Well posedness and stochastic derivation of a diffusion-growth-fragmentation equation in a chemostat

Josué Tchouanti (1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

We study the existence and uniqueness of the solution of a non-linear coupled system constituted of a degenerate diffusion-growth-fragmentation equation and a differential equation, resulting from the modeling of bacterial growth in a chemostat. This system is derived, in a large population approximation, from a stochastic individual-based model where each individual is characterized by a non-negative real valued trait described by a diffusion. Two uniqueness results are highlighted. They differ in their hypotheses related to the influence of the resource on individual trait dynamics, the main difficulty being the non-linearity due to this dependence and the degeneracy of the diffusion coefficient. Further we show that the semi-group of the stochastic trait dynamics admits a density by probabilistic arguments, that allows the measure solution of the diffusiongrowth-fragmentation equation to be a function with a certain Besov regularity.

Mots clés: Diffusion-growth-fragmentation equation coupled with resource; Stochastic Feller-type diffusion; Mild formulation; Large population approximation; Existence of density

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Josué Tchouanti. Well posedness and stochastic derivation of a diffusion-growth-fragmentation equation in a chemostat. 2021. ⟨hal-03249697v1⟩