Pathwise estimation of stochastic functional Kolmogorov-type systems with infinite delay

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.

The stochastic functional Kolomogorov-type systems for n interacting species is described by the following stochastic functional differential equation:

where represents the matrix with all elements zero except those on the diagonal which are is a scalar Brownian motion, and

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 [16] consider the global positive solution for a two-dimensional Lotka-Volterra system. Kuang [19] 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

on , where is an m-dimensional Brownian motion, and

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.

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 .

Let be the family of the probability measures μ on . Let . For any , we define a set of the probability measure

Clearly, and is continuously dependent on ε on , and as . For the arbitrary , define

The following lemma shows boundedness of polynomial functions.

Lemma 2.1For any positive constantsb, α, and the function, if, as. Then

Proof We may choose such that for any , which implies . Therefore, we have

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:

on . Here,

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.

Theorem 2.1Letandbe the global positive solution of Eq. (2.1). If there exists a vectorsuch that for anyand an integer, there are constantsK, , and probability measures () such that

for anyand. Then for any given initial data, where, the solutionof Eq. (2.1) has the following property:

Proof Define

For any , applying the Itô formula to yields

where

is a continuous local martingale with the quadratic variation

For any given and , the exponential martingale inequality yields

Since , the well-known Borel-Cantelli lemma yields that there exists an with such that for any there exists an integer , when and ,

Substituting the inequalities (2.5) into (2.4), and letting t sufficiently large, it is obtained almost surely that

where

By the condition (2.2), we obtain that

We may compute that

where

for all . Therefore,

This implies

Noting the inequality

and letting , , we obtain that

as required. □

The result (2.3) shows that for any , there is a positive random time such that, with probability one, for any ,

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.

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

; (H2) = There exist , and the probability measure , where and , such that

; (H3) = There exist , , and the probability measure , where and , such that

where .; (H4) = There 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

; (H2^′) = There exist , such that

; (H3^′) = There exist , such that

where .; (H4^′) = There 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

Theorem 3.1Under the conditions (H1) and (H3), for the global positive solutionof Eq. (1.2), if one of the following conditions

holds, then

Proof To apply Theorem 2.1, we need to compute the condition (2.2). Substituting and into the condition (2.2) yields

where . By the condition (H1),

Noting , by the condition (H3),

Substituting and into (3.3) and noting that , we get

where

If the condition (3.1a) holds, choosing to replace p in (3.4), then we have

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 .

If the condition (3.1b) holds,

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.

Corollary 3.2Under the conditions (H1) and (H3^′), for the global positive solutionof Eq. (1.3), if one of the following conditions

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.

Corollary 3.3Under the conditions (H1^′) and (H3), for the global positive solutionof Eq. (1.4), if one of the following conditions

holds, then

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.

Theorem 3.4Under the conditions (H2) and (H4), for the global positive solutionof Eq. (1.2), if

hold, then

Ifand

hold, then

Proof Repeating the proof process of Theorem 3.1 obtains Eq. (3.3). Then by the conditions (H2) and (H4), we may estimate and . By the condition (H4),

Note the elementary inequality: for any and

Then for any , by the condition (H2) and the Hölder inequality,

Substituting and into (3.3) yields

where

If the condition (3.8) holds, we can choose sufficiently small ε such that . Then, by Lemma 2.1, we have

Then, for any , Theorem 2.1 gives

Letting , we get the desired assertion (3.9).

If the condition (3.10) holds, noting and , we have

Assume that (when , the computation is obvious). Choose such that

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). □

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 ,

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.

Corollary 3.5Under the conditions (H2^′) and (H4), for the global positive solutionof Eq. (1.3), if

then (3.9) holds. Ifand

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.

Corollary 3.6Under the conditions (H2) and (H4^′), for the global positive solutionof Eq. (1.4), if

then (3.9) holds. Ifand

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).

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

which is equivalent to choosing in Eq. (1.2). Clearly, the condition (H2^′) is satisfied. Applying Corollary 3.5 to Eq. (4.1) gives the following theorem.

Theorem 4.1Under the condition (H4), if, for the global positive solutionof Eq. (4.1), then the result (3.9) holds.

Then we consider the functional Lotka-Volterra-type system of Eq. (4.1)

where and for some suitable . For any . Let , Then we have

where . Here we use the notation

which was introduced by Bahar and Mao [2]. Let us emphasize that this is different from the largest eigenvalue of the matrix M. Recall the nice property of the largest eigenvalue:

Clearly, , which implies that if the matrix M is negative definite. (4.3) shows that f satisfies the condition (H4) when we choose . The condition is equivalent to

then Theorem 4.1 implies the following corollary.

Corollary 4.2If there existssuch thatsatisfies the condition (4.4), for the global positive solutionof Eq. (4.2) has the property (3.9).

If we choose μ is the Dirac measure at some point , then Eq. (4.2) may be rewritten as

which is another Lotka-Volterra type delay equation that has been investigated by Bahar et al.[2]. Applying Corollary 4.2 to Eq. (4.5) gives the following corollary.

Corollary 4.3If there existssuch that,

then the global positive solutionof Eq. (4.5) has the property (3.9).

Now we consider another Kolmogorov-type equation

where . Clearly, here g satisfies the condition (H2^′) when we choose that and . If the matrix Q satisfies the condition

then g satisfies the condition (H3^′) when we choose that . Then Corollary 3.2 gives the following theorem.

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

is satisfied, where, then (3.2) holds.

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

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.

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 [1] 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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