Analysis of Stochastic Nicholson-Type Delay System under Markovian Switching on Patches

Based on the influence of random environmental perturbations and the patch structure, we propose a stochastic Nicholson-type delay system under Markovian switching on patches. Existence of a global positive solution is studied. .en, we show ultimate boundedness and estimation of the sample Lyapunov exponent of the solution. Furthermore, sufficient conditions for extinction of species are established, which is the main new ingredient of this paper. Finally, some numerical examples are presented. Our results improve and generalize previous related results.

In ecosystems, the pattern of complex population dynamics is inevitably subject to some kind of environmental noises. As a matter of fact, the phenomenon of stochasticity plays a critical role in understanding the evolutionary dynamics and ecological characteristics of species. Particularly, May [14] has revealed that due to environmental fluctuations, the parameters in a system should be stochastic. Environmental noises are classified into two categories: the first is white noise, and the second one is coloured noise. Stochastic population models [15][16][17][18][19][20] are more realistic compared to deterministic population models. Wang et al. [21] first studied a scalar stochastic Nicholson's blowflies delayed equation dx(t) � − αx(t) + px(t − τ)e − cx(t− τ) dt + σx(t)dB(t). (1) Notice, however, that white noise is unable to depict the phenomena that the species may be invaded by the alien population [22] or suffer sudden catastrophic shocks [23]. And in recent years, some significant progress has been made in the theory of the stochastic population models with regime switching, see [24][25][26][27] and the references therein. In [28], Zhu et al. considered a stochastic equation with Markovian switching: where continuous-time Markov chain r t t≥0 is defined on a state space S � 1, 2, . . . , m { }. On the contrary, migration is a ubiquitous phenomenon in the nature. Both continuous reaction-diffusion models and discrete patchy systems could incorporate and explain the phenomenology of spatial dispersion [29] in the literature of mathematical ecology. Objectively speaking, patchstructured models illustrate the spatial heterogeneity of species, depending on a lot of factors, such as ecological systems in different geographic types (e.g., nature reserves and other regions), various food-rich patches of habitats, and many other circumstances. Besides, models in the patchy environment include disease systems as well, such as the two-compartment model of the cancer cell population. In order to take the dispersal phenomenon into consideration, Berezansky et al. [30] introduced the Nicholson-type delay system on patches as follows: which includes the novel two-compartment models of leukemia dynamics and the systems of marine protected areas.
In particular, considering that the parameters a i of system (3) are affected by the white noise, Yi and Liu [31] formulated the stochastic diffusion system which consists of two patches: We can further model random shift in different regimes by a continuous-time Markov chain ℓ(t) Suppose that ℓ(t) { } t≥0 is irreducible and has the unique stationary distribution π � (π 1 , π 2 , . . . , π N ). Hence, we obtain the stochastic Nicholson-type system under Markovian switching on the patch structure as follows: with initial conditions where We focus on the meaning of parameters with respect to fish population in marine protected area A 1 and fishing area A 2 . x 1 (t) and x 2 (t) are the number of fish populations in A 1 and A 2 , respectively; for h � 1, 2 and i ∈ M, a 1 (i) and a 2 (i) are the mortality rate in A 1 and A 2 , respectively; let be the fish growth rates; p 1 (i) and p 2 (i) represent the maximum per adult yearly birth rate in A 1 and A 2 , respectively; c h (i) > 0; 1/c 1 (i) and 1/c 2 (i) are the number at which the reproduction at their maximum birth rate in A 1 and A 2 , respectively; τ(i) is the maturation time; B h (t) is the standard Brownian motion defined on the complete probability space (Ω, F, P); and σ h (i) ≥ 0, for any i ∈ M and h � 1, 2. We assume and B h (t) are independent of each other, h � 1, 2.
Especially, system (6) can reduce to the model in [32] if τ(i) ≡ τ, i ∈ M. By contrast, our work differs from and improves [32], which will be depicted further in detail.
In the field of ecology, it is important to use mathematics to study extinction of species, see [33,34] and the references therein. However, no work has yet been done on the problem of extinction for scalar equation (1), not to mention the scalar equation with Markovian switching (2) and system (4). In order to prove the extinction of species, the conventional method is to construct a proper Lyapunov function or functional and then estimate the upper bound of the drift term of its Ito differential. Taking system (6) for example, x 1 (t) and x 2 (t) are likely to appear in the denominator of the expression of LV, and coefficients in front of them are positive, for a general Lyapunov function V(x 1 , x 2 ). Unfortunately, this leads to some difficulties in finding the upper bound of LV. So, based on this, we give a new method for investigating extinction of species.
2 Complexity Especially, system (6) reduces to (1), (2), (4), or the system in [32] when parameters of system (6) assume some special values. at is to say, we have derived extinction of the above systems at the same time.
In this paper, system (6) is more general than the model of [21,28,[30][31][32]. In addition, our results improve and generalize the corresponding results in these literature studies.
e remainder of this paper is built up as follows. In Section 2, we show the global existence of almost surely positive solution. e asymptotic estimates for the solution, stochastically ultimate boundedness, and boundedness for the average in time of the θth moment of the solution are then constructed in Section 3. In Section 4, we discuss the pathwise properties of the solution. Sufficient conditions for extinction of species are obtained in Section 5. Numerical investigations are then given in Section 6. e last part is a conclusion.
Proof. We omit the proof since it is analogous to that of [31] by making use of the generalized Ito formula (see, e.g., eorem 1.45 in [35]) to 2 e delay stochastic Nicholson-type model under regime switching on patches (6) is a direct extension of the models in [21,28,[30][31][32]. From Lemma 1, it is worthy to point out that priori conditions α > σ 2 /2 in [21] are unnecessary. erefore, Lemma 1 improves and generalizes Lemma 2.2 in [21]. In addition, this lemma shows that both white noise and telegraph noise will not destroy a great property that the solution of (3) does not explode.

