In the last years, measure solutions to PDE, in particular those modeling populations, have drawn much attention. The talk will be devoted to the presentation of a recent, unusual result in this field, that we obtained with Pierre Gabriel. First, I will expose some wellposedness and asymptotic results for two famous population equations in the L^p and measure frameworks, and explain the critical case that interested us. Then, I will define the notion of solution we used, and if needed, recall some basic definitions about semigroups. Moving to the proof itself, I will present the main steps of the proof of the wellposedness of the problem, that relies on a duality relation used to build a solution expressed as a semigroup acting on an initial measure. Then, I will go a little more into details of the demonstration of the asymptotic behaviour. In particular, I will exhibit how we used Harris' ergodic theorem to obtain a uniform exponential convergence in (weighted) total variation norm toward an oscillating measure.