Asymptotic Behavior Analysis for a Three-Species Food Chain Stochastic Model with Regime Switching

+is paper discusses the asymptotic behavior of a class of three-species stochastic model with regime switching. Using the Lyapunov function, we first obtain sufficient conditions for extinction and average time persistence. +en, we prove sufficient conditions for the existence of stationary distributions of populations, and they are ergodic. Numerical simulations are carried out to support our theoretical results.


Introduction
In recent years, the dynamic relationship between the predator and prey has become one of the research hotspots in ecology and mathematical ecology because of its universal importance. In particular, the predator-prey model is a typical inhibition model, which greatly changes the understanding of the existence and development of basic laws in the biological community. e following model is one of the Volterra models for the three species of the predatorprey system: where x 1 (t), x 2 (t), and x 3 (t) denote the densities of prey, predator, and top-predator population at time t, respectively. e parameters a 1 , a 2 , and a 3 are positive constants that stand for the intrinsic growth rate of the species x 1 (t), the death rate of the species x 2 (t), and the death rate of the species x 3 (t), respectively. e coefficient b 11 , b 22 , and b 33 are the intraspecific competition in the resource, b 12 and b 23 represent the rate of consumption, and b 21 and b 32 represent the contribution of prey to the growth of predator. As a matter of fact, there are many extensive studies in the literatures concerned with three-species predator-prey systems (see e.g., [1][2][3][4][5][6]). For example, Krikorian [1] considered the Volterra predator-prey model in a three-species and explained the global properties of its solution. Zhou [2] investigated the existence and global stability of the positive periodic solutions of delayed discrete food chains with omnivory. Hsu [5] considers a three-species Lotka-Volterra food web model with omnivores, which is defined as feeding on more than one nutritional level.
In addition, the population system is always affected by environmental noise, which is very important for discovering the nature of random system from the biological point of view. Generally speaking, there are various types of environmental noise, e.g., white or color noise. First, let us consider a simple color noise, such as telegraph noise (see e.g., [7][8][9][10][11][12][13]). is kind of colored noise can be explained by the transformation between two or more environmental models, which can be considered to be different due to rainfall or nutrition in the population model. erefore, we can model state switching through a finite-state Markov chain. Let r(t) be a right-continuous Markov chain on the probability space, taking values in a finite-state space S � 1, 2, . . . , N { } with the generator Γ � (c uv ) N×N given by where Δt > 0 and c uv is the transition rate from state u to state v and c uv ≥ 0 if u ≠ v while c uu � − u≠v c uv . en, we can incorporate the regime switch into the three-species food chain model (1) to obtain _ x 1 (t) � x 1 (t) a 1 (r(t)) − b 11 (r(t))x 1 (t) − b 12 (r(t))x 2 (t) , _ x 2 (t) � x 2 (t) −a 2 (r(t)) + b 21 (r(t))x 1 (t) − b 22 (r(t))x 2 (t) − b 23 (r(t))x 3 (t) , _ x 3 (t) � x 3 (t) −a 3 (r(t)) + b 32 (r(t))x 2 (t) − b 33 (r(t))x 3 (t) , with initial value x i (0) � x i0 ≥ 0, r(0) � ζ ∈ S. Here, we assume that the coefficients a i (k), b ij (k) (i, j � 1, 2, 3) are all positive for k ∈ S. Next, we consider other types of environmental noise, namely, the white noise (see e.g., [14][15][16][17][18][19][20][21][22]). In particular, Mao [14] showed that different structures of white noise may have different effects on the population systems; Mao et al. [15] revealed that the environmental noise can suppress a potential population explosion. So, we assume the intrinsic growth rate a i (k) is disturbed with where _ ω i (t)(i � 1, 2, 3) are independent white noises and σ i (k) is a positive constant representing the intensity of the white noise. We assume that the Brownian motion ω(t) is independent of the Markov chain r(t). erefore, system (3) becomes a three-species food chain stochastic model under regime switching: In this paper, we show that system (5) has the following properties: (i) e solution starting from anywhere in R 3 + will remain in R 3 + with probability 1. (ii) For any given initial value x(0) ∈ R 3 + and r(0) � ς ∈ S, there exists a positive constant K(p) such that the solution x(t) of system (5) has the following property: lim sup (iii) We show that if the noise is sufficiently large, the solution to system (5) will become extinct with probability 1. is is, lim t⟶∞ x i (t) � 0, a.s. i � 1, 2, 3.
And we prove that the predator of system (5) will tend to extinction almost surly in some assumptions.
(i) e persistent in time average is investigated under certain conditions, namely, the solution x(t) of system (5) with any initial value x(0) ∈ R 3 + , r(0) � ς ∈ S has the following property: (ii) In the case of noise being relatively small, there is a unique stationary distribution μ(·, ·) with ergodic property: where μ x (·) � k∈S μ(·, k) is the marginal stationary distribution of a solution x(t) of system (5). e key method used in this paper is to construct Lyapunov functions. is Lyapunov function analysis for stochastic differential equations has been used by many authors (see [9,15]). e paper is organized as follows. In Section 2, we give the unique existence and boundedness of the solution. In Section 3, we show the sufficient conditions for extinction and persistence in time average, respectively, which have closed relations with the stationary probability distribution of the Markov chain. en, in Section 4, by using Lyapunov function, the sufficient conditions for the stationary distribution and ergodicity of the solution of system (8) are established. Finally, we illustrate our main results through an example in Section 5.

