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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 813535, 12 pages
Existence and Uniqueness for Stochastic Age-Dependent Population with Fractional Brownian Motion
School Mathematics and Computer Science, Ningxia University, Yinchuan 750021, China
Received 16 November 2011; Revised 7 January 2012; Accepted 10 January 2012
Academic Editor: Yun-Gang Liu
Copyright © 2012 Zhang Qimin and Li xining. 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.
A model for a class of age-dependent population dynamic system of fractional version with Hurst parameter is established. We prove the existence and uniqueness of a mild solution under some regularity and boundedness conditions on the coefficients. The proofs of our results combine techniques of fractional Brownian motion calculus. Ideas of the finite-dimensional approximation by the Galerkin method are used.
Stochastic differential equations have been found in many applications in areas such as economics, biology, finance, ecology, and other sciences [1–3]. In recent years, existence, uniqueness, stability, invariant measures, and other quantitative and qualitative properties of solutions to stochastic partial differential equations have been extensively investigated by many authors. For example, it is well known that these topics have been developed mainly by using two different methods, that is, the semigroup approach [4, 5] (e.g., Taniguchi et al.  using semigroup methods discussed existence, uniqueness, pth moment, and almost sure Lyapunov exponents of mild solutions to a class of stochastic partial functional differential equations with finite delays) and the variational one (e.g., Krylov and Rozovskii  and Pardoux ). On the other hand, although stochastic partial functional differential equations also seem very important as stochastic models of biological, chemical, physical, and economical systems, the corresponding properties of these systems have not been studied in great detail (cf. [8, 9]). As a matter of fact, there exists extensive literature on the related topics for deterministic age-dependent population dynamic system. There has been much recent interest in application of deterministic age-structures mathematical models with diffusion. For example, Cushing  investigated hierarchical age-dependent populations with intraspecific competition or predation.
There has been much recent interest in application of stochastic population dynamics. For example, Qimin and Chongzhao gave a numerical scheme and showed the convergence of the numerical approximation solution to the true solution to stochastic age-structured population system with diffusion . In papers [12, 13], Qi-Min et al. discussed the existence and uniqueness for stochastic age-dependent population equation, when diffusion coefficient and , respectively. Numerical analysis for stochastic age-dependent population equation has been studied by Zhang and Han . In papers [11–14], the random disturbances are described by stochastic integrals with respect to Wiener processes.
However, the Wiener process is not suitable to replace a noise process if long-rang dependence is modeled. It is then desirable to replace the Wiener process by fractional Brownian motion. But this process is not a semimartingale, so that it is not possible to apply the It calculus. A stochastic analysis with respect to fractional Brownian motion is faced with difficulties.
Next, the stochastic continuous time age-dependent model is derived. In , the nonlinear age-dependent population dynamic with diffusion can be written in the following form: where , , , , denotes the population density of age at time in spatial position, , denotes the fertility rate of females of age at time , in spatial position , denotes the mortality rate of age at time , in spatial position , denotes the Laplace operator with respect to the space variable, and is the diffusion coefficient. denotes effects of external environment for population system, such as emigration and earthquake have. The effects of external environment the deterministic and random parts which depend on , , and . is a standard Wiener process.
In this paper, suppose that is stochastically perturbed, with where is fractional Brownian motions with the Hurst constant . Then this environmentally perturbed system may be described by the equation new stochastic differential equations (1.3)-(1.7) for an age-dependent population are derived. It is an extension of (1.1).
Our work differs from these references [11–14]. In papers [11–14], the random disturbances are described by stochastic integrals with respect to Wiener processes. In this paper, we study a stochastic age-dependent population dynamic system with an additive noise in the form of a stochastic integral with respect to a Hilbert space-valued fractional Borwnian motion. It is well known that a fractional Brownian motion is a semimartingale if and only if , that is, in the case of a classical Brownian motion. For , Qimin and Chongzhao discussed the existence and uniqueness for stochastic age-dependent population equation . In this paper, we shall discuss the existence and uniqueness for a stochastic age-dependent population equation with fractional Brownian motions with . The discussion uses ideas of the finite-dimensional approximation by the Galerkin method.
