We study the long-time behaviour of a population structured by age and a phenotypic trait under a selection-mutation dynamics. By analysing spectral properties of a family of positive operators on measure spaces, we show the existence of eventually singular stationary solutions. When the stationary measures are absolutely continuous with a continuous density, we show the convergence of the dynamics to the unique equilibrium. 1 Preliminaries and Main Results 1.1 Introduction Our ultimate goal is the understanding of the long-time behaviour of a population where the individuals differ by their physical age a È R and some hereditary variable x È S R d called trait. The population evolves as follows. An individual with trait x È S and age a È R has a death rate DÔx, aÕ cN where D is the intrinsic death rate, N is the population size and c 0 the competition rate. This individual gives birth at rate BÔx, aÕ. At every birth, a mutation occurs with probability p È ×0, 1Ö and the trait of the newborn y È S is choosen according a distribution kÔx, a, yÕdy. Otherwise, the descendant inherits of the trait x È S. In his thesis [19], Tran introduced an individual-based stochastic model to describe such a discrete population. The population is described by a random point measure Z K t 1 K N K t ô i1 δ Ôx i ÔtÕ,a i ÔtÕÕ (1) which evolves as a càdlàg Markov process with values in the set M ÔS ¢ R Õ of positive finite measures on S ¢ R and each jump corresponds to birth or death of individuals. When the order K of the size of the population goes to infinity such that Z K 0 approximates a deterministic measure n 0 È M ÔS ¢ R Õ, it is shown (see [19],[20]) that the process approximates the unique weak solution (in the sense given by (10)) Ôn t Õ t0 È CÔR , M ÔS ¢ R ÕÕ of the partial differential equation t n t Ôx, aÕ a n t Ôx, aÕ ¡ ¡ DÔx, aÕ c S ¢R n t Ôy, αÕdydα © n t Ôx, aÕ, n t Ôx, 0Õ F Ön t × ÔxÕ, Ôt, x, aÕ È R ¢ S ¢ R , (2) 1