Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2012, Article ID 813535, 12 pages
doi:10.1155/2012/813535
Research Article
Existence and Uniqueness for Stochastic
Age-Dependent Population with Fractional
Brownian Motion
Zhang Qimin and Li xining
School Mathematics and Computer Science, Ningxia University, Yinchuan 750021, China
Correspondence should be addressed to Zhang Qimin, zhangqimin64@sina.com
Received 16 November 2011; Revised 7 January 2012; Accepted 10 January 2012
Academic Editor: Yun-Gang Liu
Copyright q 2012 Z. Qimin and L. 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 h ∈ 1/2, 1 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.
1. Introduction
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. 4 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
6 and Pardoux 7. 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
2
Mathematical Problems in Engineering
diffusion. For example, Cushing 10 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 11. In papers 12, 13, Qi-Min et al. discussed the
existence and uniqueness for stochastic age-dependent population equation, when diffusion
coefficient k 0 and k /
0, respectively. Numerical analysis for stochastic age-dependent
population equation has been studied by Zhang and Han 14. 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
o 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 12, the
nonlinear age-dependent population dynamic with diffusion can be written in the following
form:
∂P r, t, x ∂P r, t, x
− k1 r, tΔP r, t, x
∂t
∂r
dwt
, in QA 0, A × Q,
−μ1 r, t, xP r, t, x f1 r, t, x g1 r, t, x
dt
A
P 0, t, x β1 r, t, xP r, t, xdr, in 0, T × Γ,
0
P r, 0, x P0 r, x,
1.1
in 0, A × Γ,
on ΣA 0, A × 0, T × ∂Γ,
A
yt, x P r, t, xdr, in Q,
P r, t, x 0,
0
where t ∈ 0, T , r ∈ 0, A, x ∈ Γ ⊂ RN 1 ≤ N ≤ 3, Q 0, T × Γ, P r, t, x denotes the
population density of age r at time t in spatial position, x, β1 r, t, x denotes the fertility rate
of females of age r at time t, in spatial position x, μ1 r, t, x denotes the mortality rate of age
r at time t, in spatial position x, Δ denotes the Laplace operator with respect to the space
variable, and k1 r, t > 0 is the diffusion coefficient. f1 r, t, x g1 r, t, xdwt/dt 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 r, t, and x. wt is a standard Wiener process.
In this paper, suppose that f1 r, t, x is stochastically perturbed, with
f1 r, t, x −→ f1 r, t, x g1 r, t, xdBh t,
1.2
Mathematical Problems in Engineering
3
where Bh t is fractional Brownian motions with the Hurst constant h. Then this
environmentally perturbed system may be described by the It
o equation
∂P r, t, x ∂P r, t, x
− k1 r, tΔP r, t, x
∂t
∂r
−μ1 r, t, xP r, t, x f1 r, t, x g1 r, t, xdBh t, in QA 0, A × Q,
A
β1 r, t, xP r, t, xdr, in 0, T × Γ,
P 0, t, x 1.3
1.4
0
P r, 0, x P0 r, x,
in 0, A × Γ,
on ΣA 0, A × 0, T × ∂Γ,
A
P r, t, xdr, in Q,
yt, x P r, t, x 0,
1.5
1.6
1.7
0
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 Bh is a semimartingale if
and only if h 1/2, that is, in the case of a classical Brownian motion. For h 1/2, Qimin and
Chongzhao discussed the existence and uniqueness for stochastic age-dependent population
equation 12. In this paper, we shall discuss the existence and uniqueness for a stochastic
age-dependent population equation with fractional Brownian motions with h ∈ 1/2, 1. 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.
2. Preliminaries
Consider stochastic age-structured population system with diffusion 1.3. A is the maximal
age of the population species, so
P r, t, x 0,
∀r ≥ A.
