A degenerate nonlinear diffusion problem in age-structured population dynamics

Abstract

WE CONSIDER a degenerate nonlinear diffusion problem that arises in the modelling of agestructured populations. The derivation of such problems has been discussed in a variety of places (see [4], [5], [7] f or example) and we give only a brief description of the motivation for the equations we treat. The particular problem that we consider is discussed by MacCamy [7] who also analyses a special case of the resulting equations. Our treatment of this problem is based on the general method we developed in [l], [2] combined with variants of ideas and techniques used by Ladyshenskaya [6], MacCamy [7], Olienik [8], and Sabinina [9]. The problem is as follows. Let a denote chronological age, f denote time and x denote spatial position, and let ~(a, t, x) denote the number of individuals, per unit age and unit length, who are of age a at time t and at position X. Here, we take (a, t, x) E [0, a) x [O, Tl x (0, 1). BY FL(Q) and PC a we denote the death rate and the birth rate, per unit age, > of individuals of age a. The total population density per unit length at time t is