We consider a population structured in trait. When a birth occurs, the trait of the parent is hereditarily transmitted to the offspring unless a mutation occurs. We associate with each individual its lineage consisting of all the traits of his ancestors. The evolution of the population results from the aging, births and deaths of individuals, with a dynamics that may depend on the past history of the lineage and that allows interactions between individuals. We introduce the stochastic process that describes the system and consider its diffusion limit under the assumptions of large populations, individuals of small masses and allometric demographies.