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Discrete Dynamics in Nature and Society
Volume 2019, Article ID 1302648, 15 pages
https://doi.org/10.1155/2019/1302648
Research Article

Dynamics Analysis of a Stochastic Delay Gilpin-Ayala Model with Markovian Switching

1Department of Mathematics, Luliang University, Lishi, Shanxi 033001, China
2School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

Correspondence should be addressed to Guirong Liu; nc.ude.uxs@1975rgl

Received 1 February 2019; Accepted 16 May 2019; Published 4 June 2019

Academic Editor: Giancarlo Consolo

Copyright © 2019 Qiaoqin Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper considers a stochastic delay Gilpin-Ayala model with Markovian switching. Using Lyapunov method, we show existence and uniqueness of global positive solution. Then, by using Chebyshev’s inequality, -matrix method, and BDG’s inequality, stochastic permanence and asymptotic estimations of solutions are studied. Finally, numerical simulations illustrate the theoretical results. Our results generalize and improve the existing results.

1. Introduction

Logistic equation is one of the most important and most widely used mathematical models to describe the growth of biological species. In [1], Gilpin and Ayala introduced the following generalized logistic equation (Gilpin-Ayala model):where stands for the population size; , and are positive constants. For more details, see [1, 2].

Population systems are often subject to environmental noise and there are different types of noise. Therefore, stochastic population systems have been studied extensively. See [39] and the references therein. In [8], by using Itô’s formula and the exponential martingale inequality, the authors studied the stationary distribution, ergodicity, and extinction of the following stochastic Gilpin-Ayala model:Further, [9] obtained the stationary distribution and ergodicity of model (2) and improved the corresponding results in [8]. In addition, there is another type of environment noise, namely, colour noise, which can be demonstrated as a switching between two or more regimes of environment. See [1018] and the references therein. Reference [11] considered the following stochastic Gilpin-Ayala model under regime switching:where there exist , and studied persistence, extinction, nonpersistence, and global attractivity of (3). Further, [12] obtained the lower-growth rate and the upper-growth rate of the positive solution of (3). Reference [13] considered the following stochastic Gilpin-Ayala population model with Markovian switching:where the Gilpin-Ayala parameter is also allowed to switch. In [13], the authors established the global stability of the trivial equilibrium state of (4). In addition, they obtained extinction, persistence, and existence of a stationary distribution of (4).

On the other hand, many population systems depend on not only present states but also past states. See [1926]. In [26], the authors considered a delay logistic model under regime switchingThey obtained stochastically ultimate boundedness, stochastic permanence, and extinction as well as asymptotic estimations of solutions.

Motivated by the works mentioned above, in this paper, we consider a more general delay stochastic Gilpin-Ayala model under Markovian switchingwith the initial conditionswhere .

Throughout this paper, let be a complete probability space with filtration , where is right continuous and contains all -null sets. The one-dimensional Brownian motions and are defined in this space.

Let be a right-continuous Markov chain on the probability space, taking values in , and with infinitesimal generator given bywhere is the transitions rate from to for and for each . We assume that the Markov chain is irreducible, and , and are independent under this condition; the Markov chain has a unique stationary (probability) distribution , which can be determined by solving the following equation:subject to and .

For convenience, for any , denote

For (6), we give the following conditions:(A1);(A2);(A3);(A4);(A5);(A6);(A7);(A8);(A9);(A10);(A11);(A12);(A13).(H1); and for every , one of conditions (A1)-(A4) holds;(H2); and for every , one of conditions (A5)-(A7) holds;(H3); and for every , one of conditions (A8)-(A13) holds;(H4); and for every , (A1) or (A4) holds;(H5), ;(H6);(H7)there exists such that for any ;(H8), where , ;(H9).

2. Uniqueness of Global Positive Solution

As in (6) denotes population size at time , it should be nonnegative. Hence, we shall consider the existence and uniqueness of global positive solution of (6).

Theorem 1. If one of conditions (H1)-(H3) holds, then, for any initial value , there exists a unique solution to (6) on and will remain in with probability 1.

