A regularization scheme for an inverse problem in age-structured populations
In many models describing the evolution of species it is necessary to allow the birth and death processes to depend on the age of the individual. We then have an age-structured population where \(\rho (a,t)\) denotes the number of individuals of age \(a\) at a particular time \(t\). We denote by \(\beta = \beta (a)\) and \(\lambda = \lambda (a)\) the birth and death functions of the species -- the quantities \(\lambda (a) da\) and \(\beta (a) da\) correspond to the probabilities of an individual of age \(a\) dying or giving birth on the interval \([a,a + da]\), respectively. The usual direct problem for age-structured populations is to be given the ``birth'' and ``death'' functions \(\beta (\cdot)\) and \(\lambda (\cdot)\), as well as the initial value \(\varphi (a)\), and use these to recover \(\rho (a,t)\) for \(t > 0\). We let \(L\) denote the maximum life span of the species, that is, \(\rho (a,t) = 0\) for \(a \geq L\) and all \(t \geq 0\). In consequence, this implies that the function \(\lambda (a)\) must be singular at \(a = L\). We are concerned with the problem of determining the death function \(\lambda (a)\) or, as is more suitable to our formulation, the life table \(\pi\) from overposed data consisting of the values of the age-structure at a later time \(T\).
scientific article; zbMATH DE number 584081