Pathwise estimation of stochastic functional Kolmogorov-type systems with infinite delay
© Zhu and Xu; licensee Springer 2012
Received: 28 October 2011
Accepted: 20 July 2012
Published: 1 August 2012
In this paper, we study pathwise estimation of the global positive solutions for the stochastic functional Kolmogorov-type systems with infinite delay. Under some conditions, the growth rate of the solutions for such systems with general noise structures is less than a polynomial rate in the almost sure sense. To illustrate the applications of our theory more clearly, this paper also discusses various stochastic Lotka-Volterra-type systems as special cases.
MSC:34K50, 60H10, 92D25, 93E03.
There is an extensive literature concerned with the dynamics of this system and we here only mention [7, 9, 10]. References [7, 9, 10] study existence and uniqueness of the global positive solution of Eq. (1.1), and its asymptotic bound properties and moment average in time. The nice positive property provides us with a great opportunity to discuss further how the solutions vary in in more detail. Our interest is to discuss pathwise estimation of the global positive solutions for stochastic functional Kolomogorov-type systems with infinite delay.
There is an extensive literature concerned with the dynamics of the Kolomogorov-type systems (1.1) without the stochastic perturbation and we here only mention [3–5, 21]. As a special case, multispecies Lotka-Volterra-type systems with bounded delay and unbounded delay are studied. For example, He and Gopalsamy  consider the global positive solution for a two-dimensional Lotka-Volterra system. Kuang  examines global stability for infinite delay Lotka-Volterra-type systems. For more details about the Lotka-Volterra-type systems, we refer the reader to see [6, 11–13, 15, 20] and references therein. In fact, population systems are often subject to environmental noise. It is therefore useful to reveal how the noise affects on the Kolmogorov-type systems. Recently, the stochastic Lotka-Volterra systems have received increasing attention. References [1, 17] reveal that the noise plays an important role to suppress the growth of the solution. References [2, 18] show the stochastic system behaves similarly to the corresponding deterministic system under different stochastic perturbations, respectively. These indicate clearly that different structures of environmental noise may have different effects on Lotka-Volterra systems.
Since this paper mainly examines the pathwise estimation of the solution for stochastic functional Kolmogorov-type systems with infinite delay, we assume that there exists a unique global positive solution for all discussed equations (see [8, 14]).
In the next section, we give some necessary notations and lemmas. To show our idea clearly, Section 2 also studies the pathwise estimation for general stochastic functional differential equations with infinite delay. Applying the result of Section 2, we give various conditions under which stochastic functional Kolmogorov systems with infinite delay show the nice pathwise properties in Section 3. As in the applications of Section 2 and Section 3, Section 4 discusses several special equations, including various stochastic Lotka-Volterra systems with infinite delay.
2 A general result
Throughout this paper, unless otherwise specified, we use the following notations. Let denote the Euclidean norm in . If A is a vector or matrix, its transpose is denoted by . If A is matrix, its trace norm is denoted by . Let , , . For any , let and . Let be a complete probability space with a filtration satisfying the usual conditions, that is, it is right continuous and increasing while contains all P-null sets. Let be an m-dimensional Brownian motion defined on the complete probability space. If is an -valued stochastic process on , we let for .
The following lemma shows boundedness of polynomial functions.
as required. □
are local Lipschitz continuous and is an m-dimensional Brownian motion. Assume that Eq. (2.1) almost surely admits a unique global positive solution. Then we have the following pathwise estimation.
as required. □
In other words, with probability one, the solution will not grow faster than .
In this theorem, it is a key to compute the condition (2.2). In the following section, we apply Theorem 2.1 to examine stochastic functional Kolmogorov-type systems.
3 Main result
This section mainly applies the result of Theorem 2.1 to Eq. (1.2). For any and , we firstly list the following conditions for both f and g that we will need. (H1) = These exist , and the probability measure , where and , such that
where .. When and replace and , the above conditions may be rewritten as (H1′) = These exist , such that
where .. Here, we emphasize that the same letter represents the same parameter when we use the several conditions simultaneously. In the following sections, we always assume that the initial data . Applying the conditions (H1) and (H3) to stochastic functional Kolmogorov systems (1.2) gives
Noting and as , choose ε sufficiently small such that . Lemma 2.1 gives , which implies the condition (2.2) is satisfied. Letting , Theorem 2.1 gives the desired result under .
By the condition (3.1b), choose ε sufficiently small such that , so we have , which implies the condition (2.2) and further gives the desired result by Theorem 2.1. □
Applying Theorem 3.1 to Eq. (1.3), the condition (H3) should be replaced by (H3′), which implies . This result may be described as follows.
is satisfied, then (3.2) holds.
Applying Theorem 3.1 to Eq. (1.4), the condition (H1) should be replaced by (H1′), which is equivalent to choose in (H1). Then we may obtain the following corollary.
In these results above, the condition (H3) or (H3′) plays an important role to suppress growth of the solution. To use the condition (H3) or (H3′), it is necessary to require . To avoid the condition (H3) or (H3′), we may impose another condition on the function f to suppress growth of the solution. This idea leads to the following theorem.
Letting , we get the desired assertion (3.9).
Noting that the condition (3.10) holds, choose ε sufficiently small and u sufficiently near 1 such that . So . Applying Theorem 2.1 therefore gives the desired assertion (3.11). □
In other words, with probability one, the solution will not grow fast than .
Applying Theorem 3.4 to Eq. (1.3), the condition (H2) should be replaced by (H2′), which implies that in (H2). We may obtain the following corollary.
then (3.11) holds.
Applying Theorem 3.4 to Eq. (1.4), the condition (H4) should be replaced by (H4′), which implies that in (H4). The corollary follows.
then (3.11) holds.
Comparing Theorem 3.1 with Theorem 3.4, we conclude that in Theorem 3.1, the function g plays an important role to suppress growth of the solution in condition (H3). While in Theorem 3.4, the function f is the main factor to suppress growth of the solution in condition (H4).
4 Some special case
Theorem 4.1 Under the condition (H 4), if, for the global positive solutionof Eq. (4.1), then the result (3.9) holds.
then Theorem 4.1 implies the following corollary.
Corollary 4.2 If there existssuch thatsatisfies the condition (4.4), for the global positive solutionof Eq. (4.2) has the property (3.9).
then the global positive solutionof Eq. (4.5) has the property (3.9).
then g satisfies the condition (H3′) when we choose that . Then Corollary 3.2 gives the following theorem.
is satisfied, where, then (3.2) holds.
where , which is another stochastic functional Lotka-Volterra systems. Clearly, f satisfies the condition (H1) when we choose that , , . So, for , we have the following theorem.
which is discussed by  where Theorem 5.3 is obtained. This means that our result is more general.
This work was supported in part by the National Natural Science Foundation of China under Grant nos. 11101054, 11101434, the Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities under Grant no. 11FEFM11, the Scientific Research Funds of Hunan Provincial Science and Technology Department of China under Grant no. 2012SK3096, and Hunan Provincial Natural Science Foundation of China under Grant no. 12jj4005.
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