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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 537674, 10 pages

http://dx.doi.org/10.1155/2014/537674
Research Article

Some New Results on the Lotka-Volterra System with Variable Delay

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

Received 27 January 2014; Accepted 22 July 2014; Published 12 August 2014

Academic Editor: Zhichun Yang

Copyright © 2014 Yangzi Hu 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 discusses the stochastic Lotka-Volterra system with time-varying delay. The nonexplosion, the boundedness, and the polynomial pathwise growth of the solution are determined once and for all by the same criterion. Moreover, this criterion is constructed by the parameters of the system itself, without any uncertain one. A two-dimensional stochastic delay Lotka-Volterra model is taken as an example to illustrate the effectiveness of our result.

1. Introduction

Population systems are often subject to environment noise. In our previous papers [1, 2], we considered the following stochastic Lotka-Volterra system: and its functional form, where with representing variable delay and represents the matrix with all elements zero except those on the diagonal which are . and matrices , , , and .

Equation (1) may describe dynamics of species interaction, in which    represents the population size of th species depending both on the current states and on the past state of all population. From the point of biological view, the following three properties are very important.(A)The solution of system (1) is positive and nonexplosive; namely, for any positive initial data , (1) has a unique positive global solution .(B)The solution of system (1) is ultimately moment bounded and time average moment bounded; that is, this global solution of (1) satisfies where and are positive constants independent of . These two properties show that, in the sense of average, population size is bounded.(C)The solution of the system (2) grows at most polynomially; namely, this solution of (1) satisfies There is an extensive literature concerned with these properties of stochastic Lotka-Volterra models. For example, Mao and his coauthors [35] discussed the existence and uniqueness of the global positive solution, stochastically ultimate boundedness, and some other asymptotic properties for the stochastic Lotka-Volterra system. References [6, 7] discovered that the presence of the environmental noise may suppress the potential explosion of the solution in finite time. In our previous work [2], we showed that the environmental noise structure determined whether properties (A)–(C) were affected by the stochastic perturbation parameters or not. In our previous work [1], these three properties were also examined. In this paper, our conclusions will be improved in the following aspects.(i)In these published works, properties (A)–(C) were given under different conditions, respectively. In this paper, we will give these three properties under the same group of conditions. This is an important improvement since properties (B) and (C) do not imply each other in general.(ii)In this paper, we will present the conditions, which are easier to be verified, to guarantee properties (A)–(C). In these conditions, all parameters are from the models and do not include any uncertain parameters to be determined.

The rest of the paper is arranged as follows. In the next section, we provide some necessary notations and lemmas. Section 3 gives several lemmas to support the main results of this paper. By using Lemmas established in Section 3, Section 4 presents the conditions under which the all desired properties (A)–(C) hold. In Section 5, some simplified cases of model (1) are investigated. Although these models are less general than (1), they have wide applications and satisfy properties (A)–(C) under more simple conditions, which are provided as corollaries of the main theorems. A two-dimensional stochastic Lotka-Volterra population model will be examined as an example in Section 6.

2. Preliminaries

Throughout this paper, unless otherwise specified, we use the following notations. Let be a complete probability space with a filtration satisfying the usual conditions; that is, it is right continuous and increasing while contains all -null sets.    is a one-dimensional Brownian motion defined on .

For any given and -valued function , we always assume that For matrices , , , and in model (1), we assume that , , , and   . Assume that for ; for . Let , , and . Denote by the Euclidean norm with and is the trace norm of matrix .

Definition 1. Let satisfy condition If all eigenvalues of have positive real parts, is called an -matrix.

Lemma 2. Suppose that the matrix satisfies condition (6). Then the following conditions are equivalent (see [8]):(i) is an -matrix;(ii)there exists such that ;(iii)all of the leading principal minors of are positive.

