Markov switched stochastic Nicholson-type delay system with patch structure

Considering stochastic perturbations of white and color noises, we introduce the Markov switched stochastic Nicholson-type delay system with patch structure. By constructing a traditional Lyapunov function we show that solutions of the addressed system are not only positive, but also do not explode to infinity in finite time and, in fact, are ultimately bounded. Then we estimate its ultimate boundedness, moment, and Lyapunov exponent. Finally, we present an example of numerical simulations to verify theoretical results.


Introduction
Considering that in the models of marine protected areas and B-cell chronic lymphocytic leukemia [1] the mortality rate is perturbed by the white noise of the environment, Yi and Liu [2] and Wang et al. [3] have presented a stochastic Nicholson-type delay system with patch structure: where i ∈ I := {1, 2, . . . , n}, x i (t) is the size of the population at time t, a i is the per capita daily adult death rate, p i is the maximum per capita daily egg production, 1 γ i is the size at which the population reproduces at its maximum rate, τ i is the generation time, b ij (i = j) is the migration coefficient from compartment i to compartment j, B i (t) is an independent white noise with B i (0) = 0 and intensity σ 2 i . It is well known that the scalar Nicholson blowflies delay differential equation originated from [4,5], and Berezansky et al. [6] summarized some results and introduced several open problems to attract many scholars [7][8][9][10][11][12][13][14][15][16][17][18]. Stochastic system (1.1) can be regarded as a generalization of the deterministic Nicholson blowflies model.
In the real world, it is complex for any practical system, since besides white noises, there are color noise interferences. One type of color noises is the so-called telegraph noise, which causes the system to switch from one environmental regime to another [19] and can mostly be modeled by a continuous-time Markov chain to describe the switching process between two or more regimes. To the best of our knowledge, almost no one or a few researchers consider the Markov switched stochastic Nicholson-type delay system with patch structure. This prompts us to propose the following stochastic system: with initial conditions where ξ (t) (t ≥ 0) is a continuous-time irreducible Markov chain with invariant distribution π = (π k , k ∈ S), which takes values in a finite state space S = {1, 2, . . . , N}, and its generator Q = (ν ij ) N×N satisfies Here ν ij ≥ 0 for i, j ∈ S with i = j, and j=N j=1 ν ij = 1 for each i ∈ S, B i (t) are independent Brownian motions with B i (0) = 0 (i ∈ I), and they are independent of the Markov chain ξ (t). For i, j ∈ I and k ∈ S, the parameters τ i , a i (k), and γ i (k) are positive, and b ij (k), p i (k), and σ 2 i (k) are nonnegative constants. Since system (1.2) describes the dynamics of a Markov switched stochastic Nicholson-type delay system with patch structure, it is important to study whether or not the solution: • remains positive or never becomes negative, • does not explode to infinity in finite time, • is ultimately bounded in mean, and • to estimate the moment and sample Lyapunov exponent.
In this paper, we discuss these problems one by one. In Sect. 2, we consider the existence and uniqueness of the global positive solution of (1.2)-(1.3). Next, we study its ultimate boundedness in mean, its moment, and its sample Lyapunov exponent in Sect. 3. We carry out an example and its numerical simulation to illustrate theoretical results in Sect. 4. Finally, we provide a brief conclusion to summarize our work.

Preliminary results
In this section, we introduce some basic definitions and lemmas, which are important for the proof of the main result. Unless otherwise specified, (Ω, {F t } t≥0 , P) is a complete probability space with filtration {F t } t≥0 satisfying the usual conditions (i.e., it is right continuous, and F 0 contains all P-null sets). Let B i (t) (i ∈ I) be independent standard Brownian motions defined on this probability space. For simplicity, in the following sections, we use the following notation: Let p ≥ 1 be such that for each i ∈ I, A i (p, ξ (t)) > 0, and C i (p, ξ (t)) is bounded, where It is easy to see that for i ∈ I, A i (1, ξ (t)) = a i (ξ (t)) > 0 and by continuity we can find p > 1 such that ) is also bounded.
Proof By Lemma 1.2 of [21] the result easily follows, so we omit the proof.