In Section 2, we begin with some preliminary results which are essential for our analysis and introduce the definition of a solution with respect to stochastic age-dependent populations. In Section 3, we shall prove existence and uniqueness of solution for stochastic age-dependent population equation (1.3).
Consider stochastic age-structured population system with diffusion (1.3). is the maximal age of the population species, so By (1.7), integrating on to (1.3) and (1.5) with respect to , we obtain the following system where
where is the total population, and the birth process is described by the nonlocal boundary conditions clearly, denotes the fertility rate of total population at time and in spatial position . where denotes the mortality rate at time and in spatial position Let Then the dual space of . We denote by and the norms in and respectively, by the duality product between , , and by the scalar product in .
We consider stochastic age-structured population system with diffusion of the form where is the differential of relative to , that is, , . .
The integral version of (2.7) is given by the equation here , on .
Let () be independent centered Gaussian processes with on a given probability space , where we assume that and .
The processes are independent fractional Brownian motions with the Hurst constant and .
It follows from Kleptsyna et al. (cf. ) that where ( = 1, 2, …) are real independent Wiener processes with .
Let be a separable Hilbert space with the scalar product , and denotes a complete orthogonal system in , Then and is called a -valued fractional Brownian motion where the sum is defined mean square.
Definition 2.1. A -valued continuous stochastic process with (-a.s) is a solution of (2.7) if it holds for and all that
The objective in this paper is that we hopefully find a unique process such that (2.7) holds For this objective, we assume that the following conditions are satisfied:
(1), and are nonnegative measurable, and(2)Let and be measurable functions which are defined on with where is a positive constant.
3. Existence and Uniqueness of Solutions
Consider also the -valued fractional Brownian motion . Obviously, the following lemma holds.
Lemma 3.1. Problem (3.1) has, for fixed , a unique solution , and there exists a nonnegative random variable with finite expectation such that where for fixed , is continuous with respect to in .
Proof. The Galerkin approximations are defined by , where solves the stochastic equations
It follows from the assumption (2) that (3.3) can be solved for every by the method of successive approximation, and the iterates are measurable with respect to . Consequently, are stochastic processes since is a random -valued variable and is a stochastic process. It follows from (3.3) that
Using the chain rule, we get the following
If we set in of Qimin and Chongzhao , under assumptions (1)-(2), then this result implies that
The following result is an analogous to that of Theorem 4 in . In the Galerkin approximation, we have
for all and for . is a -valued continuous process with for all -., and is a -. unique solution.
Now let be a -valued fractional Brownian motion with and . We consider the finite-dimensional approximation
in mean square of the stochastic integral . Obviously this is a stochastic integral with respect to the -valued Brownain motion . Consequently, the corresponding Galerkin equations for (2.7) are given by Lemma 3.1 shows that these problems have solutions.
Theorem 3.2. If and , then there exists a - unique solution of (2.7) with where is a positive constant.
Proof. We choose with and define Then However, by Lemma 2.2  and assumptions -, we have Further, Consequently, in view of (3.13), Then, the Gronwall’s lemma implies that for for all . In particular, there exists a process with for , and consequently, there exists a process with for . We must now show that is solution of (2.7). We have Let be chosen arbitrary. Then there exists so that for all . Let and be the th Galerkin approximation of and , respectively. For , we have Consequently, the first term on the right-hand side of (3.10) is also less than const.. It is clear that the second term and third term on the right-hang side of (3.18) tends to zero. Then (3.18) gives for , there is is also a Cauchy sequence in for all . Let be the limit a of this sequence. Then it follows from the properties of a Gelfand triple that for , where is a positive constant. Consequently, (a.s) and it follows from (3.9) that hence, we have proved Theorem 3.2.
The authors would like to thank the referees for their very helpful comments which greatly improved this paper. The research was supported by The National Natural Science Foundation (no. 11061024) (China).
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