与年龄相关的具有扩散的随机种群系统数值解的收敛性_英文_

C
1 Preliminaries
Consider stochastic age2structured population system with diffusion (1) ～ (5) . A is the maximal age of the population species , so P ( r , t , x ) = 0 , Π r ≥A . By (5) , integrating on [ 0 , A ] to Eq. (1) and Eq. (3) with respect to r , then we obtain the following system
a family of nonlinear operators , Ft2measurable almost surely in t. g ( t , x , ・ ) :L 2H → L ( M , H) is the family of nonlinear operator , Ft2measurable almost surely in t. The integral version of Eq. ( 7 ) is given by the
Received date :2007212204 Foundation item :Suppo rted by Education Minist ry Science and Technology Research Key Foundatio n ( 208160)
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Δy d t + μ( t , x) y d t - β ( t , x) y d t = dt y - k f ( t , x , y) d t + g ( t , x , y) dW t , Γ, in Q = ( 0 , T) × y ( 0 , x) = y0 ( x) , in Γ,
deterministic and random parts which depend on r , t , x and P. There has been much recent interest in application of deterministic age2structures mathematical models with diffusion ( when g1 = 0 ) . For example , Cushing[1 ] investigated hierarchical age2dependent populations with intra2specific competition or predation when g1 = 0. Allen and Thrasher[2 ] considered vaccination strategies in age2dependent populations in the case of g1 = 0. The effects of the stochastic environmental noise considerations lead to stochastic age2structured population systems , which are more realistic. When the diffusion of the population is not considered ( k = 0 ) , Zhang showed the existence , uniqueness and exponential stability for stochastic age2dependent population equation , and numerical analysis for stochastic age2 dependent population equation has been studied by Zhang[3 - 5 ] . In general , stochastic age2dependent population equation ( 1 ) rarely has an explicit solution. Thus , numerical approximation schemes are invaluable tools for exploring its properties. In this paper , an numerical approximation method for stochastic age2dependent population equation of the type described by Eqs. (1) ～ (5) will be developed. In particular , the results extend
第29卷 第3 期 宁夏大学学报 ( 自然科学版) Vol. 29 No . 3 Journal of Ningxia U niversity (Nat ural Science Edition)
Article ID :025322328( 2008) 0320198205
y ( t , x) = P ( r , t , x) d r , in Q , ∫
0
A
∫
0
( 2) ( 3) ( 4) ( 5)
Γ < Rn , P ( r , t , x ) where t ∈( 0 , T) , r ∈( 0 , A ) , x ∈ denotes the population density of age r at time t and in the location x , β( r , t , x ) denotes the fertility rate of females of age r at time t and in spatial position x , μ( r , t , x) denotes the mortality rate of age r at time t and in the location x , Δ denotes the Laplace operator with respect to the space variable , k ( r , x ) > 0 is the dW t diffusion coefficient . f 1 ( r , t , x , P) + g1 ( r , t , x , P) dt denotes the effects of external environment for population system , such as emigration and earthquake and so on. The effects of external environment has the
( s) = sup ψ
0≤ s ≤T
p p , L V = L ( [ 0 , T ] ;V ) and
p p L H = L ( [ 0 , T ] ; H) .
We consider stochastic age2structured population system with diffusion of the form
P ( t , x) = 0 ,
Γ, on Σ = ( 0 , T) ×5
( 6)
where dt y is the differential of P relative to t , i. e. ,
) :L 2H →H be dt y = ( 5 y/ 5 t) d t. T > 0 , A > 0 , f ( t , x , ・
Consider the following stochastic age2dependent population dynamic system : 5P 5P + - k ( r, t)ΔP +μ 1 ( r, t , x) P = 5t 5r 5W f 1 ( r , t , x , P) + g1 ( r , t , x , P) , 5t ( 1) in QA = (0 , A) ×Q , A P(0 , t , x) = β 1 ( r , t , x) P( r , t , x) d r, Γ, in (0 , T) × Γ, P( r , 0 , x) = P0 ( r , x) , in Q′= (0 , A) × ( ) Σ ( ) ( ) Γ, P r, t , x = 0 , on A = 0 , A × 0 , T × 5

第 4 期 谷晓君等 : 与年龄相关的具有扩散的随机种群系统数值解的收敛性
t ho se of Zhang[ 4 ] .
199
all continuous function from [ 0 , T ] into V with sup2 norm ψ
2008年9月 Sep f Numerical Solutions to Stochastic Age2structured Population System with Diff usion
Gu X i aoj un , Z han g Qi m i n
( School of Mat hematics and Co mp uter Science , Ningxia U niversity , Yinchuan 750021 , China) Abstract : In general , mo st of stochastic age2st ruct ured pop ulatio n system wit h diff usio n do not have explicit solutio ns , t hus numerical app roximatio n schemes are invaluable tools for explo ring t heir p roperties. The main p urpo se of t his paper is to give a numerical scheme. By applying t he t heo ry of stochastic f unctional differential equation and using Gronwall lemma and Barkholder2Davis2 Gundy lemma , t he co nvergence of t he numerical app ro ximation solution to t he t rue solutio n is p roved fo r a class of stochastic age2dependent pop ulation dynamic system. In particular , t he result s are extension of t he existing result s for ordinary age2dependent pop ulation dynamic system. Key words :stochastic age2st ruct ured pop ulation system ; t rue solution ; numerical solutio ns CLC number :O175 2000 MR number :60 H15 ;34 K40 Document code :A