Research Article | Open Access
Fengying Wei, Yuhua Cai, "Global Asymptotic Stability of Stochastic Nonautonomous Lotka-Volterra Models with Infinite Delay", Abstract and Applied Analysis, vol. 2013, Article ID 351676, 8 pages, 2013. https://doi.org/10.1155/2013/351676
Global Asymptotic Stability of Stochastic Nonautonomous Lotka-Volterra Models with Infinite Delay
A kind of general stochastic nonautonomous Lotka-Volterra models with infinite delay is investigated in this paper. By constructing several suitable Lyapunov functions, the existence and uniqueness of global positive solution and global asymptotic stability are obtained. Further, the solution asymptotically follows a normal distribution by means of linearizing stochastic differential equation. Moment estimations in time average are derived to improve the approximation distribution. Finally, numerical simulations are given to illustrate our conclusions.
The impact of random factors cannot be neglected in the real world. Different kinds of random perturbations of stochastic models have been investigated in many pieces of literature. Bahar and Mao  discussed a stochastic delay Lotka-Volterra model, and they showed that environmental noise would suppress a potential population explosion and also made the solutions to be stochastically ultimately bounded. Almost at the same time, Mao  revealed that different types of environmental noise had different effects on delay population models. Meanwhile, Jiang and Shi  considered a randomized nonautonomous Logistic equation and represented the unique continuous global positive solution and positive -periodic solution.
Recently, stochastic models with delay are paid more attention by many researchers. Shen et al.  studied stochastic Lotka-Volterra competitive models with variable delay, and they obtained the unique global positive solution, stochastically ultimate boundedness, and moment average in time of the solutions. Liu and Wang  considered stochastic Lotka-Volterra models with infinite delay; they replaced the intrinsic growth rate by a random perturbation term which was dependent on the difference between the population size and the equilibrium state in their model, and then the sufficient criteria for global asymptotic stability of the solution were established. Huang et al.  extended the conclusion of Liu and Wang  to a general case. Based on a general phase space
Xu et al.  and Xu  investigated an autonomous stochastic Lotka-Volterra model with infinite delay they established the asymptotic pathwise properties of the solution to model (1), where , , , , and ; was a Banach space (see [7, 9, 10]); was probability measure defined on such that
Motivated by the works mentioned previously, we always assume that the intensity of white noise is dependent on the difference between the population size and the equilibrium state and also is dependent on time in this paper. Now, we consider a more general stochastic nonautonomous Lotka-Volterra model with infinite delay where , ( if ), and are parameter functions; noise intensities , , and are continuous bounded function on ; variable delay function and ; denotes an equilibrium state with respect to the deterministic part of model (2) in (we always assume that such exists in this paper). For simplicity, model (2) can be rewritten in the following form: Next, we will investigate the global positive solution and its asymptotic properties of model (2). Further, the approximation distribution of solutions to model (2) is explored. Our results will extend some classical deterministic results into the stochastic cases.
Let , throughout this paper unless otherwise specified, be a complete probability space with a filtration satisfying the usual conditions, and () denote the independent 1-dimensional standard Brownian motion defined on the complete probability space. We denote the nonnegative cone and positive cone by , , respectively; that is, , . If , its norm is denoted by ; if is a vector or matrix, its transpose is denoted by ; if is matrix, trace norm of matrix is denoted by ; if is negative matrix, we denote it by . Suppose that is a continuous bounded function on ; we define , , with usual assumption , where denotes the empty set.
3. The Existence and Uniqueness of Global Positive Solution
In this section, we show that model (2) has a unique global solution, and the solution will remain in with probability 1.
Theorem 1. If and hold, then there is a unique solution to model (2). Moreover, remains in with probability , where .
Proof. Clearly, the coefficients of model (2) satisfy local Lipschitz continuous but do not satisfy the linear growth condition. To show that the solution is global, a.s., is needed now, where is the explosion time.
Let be sufficiently large such that each component of initial data is lying in the interval . For each integer , define the stopping time Clearly, is increasing as . Set ; hence, a.s.. If we can prove that a.s., then a.s.. To prove this statement, let us define a -function by , where . It is easy to see that for all . Applying the Itô formula to model (2), it leads to where From the elementary inequality and Hölder's inequality, we have By the fact that , , , and , it implies that where for any . Then (6) yields that Define ; the proof is easily checked (details can be found at the appendix).
4. Global Asymptotic Stability
Theorem 2. If and hold, there exist positive numbers such that is negative definite, and then that is, is globally asymptotically stable a.s., where , , , for , and
Proof. Define a -function by . Similar to the proof of  (see Theorem 2.1), we can derive Since is negative definite, is valid along trajectories in except . The proof is complete.
5. Approximation Distribution of Solution
Proof. By the definition of equilibrium state , then . Since is stable for the deterministic part to model (2), then for .