For any given symmetric matrix , define which deduces directly that Let be the variable delay of system (1). Write with and . Then implies that and is strictly monotone increasing on . Its inverse function is defined on , which satisfies Assume that , , and . is a Banach space with the supremum norm. For any given initial data , always represents the solution of (2). When for all in the domain, we call it a positive solution; when is defined on , it is called a global solution.

Denote that Unless otherwise stated, we assume that . For any given , define If is a positive solution of (1), by the Itô formula and (12), we have that where with .

Let and    be positive constants. Define Substituting (14) into (12), together with notations in (11), yields that Particularly, when and , we have

For the sake of simplicity, let represent the following function defined on : where , and    are nonnegative constants, and is defined in (9). The following lemma plays a key role in this paper (also see [1, 9, 10]).

Lemma 3. Let be given by (18). Suppose that    is a positive solution of (1) with ; then

In this paper, always denotes a positive constant with different values at different places and exact values of these constants are insignificant.

In this paper, we often use the following inequalities:

3. Main Lemmas

In order to get the desired properties (A)–(C), we need the following three lemmas. Let us first explain that the notation : means that with for .

Lemma 4. Suppose that there exist positive constants , , , , and   , such that satisfies condition where is defined by (18). Then (1) is positive and nonexplosive; namely, for any given , (1) has a unique positive solution .

Lemma 5. Suppose that there exist positive constants , , and   , such that satisfies condition where is given by (18). Then any positive global solution    of (1) satisfies (2)-(3).

The proofs of the above two lemmas are omitted since two similar approaches can be found in [1].

Lemma 6. Suppose that there exist positive constants , , , and   , such that the following condition is satisfied: where , and are defined by (11), and is defined by (18). Then any positive global solution    of (1) satisfies

Proof. Let   . Then, By (12) and (26), we have Let ; then , where For any given and , by the exponential martingale inequality, we have that Since , we can employ the Borel-Cantelli lemma to derive that, almost surely, when is sufficiently large and , one can get that Note that . This, together with (31), (33), and (26), gives that in the sense of almost sure, when is sufficiently large, where we have used Lemma 3. This implies that in the sense of almost sure when is sufficiently large. Therefore, Obviously, is a monotony decrease function of , so can be replaced by any in condition (26). Hence we may assume that is sufficiently small. Letting and , we get that Note that for . Then (27) follows from (37).

4. The Main Results

In this section, let us apply Lemmas 46 to establish the main results of this paper. We use the denotations and .

Theorem 7. Suppose that there exist nonnegative constants , , , and , such that the following conditions are satisfied: Then for any given , (1) has a unique positive solution and this solution satisfies (2)–(4).

Proof. Let us divide this proof into the following three steps.

Step 1.  Let   . Let us test condition (25). By (8) and condition (38), for any given we have that so By (16) and (42), we get where is a function in the form of (18) with sufficiently small and By condition (40), Since is sufficiently small, we may assume that . Obviously,   , so (45) implies (25).

Now, we can apply Lemma 5 to obtain that any global positive solution    of (1) satisfies (2)-(3).

Step 2. Let   . In this step, we will test condition (24). For any given , using condition (39) yields which implies By (17), (44), and (50), where is a function in the form of (18): Condition (40) implies that . Since can be sufficiently small, we can get . So (51) can imply condition (24) (choose ). Now we can employ Lemma 4 to obtain that, for any given , (1) has a unique positive global solution .

Step 3. Choose . By (26) we have . Now we test condition (26). Note that , so by (43) we have where is a function in the form of (18), By condition (40) we have , so we may assume that . Then (54) shows that condition (26) is satisfied (choose ).

Applying Lemma 6 yields that any positive solution    of (1) satisfies (4). This completes the proof.

Theorem 8. Suppose that there exist nonnegative constants and , such that condition (38) and the following condition are satisfied: Assume that , where , , , and . Then the conclusion of Theorem 7 holds.

Proof.