2.1
4
Mathematical Problems in Engineering
By 1.7, integrating on 0, A to 1.3 and 1.5 with respect to r, we obtain the following
system
∂y
− ktΔy μt, xy − βt, xy
∂t
dBh t
ft, x gt, x
, in Q 0, T × Γ,
dt
y0, x y0 x, in Γ,
yt, x 0,
2.2
on Σ 0, T × ∂Γ,
where
βt, x ≡
A
β1 r, t, xP r, t, xdr
0
A
−1
P r, t, xdr
,
2.3
0
A
P r, t, xdr yt, x is the total population, and the birth process is described by the
A
nonlocal boundary conditions 0 β1 r, t, xP r, t, xdr clearly, βt, x denotes the fertility rate
of total population at time t and in spatial position x.
where
0
μt, x ≡
A
μ1 r, t, xP r, t, xdr
0
A
−1
P r, t, xdr
,
2.4
0
where μt, x denotes the mortality rate at time t and in spatial position x
ft, x ≡
gt, x ≡
A
0
A
f1 r, t, xdr,
2.5
g1 r, t, xdr.
0
Let
∂ϕ
∂ϕ
2
2
∈ L Γ, where
are generalized partial derivatives .
V H Γ ≡ ϕ | ϕ ∈ L Γ,
∂xi
∂xi
2.6
1
Then V H −1 Γ the dual space of V . We denote by | · | and · the norms in V and V respectively, by ·, · the duality product between V , V , and by ·, · the scalar product in H.
We consider stochastic age-structured population system with diffusion of the form
dt yt − kΔytdt μt, xytdt − βt, xytdt
ft, xdt gt, xdBh t, in Q 0, T × Γ,
y0, x y0 x,
yt, x 0,
in Γ,
on Σ 0, T × ∂Γ,
2.7
Mathematical Problems in Engineering
5
where dt yt is the differential of yt, x relative to t, that is, dt yt ∂yt/∂tdt, yt :
yt, x. T > 0, A > 0.
The integral version of 2.7 is given by the equation
yt − y0 −
t
kΔysds −
t
0
βs, x − μs, x ysds t
0
fs, xds t
0
gs, xdBh s,
0
2.8
here yt, x 0, on Σ 0, T × ∂Γ.
Let Bjh tt≥0 j 1, 2, . . . be independent centered Gaussian processes with Bjh 0 0
on a given probability space Ω, F, P , where we assume that
2
E Bjh t − Bjh s |t − s|2h μj
j 1, 2, . . . ,
2.9
∞
μj < ∞,
μj > 0,
j1
and h ∈ 1/2, 1.
The processes Bjh tt≥0 are independent fractional Brownian motions with the Hurst
2
constant h and EBjh 1 μj j 1, 2, . . ..
It follows from Kleptsyna et al. cf. 15 that
Bjh t
0
−∞
|t − r|
h−1/2
− |r|
h−1/2
dWj r t
|t − r|
h−1/2
2.10
dWj r ,
0
where Wj tt≥0 j 1, 2, . . . are real independent Wiener processes with EWj2 t μj t.
Let K be a separable Hilbert space with the scalar product ·, ·K , and ej j1,2,... denotes
a complete orthogonal system in K, Then
∞
2
E
Bjh tej t2h
K
j1
∞
μj < ∞,
2.11
j1
h
and Bh t ∞
j1 Bj tej is called a K-valued fractional Brownian motion where the sum is
defined mean square.
Definition 2.1. A H-valued continuous stochastic process ytt∈0,T with yt ∈ V P -a.s is
a solution of 2.7 if it holds for v ∈ V and all t ∈ 0, T that
yt, v
H
y0, v H t
t
0
fs, x, v
0
kΔys, v ds ds H
t
βs, xys − μs, xys, v
t
0
gs, xdBh s, v
0
H
,
P -a.s.
H
ds
2.12
6
Mathematical Problems in Engineering
The objective in this paper is that we hopefully find a unique process yt such that
2.7 holds For this objective, we assume that the following conditions are satisfied:
1 μt, x, βt, x and kr, t are nonnegative measurable, and
0 ≤ k0 ≤ kt < ∞
in 0, A × 0, T ,
0 ≤ μ0 ≤ μt, x < ∞
in 0, A × Γ,
0 ≤ βt, x ≤ β0 < ∞
in 0, A × Γ.
2.13
2 Let ft, x and gt, x be measurable functions which are defined on Q with
ft, x gt, x ≤ K,
2.14
where K is a positive constant.
3. Existence and Uniqueness of Solutions
Consider also the K-valued fractional Brownian motion Bh,n t ni1 Bih tei . Obviously, the
following lemma holds.
