Historical Processes and Their Diffusion Limits for Modeling Populations with Past Dependence
Sylvie Méléard1, Viet Chi Tran2,1
1CMAP, UMR CNRS 7641, Ecole Polytechnique, Palaiseau Cedex, France; 2Laboratoire Paul Painlevé, UMR CNRS 8524, Université Lille 1, Villeneuve d'Ascq, France

We consider a population structured in trait. When a birth occurs, the trait of the parent is hereditarily transmitted to the offspring unless a mutation occurs. We associate with each individual its lineage consisting of all the traits of his ancestors. The evolution of the population results from the aging, births and deaths of individuals, with a dynamics that may depend on the past history of the lineage and that allows interactions between individuals. We introduce the stochastic process that describes the system and consider its diffusion limit under the assumptions of large populations, individuals of small masses and allometric demographies (lifelengths and gestation periods are proportional to biomasses). The limit object is carefully described and simulations illustrate the interest of the problem.

Keywords: Historical superprocess; Interacting particle system; Large population limit

Biography: Viet Chi Tran is lecturer at the Université des Sciences et Technologies de Lille and fellow of the CMAP at Ecole Polytechnique. He is interested in stochastic models and statistical estimations for population dynamics and epidemiology.