Content area
Full Text
Abstract
We consider a certain generalization of the von Foerster-Lasota differential equation and by using the technique of infinite-dimensional cellular matrices(the so called Maclaurin differential operators) give its solution in an explicit form. We study the behaviour of corresponding motions in R in terms of ordinary and standard "Lebesgue measures".
2000Mathematics Subject Classification: Primary 03xx, 28Axx, 28Cxx, 35Exx; 35Dxx; 35Qxx Secondary 28C10, 28D10, 28D99
Keywords: Maclaurin's differential operators, Phasemotion, von Foerster-Lasota equation, a-ordinary and a-standard "Lebesgue measures"
(ProQuest: ... denotes formulae omitted.)
1. Introduction
We begin our discussion by a short history described by Antoni Leon Dawidowicz and Anna Poskrobko in [1]:
In 1926 McKendrick [2] proposed the first age-dependent model of the dynamics of a population. He assumed that the state of a population in time t is described by a function u(t, .). The number of individuals in age from the interval [x1; x2] equals R x2 x1 u(t, x)dx. From this paper the equation
... (1.1)
follows, called in the literature as McKendrick equation or more often as von Foerster equation.
McKendricks model was generalized in many ways, among others by Gurtin and Mac- Camy [3] or by the authors [5]. This equation is part of the mathematical description of a particular population, as the population of red blood cells is (see [6]). It is the model with a feedback, because a circulatory system controls the global number of erythrocytes to some quantity optimum which can change. It happens, for example, during mountain trips or in a case of any disease of the respiratory system. In their next paper [7] the authors of [6] introduced a new model of precursor cells. There the main assumption is the fact that cells mature with different intensity. The form of the equation is the following:
... (1.2)
where c : [0;1]! R and f : [0;1]×[0;+)! R are the given functions fulfilling suitable conditions. In this model x denotes the degree of cell differentiation (maturity) and R x1 x0 u(t0, x)dx is the number of cells having at time t = t0 the value x in the interval [x0; x1]. The coefficient c is the velocity of cell differentiation. Because of biological application the above equation has been the matter of interest for...