If the process ytt∈0,T is a solution of 2.7, then the process Zt yt −
t
gsdBh s solves
0
dt Zt − kΔZtdt μt, x Zt ft, xdt kΔ
t
t
gsBh s ds − βt, x Zt 0
gsdBh sdt,
t
gsdBh s dt
0
in Q 0, T × Γ,
0
Z0, x Z0 x,
Zt, x 0,
in Γ,
on Σ 0, T × ∂Γ,
3.1
where Zt : Zt, x. If Zt is a solution of 3.1, then exists a process ytt∈0,T so that Zt
t
can be written as Zt yt − 0 gs, xdBh s, and consequently yt solves 2.7.
As a result, we shall consider 3.1 instead of 2.7. It is noted that, for fixed ω ∈ Ω,
3.1 is a deterministic problem.
Lemma 3.1. Problem 3.1 has, for fixed ω ∈ Ω, a unique solution Zt, and there exists a
nonnegative random variable η with finite expectation such that
2
sup |Zs| k0
0≤s≤T
T
Zs
2 ds ≤ η,
0
where for fixed ω ∈ Ω, Zt is continuous with respect to t in H.
3.2
Mathematical Problems in Engineering
7
Proof. The Galerkin approximations are defined by Zn t the stochastic equations
Zn,i t y0, vi
H
t
kΔ
0
t
n
i1
Zn,i tvi , where Zn,i t solves
Zn,k svk , vi ds
k1
n
Zn,k svk , vi ds
βs, x − μs, x
0
n
t
k1
fs, x, vi
H
0
t
ds t
βs, x − μs, x
0
s
gu, xdBh,n u , vi ds.
kΔ
0
s
3.3
gu, xdBh,n u, vi ds
0
i 1, 2, . . . , n.
0
It follows from the assumption 2 that 3.3 can be solved for every ω by the method of
successive approximation, and the iterates are measurable with respect to ω. Consequently,
Zn,i tt∈0,T i 1, 2, . . . , n are stochastic processes since y0 is a random H-valued variable
and Bh,n tt∈0,T is a stochastic process. It follows from 3.3 that
Zn t n
y0, vi
i1
t n
v
H i
t n
kΔZn s, vi vi ds
0 i1
βs, x − μs, x
0 i1
t
n
Zn s, vi vi ds
i1
fs, x, vi
0
v ds
H i
t
βs, x − μs, x
0
gu, xdB
h,n
3.4
u, vi vi ds
0
t s
h,n
gu, xdB u , vi vi ds.
kΔ
0
s
0
Using the chain rule, we get the following
|Zn t|2 n
y0, vj
j1
2
t
2
H
2
t
kΔZn s, Zn sds
0
βs, x − μs, x Zn s, Zn s ds 2
0
2
2
t
0
t
0
βs, x − μs, x
t
0
s
gu, xdB
h,n
fs, x, Zn s ds
uds, Zn s ds
3.5
0
s
k Δ
gu, xdBh,n u , Zn s ds.
0
If we set Bt ≡ 0 in of Qimin and Chongzhao 11, under assumptions 1-2, then this result
implies that
sup |Zn s|2 k0
0≤s≤T
t
0
Zn s
2 ds ≤ η.
3.6
8
Mathematical Problems in Engineering
The following result is an analogous to that of Theorem 4 in 1. In the Galerkin
approximation, we have
2
E|Zn t − Zt| E
t
Zn s − Zs
2 ds −→ 0
3.7
0
for all t ∈ 0, T and for n → ∞. Zt is a H-valued continuous process with Zt ∈ V for all
t ∈ 0, T P -a.s., and Zt is a P -a.s. unique solution.
Now let Bh tt∈0,T be a H-valued fractional Brownian motion with ∞
j1 λj μj < ∞
∞
1/2
and j1 λj μj < ∞. We consider the finite-dimensional approximation
n t
0
j1
3.8
gs, xdBjh svj ds
t
gu, xdBh u. Obviously this is a stochastic
n
integral with respect to the V -valued Brownain motion Bh,n u Bjh uvj . Consequently,
in mean square of the stochastic integral
0
the corresponding Galerkin equations for 2.7 are given by
j1
dt ym t − kΔym tdt μt, xym tdt − βt, xym tdt
ft, xdt gt, xdBh,m t,
m
y0 , vj vj , in Γ,
ym 0 in 0, T × Γ,
3.9
j1
y t, x 0,
m
on Σ 0, T × ∂Γ,
dt yn t − kΔyn dt μt, xyn tdt − βt, xyn tdt
ft, xdt gt, xdBh,n t,
n
y0 , vj vj , in Γ,
yn 0 in 0, T × Γ,
3.10
j1
yn t, x 0,
on Σ 0, T × ∂Γ.
