| The population of a region or a country and even the entire world is a dynamic system, which is often referred to the population evolution system. For its own operation rules, the development of population can be accurately described with some mathematical models which are expressed with some problems involved in partial differential equations. Since the establishment of the mathematical model for population, many experts have achieved a lot of results which are valuable both theoretical and practical applications. However, it is usually difficult to study steady population models since the boundary conditions are generally given with some integrals of complicated functions which have some practical significance to the real problems. In 2007, W. Wang applied Semi-Dispersed Algorithm, which was originally used in study some problems in physics and engineering, to population evolution equations and hence overcame the difficulties arised in previous work. In the same year, W. Tian also studied the steady age-structure population evolution equation on [ ]L1 0, rm and showed that semi-dispersed algorithm method is feasible.In this paper we discuss semi-dispersed method and apply the method to the study of population evolution equation, and showed that semi-dispersed algorithm method is feasible for the population evolution equation in Lp [ 0,rm ].The steady age-structure population evolution equation is the following: )where p (r , t is the population that younger than r at time t ,μ(r ) is the death rate, is the birth rate, is the maximum age of population living up, is the range of child-bearing age. b( r )rm[r1 , r2]Referring the physical background, we make the following assumptions:1.(1) b( r ) is non-negative continuous function on [ ]r1 , r2; number and2.μ(r ) >0 and for r < rm,∫( ) <∞∫( )=∞. r drmd0μττ, 0μττThen the semi-dispersed model may be transformed to an abstract differential equation on Lp [ 0,rm ].Disperse Equation (1) on [ ]r1 , r2 for b( r ) with boundary conditions p (0 , t) by picking n ?2 points on [ ]r1 , r2,and for function b(r ) we define the following two step functionsAccording to the definition of step functions, we have:Thus we can obtained two dispersion equations:Because the two equations are similar in form and have the same conditions, we only discuss equation (2) in the course of the study. In the following, we will transform equation (2), (3) into some abstract ordinary differential equations. For equation (2), we use the abstract Cauchy problem on space to describe the equation. DenotingBanachA = ?ddr?μ(r ) and select state-space X = Lp [ 0,rm], set for ( ) ( ) ( )obvious that X is a space, the domain of operator A is defined as: Banach is an absolutely continuous function, and p b( ) rrb(r )p (r )d r}. =∫Then System (2) can be described as that following abstract Cauchy problem on Banach space X .For System (3), denote An = ?ddr?μ(r )with ( )X . The domain of the operator An is as the following: absolutely continuous function, and p nb( ) rr bn(r ) pn(r )d r}=∫System (3) can be described as the following abstract Cauchy problem on Banach space X . Finally we prove that the two solutions obtained with the two semi-dispersed equations may approach the solutions to the solution of the original equation from both sides with the use of the theory in functional analysis and mathematical analysis. At the end, we illustrate that the semi-dispersed method is feasible in the study of the population evolution equation.
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