Living systems are characterized by variability; in the view of C. Darwin, they are subject to
constant evolution through the three processes of population growth, selection by nutrients
limitation and mutations.
Several mathematical theories have been proposed in order to describe the dynamics generated
by the interaction between their environment and the trait selection of the `fittest'. One can
use stochastic individual based models, dynamical systems, game theory considering traits
as strategies. From a populational point of view, the population obeys an integro-partial-differential
equation for the density number.
We will give a self-contained mathematical model of such dynamics and show that an asymptotic method
allows us to formalize precisely the concepts of monomorphic or polymorphic population. Then, we can
describe the evolution of the fittest trait and eventually to compute various forms of branching points
which represent the cohabitation of two different populations.
The concepts are based on the asymptotic analysis of the above mentioned parabolic equations
once appropriately rescaled. This leads to concentrations of the solutions and the difficulty
is to evaluate the weight and position of the moving Dirac masses that describe the population.
We will show that a new type of Hamilton-Jacobi equation, with constraints, naturally
describes this asymptotic.
Recent developments concern non-proliferative advantages and lead to define the notion of
This talk is based on collaborations with G. Barles, O. Diekmann, M. Gauduchon, S. Genieys,
P.-E. Jabin, A. Lorz, S. Mirahimmi, S. Mischler and P. E. Souganidis.