Approches probabilistes et numériques de modèles individus-centrés du chemostat

Dans une premiere partie, nous proposons un nouveau modele de chemostat dans lequel la population bacterienne est representee de maniere individu-centree, structuree en masse, et la dynamique du substrat est modelisee par une equation differentielle ordinaire. Nous obtenons un processus markovien que nous decrivons a l'aide de mesures aleatoires. Nous determinons, sous une certaine renormalisation du processus, un resultat de convergence en loi de ce modele individu-centre hybride vers la solution d'un systeme d'equations integro-differentielles. Dans une seconde partie, nous nous interessons a des modeles de dynamiques adaptatives du chemostat. Nous reprenons le modele individu-centre etudie dans la premiere partie, auquel nous ajoutons un mecanisme de mutation. Sous des hypotheses de mutations rares et de grande population, les resultats asymptotiques obtenus dans la premiere partie nous permettent de reduire l'etude d'une population mutante a un modele de croissance-fragmentation-soutirage en milieu constant. Nous etudions la probabilite d'extinction de cette population mutante. Nous decrivons egalement le modele deterministe associe au modele individu-centre hybride avec mutation et nous comparons les deux approches, stochastique et deterministe; notamment nous demontrons qu'elles menent au meme critere de possibilite d'invasion d'une population mutante dans une population residente.Nous presentons des simulations numeriques illustrant les resultats mathematiques obtenus.

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