  Following previous works about integro-differential equations of parabolic
type modelling the Darwinian evolution of a population, we study a
two-population system in the cooperative case. First, we provide a theoretical
study of the limit of rare mutations and we prove that the limit is described
by a constrained Hamilton-Jacobi equation. This equation is given by an
eigenvalue of a matrix which accounts for the diffusion parameters and the
coefficients of the system. Then, we focus on a particular application: the
understanding of a phenomenon called Adaptation to DNA damage. In this
framework, we provide several numerical simulations to illustrate our
theoretical results and investigate mathematical and biological questions.
