Selection, mutation, adaptive dynamics: an asymptotic point of view

Living systems are subject to constant evolution. The environment can be considered as a nutriment shared by all the population. This alllows certain individuals, charaterized by a ’trait’ to better develop but also to change the environment. This leads to select the ’best adapted trait’ in the population. On the other hand the population can develop with small variance on the trait (mutations). We will give a mathematical model of such dynamics and show that an asymptotic method allows us to describe the evolution of the ’best adapted trait’ and eventually to compute bifurcations which lead to the cohabitation of two different traits.