property of (2) and (4), we only need to prove it for (4). Define where On the other hand, Set q � ((c 1 + b 2 )/ � r 2 ), and similarly we have 4 Journal of Mathematics Define where q � (c It is easy to observe that and Define a bounded closed set as follows: Journal of Mathematics where ε is a sufficiently small number, and then the set U C � (R 2 + /U) contains the following four domains: Take ε sufficiently small enough such that where η 1 , η 2 are defined later. Next, we verify By (17), (18), and (21), we have 6 Journal of Mathematics Case 2. If (u, v) ∈ U 2 ε , namely, − ∞ ≤ v ≤ ln ε, then e u+v ≤ εe u ≤ ε(1 + e 3u ), and similarly we have Case 3. If (u, v) ∈ U 3 ε , then we derive from (17) and (22) that where Case 4. If (u, v) ∈ U 4 ε , similarly, from (17) and (22) we have where Consequently, we deduce that LV(u, v) ≤ − 1 on all (u, v) ∈ U C . Obviously, the other condition of Lemma 1 holds too, so we conclude from Lemma 1 that system (4) is ergodic and has a unique stationary distribution in R 2 + × S; that is, system (2) is ergodic and has a unique stationary distribution in R 2 + × S. is completes the proof. For (2), if the state Markovian chain α(t) takes value in space S � 1 { }, namely, there is no switching, then (2) turns to the following subsystem: For (27), from eorem 1, we can easily obtain the following conclusion.

Stochastic Permanence
For (2), if we consider the birth rate instead of the death rate of predator, then (2) turns to the following model: where r 2 (·) > 0 is the birth rate of species y(t) and other parameters are the same as before. Now, we consider the stochastic permanence of (29).
e proof is rather standard. Readers may refer to the details in [24] or [21,23].
Obviously, if there is no switching, we can similarly obtain the following corollary.

Corollary 2.
For any initial value (x(0), y(0)) ∈ R 2 + , the subsystem of (29) is stochastically permanent if Remark 3. eorem 2 reveals that when some subsystems of (2) are no stochastic permanent, if we give a suitable switching, then switching system (2) may be stochastic permanent, which implies the switching has very important influence to the dynamics of (2). By simulation, we can verify it directly, see Figure 1.

Conclusions and Discussion
In this paper, we study a stochastic predator-prey system with regime switching and Holling-type II functional responses. eorems 1 and 2 give the sufficient conditions of stationary distribution and the stochastic permanence of (2). Finally, some examples are given to verify the main results. Our numerical examples reveal that switching and random factors bring much influence to the dynamics of this system.
By comparison analysis, we give Remarks 1 and 2 to show that our main results improve or generalize the corresponding results in [3]. e main method applied in this paper is by constructing some suitable functionals instead of stochastic analysis techniques to study the stationary distribution, which is less applied for switching system. In the process of our analysis, Holling-type II functional response brings some difficulties and we apply inequality techniques to overcome them.
As Arditi and Ginzburg [23] pointed out that the predator-dependent functional response can provide better description in some cases, then how to deal with predatordependent functional response such as Beddington-DeAngelis type? Furthermore, studies have shown that the mental state of the adolescent prey can be mediated by fear induced from predators and the alternation causes deleterious outcomes on their adult's survival [24] and then how fear will impact our system? All these are interesting for us to study in the future.

Data Availability
No data were used to support the findings of this study.