Preliminaries
roughout this paper, unless otherwise specified, let (Ω, F, F t t≥0 , P) be a complete probability space with a filtration F t t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F 0 contains all P-null sets). Let R 3 + denote the positive cone of R 3 , namely, For convenience and simplicity in the following discussion, denote , (i, j � 1, 2,3).C 1,2 (R + × R 3 × S) denote the family of all nonnegative real-value function V(t, x, k) which are continuously twice differentiable in x and once in t.
Furthermore, as a standing hypothesis, we assume that the Markov chain r(t) is irreducible in this paper. is is very reasonable as it means that the system will switch from any regime to any other regime. is is equivalent to the condition that, for any u, v ∈ S, one can find finite numbers Note that Γ always has an eigenvalue 0. e algebraic interpretation of irreducibility is rank (Γ) � N − 1. Under this condition, the Markov chain has a unique stationary distribution π � (π 1 , π 2 , . . . , π N ) ∈ R 1×N , which can be determined by solving the following linear equation πΓ � 0 subject to N k�1 π k � 1π k > 0, ∀k ∈ S. For convenience, system (5) can be rewritten into the following form: where For any twice continuously differentiable function V(t, x, k) ∈ C 1,2 (R + × R 3 × S), we define L by where From the biological sense, we are only interested in the positive solutions of equation (5). erefore, we first prove the existence and uniqueness of the global positive solution. Theorem 1. For any given initial value x(0) ∈ R 3 + , r(0) � ς ∈ S, there is a unique positive solution x(t) of system (5), and the solution will remain in R 3 + with probability 1.
Proof. Note that the coefficients of system (5) are local Lipschitz continuous for the given initial value where τ e is the explosion time (see [8,15]). To show this solution is global, we need to show that τ e � ∞ a.s. Since the initial value is positive and bounded, there is a number m 0 > 0 large enough such that where infϕ � ∞ (as usual ϕ denotes the empty set). Clearly, τ m is increasing as m ⟶ ∞. Set τ ∞ � lim m⟶∞ τ m , hence τ ∞ ≤ τ e a.s. If we can show that τ ∞ � ∞ a.s., then τ e � ∞ a.s. and x(t) ∈ R 3 + a.s. for all t ≥ 0. In other words, to complete the proof all we need to show that τ ∞ � ∞ a.s. If this Mathematical Problems in Engineering 3 statement is false, there is a pair of constants T > 0 and ε ∈ (0, 1) such that P τ ∞ ≤ T > ε. Hence, there is an integer where are positive constants. e nonnegativity of this function can be seen from x − 1 − logx ≥ 0 on x > 0. en, from the generalized Itô formula (10), we have us, In addition, it is straightforward to see that lim |x|⟶∞ H(x, k) < 0 and there exists a constant M(k) > 0 such that max x∈R 3 + H(x, k) ≤ M(k). en, it follows from (15) and (17)  that   4 Mathematical Problems in Engineering where M * � max M(k), q N l�1 c kl is a positive constant. en, from Itô formula (10), we have e Gronwall inequality [9] implies that Set Ω m � τ m ≤ T for m ≥ m 1 and by (13), we know P(Ω m ) ≥ ε. Note that, for every ω ∈ Ω m , there is some It then follows from (20) that is completes the proof.
□ Theorem 2. For any given initial value Proof. By eorem 1, the solution x(t) will remain in R 3 + for all t ≥ 0 with probability 1.
Define a function: where are positive constants. By virtue of the generalized Itô's formula (10), we have is is

