A microscopic interpretation for adaptive dynamics trait substitution sequence models

Nicolas Champagnat (1)

1 WIAS - Weierstraß-Institut für Angewandte Analysis und Stochastik = Weierstrass Institute for Applied Analysis and Stochastics [Berlin]

We consider an interacting particle Markov process for Darwinian evolution in an asexual population with non-constant population size, involving a linear birth rate, a density-dependent logistic death rate, and a probability $\mu$ of mutation at each birth event. We introduce a renormalization parameter $K$ scaling the size of the population, which leads, when $K\rightarrow+\infty$, to a deterministic dynamics for the density of individuals holding a given trait. By combining in a non-standard way the limits of large population ($K\rightarrow+\infty$) and of small mutations ($\mu\rightarrow 0$), we prove that a time scales separation between the birth and death events and the mutation events occurs and that the interacting particle microscopic process converges for finite dimensional distributions to the biological model of evolution known as the ``monomorphic trait substitution sequence'' model of adaptive dynamics, which describes the Darwinian evolution in an asexual population as a Markov jump process in the trait space.

Mots clés: measure-valued process; interacting particle system; mutation-selection processes; Darwinian evolution; trait substitution sequence; adaptive dynamics; finite dimensional distributions convergence; time scale separation; stochastic domination; branching processes; large deviations

Domaines: Probabilités [math.PR]; Evolution [q-bio.PE]

Nicolas Champagnat. A microscopic interpretation for adaptive dynamics trait substitution sequence models. Stochastic Processes and their Applications, 2006, 116 (8), pp.1127-1160. ⟨10.1016/j.spa.2006.01.004⟩. ⟨hal-00015130⟩