Boundedness
Because of resource constraints, asymptotic boundedness is the core of the research in ecosystems. And it is the main purpose of the present section. For simplicity, we use the following notations. For any i ∈ M, denote Firstly, inspired by the work of Wang and Chen [32], we give this theorem.

Proof.
Define e generalized Ito formula, together with the fact erefore, for t > 0, en, (16) implies Noting that the Markov chain ℓ(t) has an invariant distribution π � (π i .i ∈ M) and applying the ergodic property of the Markov chain, it yields Hence, Consequently, we infer immediately that (12) holds. On the contrary, according to (12), (18), and the fact that it follows that (11) and (13) hold. e proof is therefore complete. □ Remark 2. In eorem 1, the parameter θ is greater than 1 in the result. Although ultimate boundedness in the θth moment was derived for θ restricted to the precondition θ > 1, θth moment of system (6) can be obtained when θ ≤ 1 by Hölder's equality.

Theorem 2. Given any initial values
where λ � min a 1 , a 2 . at is, (6) is ultimately bounded in mean.
eorem 3 can be seen as the extension and improvement of [31,32].

Asymptotic Pathwise Estimation
We shall estimate a sample Lyapunov exponent in what follows.
By the properties of quadratic functions, the proof of this lemma is easy and so is omitted. In the process of finding K(a), we know that the precondition is a − K(a) < 0. In this case, we can choose K(a) which satisfies K(a) � (a + ������ a 2 + b 2 √ /2). We have to mention that it has no relation with the sign of parameter a. If a < 0, we get (a + ������ a 2 + b 2 √ /2) < − (b 2 /4a) by simple computation. So, this lemma is an improvement of Lemma 1.2 in [28] and Lemma 2.1 in [32]. (7), solution x(t) of (6) satisfies

Theorem 4. Given any initial values
Complexity 5 for any positive constant ε.

Extinction
Sufficient conditions for extinction are the subject of this section. Unless otherwise stated, we hypothesize τ(i) ≡ τ, i ∈ M in this section. We first rewrite (6) as follows: where the operator f 1 : the operator f 2 : is defined as We first note that for i ∈ M, whence (6) admits a trivial solution corresponding to φ(0) � 0. Before our result, we give a lemma.

Lemma 3.
For system (36), the terms f 1 (x, y, i) and f 2 (x, i) are locally bounded in (x, y) while uniformly bounded in i. at is, for any m > 0, there is K m > 0 satisfying for all i ∈ M, x, y ∈ R 2 + with |x| ∨ |y| ≤ m.
e proof is not particularly difficult, so we omit the proof.

Theorem 5. Assume that
en, the solution of (36) satisfies lim t⟶∞ x(t) � 0, a.s., for any initial values (7). at is, all populations in system (36) go to extinction with probability one.
□ Corollary 1. Assume that are nonnegative constants, τ r t ≡ τ ≥ 0, and 2 min en, solution x(t) of (2) obeys for any initial value . at is, all populations in equation (2) go to extinction with probability one.

Corollary 2.
Assume that a h , b h , p h , c h , σ h , τ are nonnegative constants, h � 1, 2, and en, solution x(t) of (4) obeys lim t⟶∞ x(t) � 0, a. s., for any initial value . at is, all populations in model (4) go to extinction with probability one.

Remark 7.
is theorem reveals that the solutions of (6) will all tend to the origin asymptotically with probability one when the intensities of noises and the parameters satisfy condition (40). However, [21,28,31,32] do not study extinction of populations. Besides, this method can be extended to research extinction in the above literature studies. 10 Complexity Corollaries 1 and 2 give the conditions of extinction of (2) and (4), respectively. erefore, our work is the extension of [21,28,31,32].

Numerical Simulations
Based on [38], we show numerical simulations in the present section.
In Figure 1, we give a simulation of the sample path of ℓ(t) t≥0 with ℓ(0) � 3.
In Figure 2 Figure 3 clearly supports this result.
In Figure 4 erefore, conditions of eorem 5 have been checked. us, from eorem 5, all species become extinct. Figure 4 clearly supports this result.

Conclusions
By the conclusion of Lemma 1, it is worthy to point out that the Brownian noise and colored noise will not destroy a great property that the solution of (6) may not explode. Especially, system (6) reduces to (1)-(4) or the model in [32] when parameters of system (6) take some special values. From Lemma 1, the condition α > (σ 2 /2) in [21] is too strict and unnecessary. In eorem 1, we comprehensively analyze ultimate boundedness in the θth moment and boundedness for the average in time of the θth moment of solution, which is the improvement of eorem 3.1 in [21], eorem 2.2 in [28], eorem 3.3 in [31], and eorem 3.2 in [32]. In eorem 4, we find an upper bound Q/2 of the sample Lyapunov exponent. When parameters of system (6) take some special values, we compute that the upper bound Q/2 is less than the corresponding upper bound in [21,28]. Furthermore, we find that the condition 2α 1 [31] is not necessary. Despite all this, if we let parameter ϵ satisfy the above conditions, we compute that Q/2 is less than the upper bound in [31]. One point should be stressed is that the method for extinction in eorem 5 can be used successfully for the models in [21,28,31,32]. And then, Corollaries 1 and 2 give the conditions of extinction of (2) and (4), respectively. From Remarks 1-7, our work is a generalization and promotion of the corresponding work in [21,28,[30][31][32]. To some extent, our proposed approaches are both more robust and more efficient than the existing methods.

Data Availability
No data were used to support this study.

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

Authors' Contributions
All the authors contributed equally and significantly to writing this paper. All authors read and approved the final manuscript.  (6)