Abstract
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. Then, 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.
1. Introduction
In 1980, Gurney et al. [1] established Nicholson’s blowflies equation according to experimental data of Nicholson [2]. In recent decades, there have been a large amount of results related to the dynamical behaviors for this model and its modification, see [3–13].
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 [15–20] are more realistic compared to deterministic population models. Wang et al. [21] first studied a scalar stochastic Nicholson’s blowflies delayed equation
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 [24–27] and the references therein. In [28], Zhu et al. considered a stochastic equation with Markovian switching:where continuous-time Markov chain is defined on a state space .
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, patch-structured 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 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 defined on a state space . Let be right-continuous and be its generator of , i.e.,where , for , and . Suppose that is irreducible and has the unique stationary distribution . Hence, we obtain the stochastic Nicholson-type system under Markovian switching on the patch structure as follows:with initial conditionswhere and for and .
We focus on the meaning of parameters with respect to fish population in marine protected area and fishing area . and are the number of fish populations in and , respectively; for and , and are the mortality rate in and , respectively; let be the fish growth rates; and represent the maximum per adult yearly birth rate in and , respectively; ; and are the number at which the reproduction at their maximum birth rate in and , respectively; is the maturation time; is the standard Brownian motion defined on the complete probability space ; and , for any and . We assume is -adapted. Nevertheless, suppose and are independent of each other, .
Especially, system (6) can reduce to the model in [32] if . 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 It differential. Taking system (6) for example, and are likely to appear in the denominator of the expression of , and coefficients in front of them are positive, for a general Lyapunov function . Unfortunately, this leads to some difficulties in finding the upper bound of . So, based on this, we give a new method for investigating extinction of species. Especially, system (6) reduces to (1), (2), (4), or the system in [32] when parameters of system (6) assume some special values. That 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, 30–32]. In addition, our results improve and generalize the corresponding results in these literature studies.
The remainder of this paper is built up as follows. In Section 2, we show the global existence of almost surely positive solution. The asymptotic estimates for the solution, stochastically ultimate boundedness, and boundedness for the average in time of the th 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. The last part is a conclusion.
2. Preliminary Results
To simplify, denote the solution of (6) with initial values (7):where . Let
We denote , , and . For any , let and . Let denote Euclidean norm in . Denote the trace norm for matrix .
Lemma 1. Given any initial values (7), system (6) has a unique solution for all almost surely.
Proof. We omit the proof since it is analogous to that of [31] by making use of the generalized It formula (see, e.g., Theorem 1.45 in [35]) to
Remark 1. The delay stochastic Nicholson-type model under regime switching on patches (6) is a direct extension of the models in [21, 28, 30–32]. From Lemma 1, it is worthy to point out that priori conditions in [21] are unnecessary. Therefore, 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.
3. 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 , denote
Firstly, inspired by the work of Wang and Chen [32], we give this theorem.
Theorem 1. Let such that . Given any initial values (7), solution of (6) satisfiesand
In particular,
That is, system (6) is ultimately bounded.
Proof. DefineThe generalized It formula, together with the fact and the elementary inequality for any and , yieldsTherefore, for ,Then, (16) impliesNoting that the Markov chain has an invariant distribution and applying the ergodic property of the Markov chain, it yieldsFurthermore, we haveHence,Consequently, we infer immediately that (12) holds. On the contrary, according to (12), (18), and the fact thatit follows that (11) and (13) hold. The proof is therefore complete.
Remark 2. In Theorem 1, the parameter is greater than 1 in the result. Although ultimate boundedness in the th moment was derived for restricted to the precondition , th moment of system (6) can be obtained when by Hölder’s equality.
Remark 3. Without regime switching or without migration and regime switching, Theorem 1 improves the corresponding results in [21, 31]. If , system (6) is a direct extension of the model in [32]. Besides, no proof of ultimate boundedness in the th moment is given in [32], which is shown in Theorem 1. Therefore, this theorem extends and improves Theorem 3.1 in [21], Theorem 2.2 in [28], Theorem 3.3 in [31], and Theorem 3.2 in [32].
Theorem 2. Given any initial values (7), solution of (6) satisfieswhere . That is, (6) is ultimately bounded in mean.
Proof. Let Then,Finally, (22) follows by letting . The proof is therefore complete.
Remark 4. Compared with Theorem 1, this theorem describes the case that , which does not require any conditions. If , we get , where is defined in [32]. So, this theorem improves and extends Theorem 3.1 in [21] and Theorem 3.1 in [32].
Theorem 3. System (6) is stochastically ultimately bounded.
Proof. By (22), we deriveBy the Chebyshev inequality, it yields, for any ,where . The proof is therefore complete.
Remark 5. Theorem 3 can be seen as the extension and improvement of [31, 32].
4. Asymptotic Pathwise Estimation
We shall estimate a sample Lyapunov exponent in what follows.
Lemma 2. If and , then for , where
By the properties of quadratic functions, the proof of this lemma is easy and so is omitted. In the process of finding , we know that the precondition is . In this case, we can choose which satisfies We have to mention that it has no relation with the sign of parameter . If , we get by simple computation. So, this lemma is an improvement of Lemma 1.2 in [28] and Lemma 2.1 in [32].
Theorem 4. Given any initial values (7), solution of (6) satisfieswhere , withfor any positive constant .