Lemma 2.2 For any given initial conditions
Proof Because all coefficients of system (1.2) are locally Lipschitz continuous, for any given initial condition (1.3), there exists a unique maximal local solution where ν e is the explosion time.
Firstly, we prove that x(t) is positive on [0, ν e ) almost surely. For t ∈ [0, τ ], system (1.2) with initial conditions given in (1.3) becomes the system of stochastic linear differential equations: From the stochastic comparison theorem [22], are the solutions of the stochastic differential equations For t ∈ [0, τ ], system (2.2) has the explicit solutions Using the same method, we have x i (t) > 0 a.s. for t ∈ [τ , 2τ ], i ∈ I. Moreover, repeating this procedure, we also have x i (t) > 0 (i ∈ I) a.s. on [mτ , (m + 1)τ ] for any integer m ≥ 1. Thus system (1.2) with initial conditions (1.3) has the unique positive solution x(t) almost surely for t ∈ [0, τ e ).
Next, we prove that x(t) exists globally. Let m 0 ≥ 1 be sufficiently large such that max -τ ≤t≤0 ϕ i (t) < m 0 , i ∈ I. For every integer m ≥ m 0 , define the stopping time where throughout this paper, inf ∅ := ∞. Obviously, ν m is increasing as m → ∞. Set ν ∞ = lim m→∞ ν m , where ν ∞ ≤ ν e a.s. If we can prove that ν ∞ = ∞ a.s., then ν e = ∞ and x(t) ∈ R n + for all t ≥ 0 a.s. For this purpose, we need to show that ν ∞ = ∞ a.s. Define , it is easy to show by Itô formula that where m ≥ m 0 and T > 0 are arbitrary, and In the last inequality, we used the fact that sup x≥0 xe -x = 1 e . For any m ≥ m 0 , integrating both sides of (2.3) from 0 to ν m ∧ T and taking expectations yield that (2.5) Since for every ω ∈ {ν m ≤ T}, there exists at least one of x i (ν m , ω) (i ∈ I) equal to m, we have that V (x(ν m ∧ T)) ≥ (m -1 -ln m). Then from (2.5) it follows that where I {ν m ≤T} is the indicator function of {ν m ≤ T}. Letting m → ∞ gives lim m→∞ P{ν m ≤ T} = 0, and hence P{ν ∞ ≤ T} = 0. Since T > 0 is arbitrary, we must have P{ν ∞ < ∞} = 0. So P{ν ∞ = ∞} = 1 as required, which completes the proof of Lemma 2.2.

Main results
By Lemma 2.2, we show that the solution of the Markov switched stochastic Nicholsontype delay system (1.2) with initial conditions (1.3) remain in R n + almost surely and do not explode to infinity in finite time. This good property gives a great opportunity to study more complicated dynamic behaviors of system (1.2). In this section, we study the remaining problems: estimating the ultimate boundedness in mean, the average in time of the pth moment, and a sample Lyapunov exponent for system (1.2).
Proof By Lemma 2.2 the global solution x(t) of (1.2) is positive on t ≥ 0 with probability one. It follows from (1.2) and the fact sup x≥0 xe -x = 1 e that which, together with Itô's formula, implies that Integrating both sides of (3.3) from 0 to t and then taking the expectations, we have , it is easy to get lim sup t→∞ E|x(t)| ≤ nc a , which is the required statement (3.1). The proof is now completed.
Proof In view of Itô's formula, Young's inequality, and the fact sup x≥0 xe -x = 1 e , from (1.2) it follows that Since the Markov chain ξ (t) has an invariant distribution π = (π i , i ∈ S), this implies lim sup Using Itô's formula, the Young and Cauchy inequalities, and the fact sup x≥0 xe -x = 1 e again, from (1.2) and Lemma 2.1 we get that Meanwhile, the exponential martingale inequality (Theorem 1.7.4 of [25]) implies that, for every l > 0, Using the convergence of ∞ l=1 1 l 2 and the Borel-Cantelli lemma (Lemma 1.2.4 of [25]), we obtain that there exists a set Ω 0 ∈ F with P(Ω 0 ) = 1 and a random integer l 0 = l 0 (ω) such that, for every ω ∈ Ω 0 , for all 0 ≤ t ≤ l, l ≥ l 0 , i ∈ I. Substituting (3.9) into (3.8), for any ω ∈ Ω 0 , l ≥ l 0 , 0 < l -1 ≤ t ≤ l, we have Letting l → ∞ and recalling that the Markov chain ξ (t) has an invariant distribution π = (π j , j ∈ S), we get that lim sup By Itô's formula, from system (1.2) we obtain that where i = 1, 2, 3, t = 0.01, k = 100, ξ n ∈ S ( Fig. 1(b)) is a 3-state Markov chain with generator P, {U n i } is a sequence of mutually independent random variables with EU n i = 0 and E(U n i ) 2 = 1, independent of the Markov chain ξ n . Furthermore, from Theorems 3.1 and 3.2 we have the following estimates:

Conclusions
This paper is concerned with n connected Nicholson's blowflies models under perturbations of white and color noises. Using a traditional Lyapunov function, we show that the solution of the Markov switched stochastic Nicholson-type delay system with patch structure remains positive and does not explode in finite time. Meanwhile, we estimate its ultimate boundedness, pth moment, and Lyapunov exponent. From Remarks 2.1 and 3.1 we find that the results obtained in this paper extend and improve some results in [2,3,21,23,24,28,29]. Inspired by the latest stochastic models in [30,31], in the future work, we will deeply study dynamic behaviors of the addressed system, such as persistence, extinction, and so on.