Proof. Since the coefficients of (6) are locally Lipschitz continuous, then there is a unique maximal local solution on , where is the explosion time. Let satisfyingFor each integer , define the stopping timeHere, set . Obviously, is increasing as and a.s. If a.s., then , for and a.s. Thus, to prove this theorem, we need to prove a.s. LetClearly, for any . Applying Itô’s formula to , we havewhere is defined by By Young’s inequality, where If one of conditions (H1)-(H3) holds, then is bounded above; i.e., there exists such that . Substituting these into (14) yieldsFor any and , integrating both sides of (18) from to and taking the expectations yieldsIn addition,and Substituting (20) and (21) into (19) givesNote that or . From (22), for any , which implies Then, . Further, . Since is arbitrary, we must have . That is, a.s. Therefore, Theorem 1 holds.

Corollary 2. Assume that (H9) holds. Then, for any initial value , there exists a unique solution to (6) on and will remain in with probability 1.

Remark 3. If , then (6) transforms to (2). Clearly, if , then (H1) holds. Hence, Theorem 1 generalizes Lemma 2 in [9].

Remark 4. If , then (6) reduces to (3). Clearly, if , then one of conditions (H1)-(H3) holds. Hence, Theorem 1 generalizes Theorem 1 in [11].

3. Stochastic Permanence

Definition 5. Equation (6) is stochastically permanent if for any , there exist constants and such that for any solution of (6),

Lemma 6 (Lemma 3, [12]). If (H7) and (H8) hold, then there exists a constant such that the matrixis a nonsingular -matrix, where .

Lemma 7 (Theorem 2.10, [27]). If , then the following results are equivalent.(1) is nonsingular -matrix.(2) is semipositive; that is, there is in satisfying .

Lemma 8. Assume that one of the following conditions holds:(i) and (H4) holds(ii), and (H5) holds(iii) and (H6) holds For any initial value , let be the solution of (6) with the initial conditions (7). Then there exists such that

Proof. From Theorem 1, if one of conditions (H4)-(H6) holds, then (6) exists as a unique global positive solution . Now, we prove Lemma 3 if (ii) holds. For , letIt follows from (ii) that . Let satisfying . Applying Itô’s formula to yieldswhere is defined byApplying Young’s inequality and (H5), we have where From condition (ii) and , one can show thatHence, there exists such that . Thus,In addition,From (29), (34), and (35), we have which yields That is, (27) holds. If (i) or (iii) holds, the proof is similar and omitted. Hence, Lemma 8 holds.

Remark 9. If and (H6) holds, then condition (A1’) in [26] holds. Hence, Lemma 8 generalizes Theorem 3.3 in [26].

Remark 10. If , then (H6) holds. Hence, Lemma 8 improves and generalizes Lemma 6 in [11].

Theorem 11. Assume that (H7) and (H8) and one of conditions (H4)-(H6) hold. If and , then (6) is stochastically permanent.

Proof. Clearly, if one of conditions (H4)-(H6) holds, then one of conditions (H1)-(H3) holds. From Theorem 1, for any given , let be the solution of the initial value problem (6) and (7). Define Using the generalized Itô’s formula, we haveFrom Lemmas 6 and 7, there is such that . Further, for any ,Define . By the generalized Itô’s formula again,where is defined by It follows from (40) that there exists such that for any ,From the generalized Itô’s formula, we have In addition, where Further, Then there is such that . Thus, for any , Further,That is, . For any , let . From Chebyshev’s inequality,Further, . This yieldsNext we claim that for any , there exists such that . Let . From Lemma 8, there exists such that .
For any , let . Using Chebyshev’s inequality yieldswhich means thatHence,From (51), (54), and Definition 5, Theorem 11 holds.

4. Asymptotic Estimation

In this section, we study some asymptotic properties of (6).

Lemma 12 ((Theorem 2.13, [27]) (BDG’s inequality)). Let . Define, for , and . Then for every , there exist universal positive constants , which are only dependent on , such thatfor all . In particular, one may take

Theorem 13. Assume that one of conditions (H4)-(H6) holds. For any , let be the solution of the initial value problem (6) and (7).
(i) If (H4) holds, then(ii) If (H5) holds and or (H6) holds, then

Proof. (i) Assume that (H4) holds. Applying the generalized Itô’s formula to yieldswhere