Proof. The generalized It formula, together with Lemma 2 and the Cauchy–Schwarz inequality, yieldswhere for any ,with the quadratic variationAccording to the exponential martingale inequality (see, e.g., [36]), for any integer , we haveSince and Borel–Cantelli’s lemma (see, e.g., [36]), there exist with and an integer such thatfor all , . Substituting the above inequality into (28), for any , we havewhich yieldsfor all . Letting and using the inequality for any , we obtainThe proof is therefore complete.
Remark 6. Without migrations, we get By comparison, we find that , where is defined in [28]. In addition, without migration and regime switching, we can get in Theorem 4 is less than , where is defined in [21]. Furthermore, the condition in [31] means that the parameter needs to be satisfied: . However, we know that this condition is unnecessary from the above theorem. Despite all this, if we let the parameter satisfy , we compute that in Theorem 4 is less than in [31]. Therefore, the above work is a promotion of Theorem 4.1 in [21], Theorem 2.2 in [28], and Theorem 4.1 in [31].
5. Extinction
Sufficient conditions for extinction are the subject of this section. Unless otherwise stated, we hypothesize in this section. We first rewrite (6) as follows:where the operator is defined asthe operator is defined as , and . We first note thatfor , whence (6) admits a trivial solution corresponding to .
Before our result, we give a lemma.
Lemma 3. For system (36), the terms and are locally bounded in while uniformly bounded in . That is, for any , there is satisfyingfor all , with .
The proof is not particularly difficult, so we omit the proof.
Theorem 5. Assume that
Then, the solution of (36) satisfies , for any initial values (7). That is, all populations in system (36) go to extinction with probability one.
Proof. Step 1: let Obviously, is positive-definite and radially unbounded. That is, The generalized It formula yields By computation, we know where and . It is straightforward to see from (40) that For simplicity, we let By condition (40) again, we obtain that Applying (44) and (46), we derive Substituting the preceding equality into (43), it yields Then, the nonnegative semimartingale convergence theorem (see, e.g., [37]) implies Moreover, we obtain from (48) that Then, letting , together with the Fubini theorem, we have Let , where . Obviously, . Combining Chebyshev’s inequality and (51), we see that . By Borel–Cantelli’s lemma, one can show that , that is, Step 2: from (52), we observe One now needs to consider If the above conclusion would not hold, then . So, there is satisfying where Noting that Lyapunov function and the solution of (6) are all continuous, together with (49), it yields Define Clearly, In addition, by (42), we get So, Recalling (7), we know that the initial values satisfy for . We therefore could find an integer , depending on , sufficiently large for for almost surely, while where . By (55) and (62), one implies where is the complement of . Let From (53) and the definitions of and , we have Hence, we define for , and its differential is where Here, is the indicator function of . Recalling Lemma 3, we know for any , , and . Moreover, the definition of yields , . We also note that, for all and , In the close ball , is uniformly continuous. Therefore, there exists small sufficiently such that For , we emphasize that if for some and , then . Furthermore, let the number be small enough such that By (63) and (65), we can obtain that In particular, if , then . Hence, the definition of implies . So, for all . Then, from the Hölder inequality and the Burkholder–Davis–Gundy inequality, it follows that which together with (71) and the Chebyshev inequality, imply easily that Obviously, we observe that if . Hence, by (72) and (75), we can derive that This, together with (70), we can get that Recalling (51), (69), and (77), for all and , we compute which is a contradiction. Consequently, we infer (54). Step 3: by (46) and (54), we now derive The proof is therefore complete.
Corollary 1. Assume that are nonnegative constants, , and
Then, solution of (2) obeysfor any initial value . That is, all populations in equation (2) go to extinction with probability one.
Corollary 2. Assume that are nonnegative constants, , and
Then, solution of (4) obeysfor any initial value . That is, all populations in model (4) go to extinction with probability one.
Remark 7. This 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. Corollaries 1 and 2 give the conditions of extinction of (2) and (4), respectively. Therefore, our work is the extension of [21, 28, 31, 32].
6. Numerical Simulations
Based on [38], we show numerical simulations in the present section.
Here, we consider model (6) with the same initial data and the same on with
Then, we know the Markov chain is irreducible and has a unique stationary distribution .
In Figure 1, we give a simulation of the sample path of with .
In Figure 2, we can choose ; ; ; ; ; . It is easy to see that when . There is a good agreement between Lemma 1 and Figure 2. By Theorem 2, we know . Furthermore, we get by calculation. Therefore, conditions of Theorem 4 have been checked. So, .
In Figure 3, we can choose Let . Then, conditions of Theorem 1 could be checked. By calculation, we get . Figure 3 clearly supports this result.
In Figure 4, we can choose ; ; ; ; . By calculation, we get , , and . Therefore, conditions of Theorem 5 have been checked. Thus, from Theorem 5, all species become extinct. Figure 4 clearly supports this result.
7. 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 in [21] is too strict and unnecessary. In Theorem 1, we comprehensively analyze ultimate boundedness in the th moment and boundedness for the average in time of the th moment of solution, which is the improvement of Theorem 3.1 in [21], Theorem 2.2 in [28], Theorem 3.3 in [31], and Theorem 3.2 in [32]. In Theorem 4, we find an upper bound of the sample Lyapunov exponent. When parameters of system (6) take some special values, we compute that the upper bound is less than the corresponding upper bound in [21, 28]. Furthermore, we find that the condition in [31] is not necessary. Despite all this, if we let parameter satisfy the above conditions, we compute that is less than the upper bound in [31]. One point should be stressed is that the method for extinction in Theorem 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, 30–32]. 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
The 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.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (no. 11971279).