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.
Keywords:stochastic functional Kolmogorov systems; Lotka-Volterra systems; pathwise estimation; infinite delay
The stochastic functional Kolomogorov-type systems for n interacting species is described by the following stochastic functional differential equation:
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.
However, little is yet known about the pathwise property of the stochastic functional Kolmogorov-type systems with infinite delay although they may be seen as a generalized stochastic functional Lotka-Volterra system. This paper will examine the pathwise estimation of the stochastic functional Kolmogorov-type systems under the general noise structures. Consider the n-dimensional unbounded delay stochastic functional Kolmogorov-type systems
where the initial data space denotes the family of bounded continuous -value functions φ defined on with the norm . Here, f and g are local Lipschitz continuous. Clearly, Eq. (1.2) includes the following forms:
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. □
To show our idea about the pathwise estimation of solutions clearly, we first consider the following general n-dimensional stochastic functional differential equation:
is a continuous local martingale with the quadratic variation
Substituting the inequalities (2.5) into (2.4), and letting t sufficiently large, it is obtained almost surely that
By the condition (2.2), we obtain that
We may compute that
Noting the inequality
as required. □
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
, and the probability measure , where and , such that
, , and the probability measure , where and , such that
, such that
, 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
If the condition (3.1b) holds,
is satisfied, then (3.2) holds.
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.
The result (3.9) is a very strong result. It shows that growth of the solution is so slow that it is near to be bound. By this result, we know that for any , there is a positive random time such that, with probability one, for any ,
then (3.11) holds.
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
In this section, some special Kolmogorov-type equations are considered by applying the results in the previous two sections to obtain some pathwise estimation, where some special noises are chosen. We firstly consider the n-dimensional stochastic functional equations
Then we consider the functional Lotka-Volterra-type system of Eq. (4.1)
which was introduced by Bahar and Mao . Let us emphasize that this is different from the largest eigenvalue of the matrix M. Recall the nice property of the largest eigenvalue:
then Theorem 4.1 implies the following corollary.
which is another Lotka-Volterra type delay equation that has been investigated by Bahar et al.. Applying Corollary 4.2 to Eq. (4.5) gives the following corollary.
Now we consider another Kolmogorov-type equation
Theorem 4.4Under the conditions (4.8) and (H1), for the global positive solutions of Eq. (4.7), if one of the following two conditions
Roughly speaking, Theorem 4.4 shows that under the condition (4.8), if the growth orderα of f is smaller than 2 (or under some conditions), (3.2) is satisfied for all . To satisfy (3.2) under , the condition (4.8) plays a crucial role. But here f is only required to satisfy an unrestrictive condition, which includes the linear growth condition. If f is the linear growth function, namely,
then Eq. (4.7) may be rewritten as
Theorem 4.5Under the condition (4.8), the global positive solution of Eq. (4.10) has the property
We further choose that μ is a Dirac measure at the point τ, then Eq. (4.10) is written as the delay stochastic Lotka-Volterra systems
which is discussed by  where Theorem 5.3 is obtained. This means that our result is more general.
The authors declare that they have no competing interests.
The authors contributed equally and significantly in writing this article. The authors read and approved the final manuscript.
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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