Mathematical Problems in Engineering
Furthermore, for any given positive constant p > 1, we have Note that Similar to (17), we obtain Combining (28)-(30), we can obtain

Mathematical Problems in Engineering
Hence, By comparison theorem [9], we obtain which implies that there is a T > 0 such that In addition, E[(x 1 (t) + x 2 (t) + x 3 (t)) p ] is continuous and there exists a ι(p) such that e proof is complete.

Extinction and Persistence in Time Average
In the previous section, we have proved that the solution of system (5) with a positive initial value remains in the positive cone R 3 + . In order to further study asymptotic properties of system (5), in this section, we investigate the persistence in time average and extinction of system (5) under a certain condition. We first give some assumptions and related definitions.

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Particularly, Proof. By the generalized Itô formula (10), we yield By Assumption 1 and basic inequality ( where Noticing that by the strong law of large numbers for martingales [9,15], we therefore have Similarly, lim t⟶∞ M i (t)/t � 0, a.s. i � 2, 3. It finally follows from (41), by dividing by t on both sides and then letting t ⟶ ∞, and we obtain lim sup So, we obtain the desired assertion. Let us consider the following system: According to Luo [11], we have shown that By the comparison principle, we know that x 1 (t) ≤ ϕ(t), t ≥ 0. □ Theorem 4. If Assumption 2 holds, then for any initial value x(0) ∈ R 3 + , r(0) � ς ∈ S, the predator of system (5) will tend to extinction almost surly.
e result is confirmed.
□ Remark 1. It can be seen from the proof process of eorem 4 that when the following conditions are met Species x 1 and x 2 coexist and species x 3 is extinct.

Lemma 1. If Assumption 3 holds, for any initial value
x(0) ∈ R 3 + , r(0) � ς ∈ S, the solution x(t) of system (5) has following property: Proof. e proof is a modification of that of Mao [16], Zhu and Yin [12], or Luo [11]. So, we omit it. □ Theorem 5. If Assumption 3 holds, the solution x(t) of system (5) with any initial value x(0) ∈ R 3 + , r(0) � ς ∈ S has the following property: Mathematical Problems in Engineering 9 lim inf is the only nonnegative solution of the following equation: (59) Proof. From system (5), we have where Integrating the both sides of (60) and dividing by t on both sides, we yield According to Lemma 1 and lim t⟶∞ ω i (t)/t � 0, i � 1, 2, 3, lim inf (63) e result is confirmed.