Lemma 3.1 shows that these problems have solutions.
∞
1/2
Theorem 3.2. If ∞
< ∞, then there exists a P -a.s unique solution
j1 λj μj < ∞ and
j1 λj μj
ytt∈0,T of 2.7 with
2
Eyt k0 E
t
0
where Mt,h is a positive constant.
Zs
2 ds ≤ Mt,h ,
∀t ∈ 0, T ,
3.11
Mathematical Problems in Engineering
9
Proof. We choose n > m with n m p and define
Zm,p t ymp t − ym t −
t
gu, xdBh,m u 0
t
gu, xdBh,mp u.
3.12
0
Then
|Z
m,p
2
t| ≤ ymp 0 − ym 0 2
2
2
t
t
kΔZm,p s, Zm,p sds
0
βs, x − μs, x ymp s − ym s , Zm,p s ds
0
2
t
3.13
fs, x, Zm,p sds
0
s
t h,mp
h,m
m,p
2 kΔ gu, xd B
u − B u , Z s ds.
0
0
However, by Lemma 2.2 14 and assumptions 1-2, we have
s
t kΔ gu, xd Bh,mp u − Bh,n u , Zm,p ds
0
0
t s
n
h
m,p
≤ k0
λj E gu, xdBj u vj , Z s ds
jm1
0
0
t
t s
2
1
h
ds 1 E|Zm,p s|2 ds
≤ k0
λj E
gu,
xv
dB
u
j
j
2 jm1
2 0
0
0
n
1
1 t
≤ k0 T 2h K 2 T
λj μj E|Zm,p s|2 ds.
2
2
0
jm1
n
3.14
Further,
E βs, x − μs, x ymp s − ym s , Zm,p s ds
s
≤ E β0 − μ0 |Zm,p s|2 β0 − μ0 gu, xd Bh,mp u − Bh,n u |Zm,p s|
0
mp
μj .
≤ 2β0 − μ0 E|Zm,p s|2 β0 − μ0 K 2 T
jm1
3.15
Consequently, in view of 3.13,
E|Zn s|2 2k0 E
t
t
E|Zn s|2 ds
Zn s
2 ds ≤ 2β0 − μ0 K 2 1
0
0
k0 Ch K T
2
mp
mp
2
λj μj 2 β0 − μ0 T K
μj .
jm1
jm1
3.16
10
Mathematical Problems in Engineering
Then, the Gronwall’s lemma implies that
t
E|Zm,p s|2 −→ 0,
E
Zm,p s
2 −→ 0
3.17
0
for m, p → ∞ for all t ∈ 0, T . In particular, there exists a process Ztt∈0,T with E|Zm t −
Zt|2 → 0 for m → ∞, and consequently, there exists a process yt with E|ym t−yt|2 → 0
for m → ∞. We must now show that ytt∈0,T is solution of 2.7. We have
t
s
2
2
h,mp
h,n
Eyn t − ym t gu, ad B
u − B u 2k0 E yn s − ym s ds
0
0
t s
≤ 2E k Δ yn s − ym s , gu, xd Bh,n u − Bh,m u ds
0
2E
t
0
βs, x − μs, x y − y s, y s − y s n
m
0
2E
t
fs, x, yn s − ym s 0
s
n
m
s
gu, xd Bh,n u − Bh,m u ds
0
gu, xd Bh,mp u − Bh,n u ds.
