A class of nonlinear diffusion problems in age-dependent population dynamics

Abstract

IN THIS paper we study the existence, the uniqueness and the asymptotic behaviour of a class of nonlinear diffusion problems. These problems are motivated by models of age-dependent population dynamics which originate in the article by Gurtin & MacCamy [ll]. Only a few special cases of these models have been analyzed to date, with the most recent published work being that of MacCamy [17], Gurtin & MacCamy [12] and Langlais [15, 161. The cases treated in [12, 171 are restricted to special simple forms of the birth and death moduli that occur in these models and to a specific form of the nonlinear diffusion mechanism. On the other hand, Langlais [16] considers a different nonlinear diffusion mechanism and obtains a type of generalized solution for general forms of the birth and death moduli. In this work, we consider a general form of the nonlinear diffusion mechanism and we develop a method for establishing the existence and uniqueness of solutions of the resulting equations. Our method allows us to obtain classical solutions even with general forms of the birth and death moduli. We also describe the asymptotic behaviour of the solutions when the time variable becomes large, and apply our results to some specific examples. The equations that we study are motivated as follows. Consider a population that can disperse in a spatial domain R. For simplicity we take Q = J, where J is an open interval in R, but it will be clear from the context that more general regions Q of R” can be considered. Let ~(a, t, x) denote the number of individuals, per unit length and unit age, that are of age a at time t and are located at the position x E J. The total population, per unit length, at time t and at position x is given by