Stationary Distribution and Ergodicity
From eorem 3, we know that three-species tend to extinct almost surely if the noise intensity is sufficiently large. However, in the section, our main aim here is to find out what happen if the noise is relatively small. Namely, we shall discuss the sufficient conditions under which system (5) admits a unique ergodic stationary distribution. To this end, consider system (8): Next, we give the conditions for the unique existence and of stationary distribution (see [13,[23][24][25][26]).
Lemma 2 (see [23]). System (8) is positive recurrent if the following conditions are satisfied: (2) For each k ∈ S, with some constant λ ∈ (0, 1] for all x ∈ R 3 , (3) ere exists a bounded open subset D of R 3 with a smooth boundary satisfying that, for each k ∈ S there exists a nonnegative function function V(·, k): D c ⟶ R such that V(·, k) is twice continuously differentiable and that for some κ > 0 Moreover, the positive recurrent process (x(t), r(t)) has a unique ergodic stationary distribution μ. at is, if Z(x(t), r(t)) be a function integrable with respect to the measure μ, then P lim We assume that the following condition holds: en, we can prove.

Theorem 6.
If Assumption 1 and 4 are satisfied, then system (8) with initial value in R 3 + × S is positive recurrent and has a unique stationary distribution μ(·, ·) with ergodic property: where μ x (·) � N k�1 μ(·, k) is the marginal stationary distribution of a solution x(t) of system (8).
On the contrary, Next, we define a vector Ξ � (Ξ 1 , Since the generator matrix Γ is irreducible and Lemma 2.3 in [13], for Ξ k there exists a solution, for the following system: us, we obtain Combining (74)-(79), we obtain It then follows from Assumption 4 that erefore, from (81), there exists a sufficiently small ρ such that It then follows Lemma 2 that the solution of system (8) is positive recurrent with respect to the domain E ρ and has a unique stationary distribution. Moreover, the ergodic property follows from the moments estimation of eorem 2.

Numerical Simulations
In order to verify the results above, we numerically simulate the solution of the system (5). By Euler-Maruyama scheme [27,28], we use discretized Brownian paths over [0, T] and write efficient Matlab code, then obtain the simulation figures. en, we can get the stationary distribution of r(t) is π � (0.6, 0.4). Take the step size Δt � 0.005 and the following setting for parameters. Case 1. When r(t) � 1, when r(t) � 2, e above parameters satisfy the condition of eorem 3, that is, b 12 As can be seen from Figure 1, when Assumption 1 holds, all three species are extinct.
when r(t) � 2, It is easy to calculate (87) Figure 2 shows that, under the condition of Assumption 2, predators are almost inevitably extinct, and only the lowest predators survive.
Case 3. When r(t) � 1, when r(t) � 2, It is easy to see that    Figure 3 shows that when Assumption 3 holds, the three species live for an average of time.
In order to further expound the influence of the regime switching on the stationary distribution of system (5), we give the following example.
when r(t) � 2, en, we can get the stationary distribution of r(t) is π � (0.5, 0.5). It is easy to check that Assumption 4 is satisfied and π � 0.1229 > 0. e stationary distribution of the Markov chain implies that the populations have an equal chance living in the component-wise environment 1 or 2. Figure 4 shows that the stable distribution of Markov chain shows that species x 1 can survive in different environments. In Figure 4(a), the blue line, red line, and black line, respectively, represent the density track of species x 1 under two switching states. In Figure 4(b), the black line, red line, and blue line, respectively, represent the probability density function curve of species x 1 in two switching states. Figure 5 shows the density trajectory and probability density function  curve of predator x 2 under Markov chain switching. Figure 6 shows the density trajectory and probability density function curve of predator x 3 under Markov chain switching.

Conclusion
ree-species food chain model has been studied by many scholars recently. In particular, many asymptotic estimates on the sample average in time have been obtained (see e.g., [8,29]). However, to the best of our knowledge, there is rare result about the stationary distribution of a three-species food chain stochastic model under regime switching. In this paper, we develop and analyse a three-species food chain stochastic model, which takes both white and colored noises into account. We first prove the existence of the global positive solution of the model. en, using the stochastic Lyapunov functions, we investigate extinction in probability and persistence in time average. Furthermore, we obtain a stationary distribution of the solution. Moreover, some interesting questions deserve further investigation, such as incorporating intervention strategies into the system. We leave this for future consideration.

Data Availability
In addition, the numerical simulation in our paper is a verification of our conclusion. At present, no data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.