0
3.18
mp
Let ε > 0 be chosen arbitrary. Then there exists p0 > 0 so that jp1 λj μ1/2
< ε for all p > p0 .
j
Let yn,r t rj1 yjn,r tvj and ym,r t rj1 yjm,r tvj be the rth Galerkin approximation of
yn t and ym t, respectively. For r m p, we have
t
n,r
s
m,r
h,n
h,m
E k Δ y s − y s , gu, xd B u − B u ds
0
0
mp
t n,r
s
≤ k0 E
λi yi s − yim,r s
gu, xdBih uds
ip1 0
0
1/2
t s
2 1/2
mp
t n,r
2
m,r
h
y s − y s ds
≤ k0 E
λi E gu, xdBi u ds
i
i
0
ip1
mp
≤ const. k0 E
0
3.19
0
λi μ1/2
i
ip1
< const. × ε.
Consequently, the first term on the right-hand side of 3.10 is also less than const. × ε. 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
t
E
0
ymp s − yp s
2 −→ 0
3.20
Mathematical Problems in Engineering
11
for m, p → ∞, there is ym t is also a Cauchy sequence in L2V Ω × 0, T for all t ∈ 0, T .
Let y be the limit a of this sequence. Then it follows from the properties of a Gelfand triple
that
t
E
yn s − ys2 ≤ ME
0
t
yn s − ys2 −→ 0
3.21
0
for n → ∞, where M is a positive constant. Consequently, ys ysa.s and it follows
from 3.9 that
dt yt − kΔytds − βs, x − μs, x ytds fs, xds gs, xdBh t,
3.22
hence, we have proved Theorem 3.2.
Acknowledgments
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.
References
1 T. Caraballo, K. Liu, and A. Truman, “Stochastic functional partial differential equations: existence,
uniqueness and asymptotic decay property,” Proceedings Royal Society of London A, vol. 456, no. 1999,
pp. 1775–1802, 2000.
2 P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, New York,
NY, USA, 1992.
3 X. Mao, Stochastic Differential Equations and Applications, Ellis Horwood, New York, NY, USA, 1997.
4 T. Taniguchi, K. Liu, and A. Truman, “Existence, uniqueness, and asymptotic behavior of mild
solutions to stochastic functional differential equations in Hilbert spaces,” Journal of Differential
Equations, vol. 181, no. 1, pp. 72–91, 2002.
5 P. Kotelenez, “On the semigroup approach to stochastic evolution equations,” in Stochastic Space Time
Models and Limit Theorems, Reidel, Dordrecht, The Netherlands, 1985.
6 N. V. Krylov and B. L. Rozovskii, “Stochastic evolution equations,” Journal of Soviet Mathematics, vol.
16, pp. 1233–1277, 1981.
7 E. Pardoux, “Stochastic partial differential equations and filtering of diffusion processes,” Stochastics,
vol. 3, no. 2, pp. 127–167, 1979.
8 T. Caraballo, “Asymptotic exponential stability of stochastic partial differential equations with delay,”
Stochastics, vol. 33, no. 1-2, pp. 27–47, 1990.
9 T. Taniguchi, “Almost sure exponential stability for stochastic partial functional-differential
equations,” Stochastic Analysis and Applications, vol. 16, no. 5, pp. 965–975, 1998.
10 J. M. Cushing, “The dynamics of hierarchical age-structured populations,” Journal of Mathematical
Biology, vol. 32, no. 7, pp. 705–729, 1994.
11 Z. Qimin and H. Chongzhao, “Exponential stability of numerical solutions to stochastic agestructured population system with diffusion,” Journal of Computational and Applied Mathematics, vol.
220, pp. 22–33, 2008.
12 Z. Qimin and H. Chongzhao, “Exponential stability of numerical solutions to a stochastic agestructured population system with diffusion,” Applied Mathematical Modelling, vol. 32, pp. 2297–2206,
2008.
13 Z. Qi-Min, L. Wen-An, and N. Zan-Kan, “Existence, uniqueness and exponential stability for
stochastic age-dependent population,” Applied Mathematics and Computation, vol. 154, no. 1, pp. 183–
201, 2004.
12
Mathematical Problems in Engineering
14 Q. Zhang and C. Han, “Convergence of numerical solutions to stochastic age-structured population
system with diffusion,” Applied Mathematics and Computation, vol. 186, pp. 1234–1242, 2006.
15 M. L. Kleptsyna, P. E. Kloeden, and V. V. Anh, “Existence and uniqueness theorems for FBM stochastic
differential equations,” Information Transmission Problems, vol. 34, no. 1, pp. 41–51, 1998.
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Optimization
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