Linearizing the th equation of model (2) by Taylor expansion at , then we have Denote and , . Since and , (13) can be simplified as where , . Then (14) implies that From the definition of 1-dimensional Brownian motion, we can derive that , . According to the conditions of Theorem 2, when , one can find that a.s.; thus, ; in other words, asymptotically follows 1-dimensional normal distribution. When tends to , the mean of solution to model (2) is ; it is just consistent with the conclusion of Theorem 2. However, the deviation between solution and mean may approach infinity; it is bad information for further analysis. Now, if the variance can be evaluated, the disadvantage will be improved.
6. Moment Estimation
First, let us prove one useful moment estimation.
Theorem 4. If and hold, then there is a positive constant , such that , where .
Proof. Define a -function by . For any given , applying the Itô formula to and taking expectation, it yields that
By fundamental inequality ,
Noting that is bounded in ; namely, . Substituting this into (16), it thus follows that
By and , we have
Hence, we derive
This implies that .
Again, by the fact that , it then follows that , and we have the assertion follows by setting . The proof is complete.
Remark A. If and are valid, then ; that is, solution to model (2) is stochastically ultimately bounded.
If we take , then the result is valid by using of Theorem 4 and Chebyshev's inequality. The proof is omitted herewith.
Theorem 5. If and hold, there exists a positive constant , and then
Proof. Rewrite (20) as with Clearly, is bounded in ; namely, . So By (19), we have Integrating both sides of (29) from 0 to ( is arbitrary) and then taking expectations, we obtain that Similar to (22), by the definition of , we can derive Substituting (31) into (30), then we have By the fact that and , the conclusion is obviously valid. The proof is complete.
7. Example and Numerical Simulation
Now, we will simulate asymptotic behaviors of solutions to model (2). Let us consider the following 1-dimensional stochastic autonomous Logistic model: with initial data , where , , , , , , , , and are all positive numbers, and the equilibrium state is . By Theorem 2, if , then the equilibrium state of model (33) is globally asymptotically stable a.s.. According to Milstein method mentioned in Higham , the initial data of model (33) is given by and , ; the difference equation is followed next where is the Gaussian random variable which follows the standard normal distribution . If we fix the parameters , , , , , and , then . When , model (33) becomes a deterministic one; then the equilibrium state is globally asymptotically stable a.s. by Theorem 2 (see Figure 1). When and , the intrinsic growth rate of model (33) is perturbed by white noise; if , the equilibrium state is globally asymptotically stable a.s. (see Figure 2). When and , that is, each parameter of model (33) is fluctuated by white noise, if and are satisfied, then the equilibrium stable is globally asymptotically stable a.s. (see Figure 3).
From the definition of , we obtain where It is easy to check that . Equation (A.1) becomes Integrating both sides of (A.3) from 0 to ( is arbitrary) and then taking expectations, it yields that Noting that for every , by the definition of stopping time , or for some , . It then follows that
Thus, . Since is arbitrary, we must have ; then as required.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the Natural Science Foundation of China (no. 11201075), the Natural Science Foundation of Fujian Province of China (no. 2010J01005), and the Technology Innovation Platform Project of Fujian Province (no. 2009J1007).
- A. Bahar and X. Mao, “Stochastic delay Lotka-Volterra model,” Journal of Mathematical Analysis and Applications, vol. 292, no. 2, pp. 364–380, 2004.
- X. Mao, “Delay population dynamics and environmental noise,” Stochastics and Dynamics, vol. 5, no. 2, pp. 149–162, 2005.
- D. Jiang and N. Shi, “A note on nonautonomous logistic equation with random perturbation,” Journal of Mathematical Analysis and Applications, vol. 303, no. 1, pp. 164–172, 2005.
- Y. Shen, G. Zhao, M. Jiang, and X. Mao, “Stochastic Lotka-Volterra competitive systems with variable delay,” in Advances in Intelligent Computing, vol. 3645 of Lecture Note in Computer Science, pp. 238–247, 2005.
- M. Liu and K. Wang, “Global asymptotic stability of a stochastic Lotka-Volterra model with infinite delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 8, pp. 3115–3123, 2012.
- Y. Huang, Q. Liu, and Y. Liu, “Global asymptotic stability of a general stochastic Lotka-Volterra system with delays,” Applied Mathematics Letters, vol. 26, no. 1, pp. 175–178, 2013.
- Y. Xu, F. Wu, and Y. Tan, “Stochastic Lotka-Volterra system with infinite delay,” Journal of Computational and Applied Mathematics, vol. 232, no. 2, pp. 472–480, 2009.
- Y. Xu, Stochastic Differential Equations with Infinite Delay and Applications, Huazhong University of Science and Technology, Wuhan, China, 2009.
- J. Hale and J. Kato, “Phase space for retarded equations with infinite delay,” Funkcialaj Ekvacioj, vol. 21, no. 1, pp. 11–41, 1978.
- F. Wei, The Basic Theory of Stochastic Functional Differential Equations with Infinite Delay, Northeast Normal University, Changchun, China, 2006.
- D. Higham, “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM Review, vol. 43, no. 3, pp. 525–546, 2001.
Copyright © 2013 Fengying Wei and Yuhua Cai. 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.