Step 1. By Lemma 2, condition (60) can imply that is an -matrix. Thus, there exists such that . Let   , , where is sufficiently small. Now we test condition (24). By (15) we have that where is a function in the form of (18). Choose sufficiently large; then where we have used inequalities (20)–(22), , ; consider is a function in the form of (18): when , , and , The last inequality is based on the condition . Thus we may assume that and are sufficiently small, while is sufficiently large; then   . Substituting (63) into (61) yields that where is a function in the form of (18). Clearly, (67) shows that condition (24) is satisfied.

Now, we can use Lemma 4 to obtain that, for any given , (1) has a unique global positive solution .

Step 2. Let . In this step we test condition (25); for that, we only need to show that conditions (38) and (59) hold. The method is similar to the proof of Theorem 7, Step 1.

Step 3. Taking any , now we test condition (26). We can replace by : Obviously, Letting be sufficiently large, then inequality (21) gives that By condition (58) we have thus , which implies Condition (57) derives that hence, So Combining (69)–(76) yields where is a function in the form of (18), When and , By condition (59), we have ; therefore, . Since we may assume that is sufficiently small and is sufficiently large, there must be . Thus, condition (77) deduces that condition (26) is satisfied.

Now, we can apply Lemma 6 to obtain that any positive solution    of (1) satisfies (27). And then we can get that satisfies (4) by letting . This completes the proof.

Remark 9. Observing and comparing the conditions of Theorems 7 and 8, the condition they have in common is (38), which only involves parameters from the drift coefficient . Condition (39) in Theorem 7 corresponds to conditions (57), (58), and (60) in Theorem 8 which depend on stochastic disturbances of system (1). Both of them can guarantee the existence and uniqueness of the solution. But it seems that the three conditions of Theorem 8 are more precise than condition (39). Hence, we may expect that Theorem 8 can give more accurate results. However, it needs condition , which is not requested in Theorem 7. So Theorems 7 and 4.2 have their own strengths and weaknesses.

Remark 10. Theorems 7 and 8 give two classes of conditions under which the desired properties (A)–(C) hold. This is an improvement for our previous results ([1, 2]), since we only established these three results in different conditions, respectively. Moreover, conditions of the two theorems are directly dependent on the parameters of system, except and . This implies that these conditions are easier to be verified.

5. Some Corollaries

In (1), letting , , and , one can get the following “defective” LV systems: where (83) is equivalent to taking in (1). For (81)–(83), we can simplify the conditions of Theorems 7-8 and then obtain corollaries as follows.

Corollary 11. Suppose that there exist nonnegative constants , , and , such that condition (38) and the following conditions are satisfied: Then for any given , (81) has a unique global positive solution , which satisfies (2)–(4).

Taking in Theorem 7, (84) deduces (39)-(40) directly. The following corollary can be found in [3, 4].

Corollary 12. Let ,   . Then for (81), the conclusion of Corollary 11 holds.

This corollary can be deduced from Theorem 8. First, let such that condition (57) is satisfied. Second, when is sufficiently small, conditions (58)-(59) are satisfied.

Clearly, Theorem 8 cannot be applied on system (82), but employing Theorem 7 we have the following.

Corollary 13. Suppose that there exist nonnegative constants and , such that (38) and the following condition are satisfied: Then for any given , (82) has a unique global positive solution , which satisfies (2)–(4).

Note that when , we should take such that condition (39) is satisfied.

Applying Theorem 7 on (83) yields the following.

Corollary 14. Suppose that there exist nonnegative constants , , and such that conditions (84) and the following condition are satisfied: Then for any given , (83) has a unique global positive solution , which satisfies (2)–(4).

6. Examples

Consider the following 2-dimensional LV system: where and are nonnegative constants,   , and . Let By (88), we can compute Then, by (38) and (39) we have

Apply Theorem 7. For any given , we have It can be seen that, taking and , we have , and then . Next, Clearly, when and ,