Section: New Results

Stochastic and hybrid models

We continued our study of stochastic modeling of the chemostat. In a first study we establish the Fokker-Planck equation of the law of the diffusion process. This equation features relevant boundary condition for the washout. We propose specific finite difference schemes to account for this feature [18] . In a second work we adopt the same approach to more accurately study the logistic model [64] which allowed us to propose estimation procedures to take into account the extinction (see Section 6.3.2 ).

We proposed a chemostat model where the bacterial population is individually-based (IBM), each bacterium is explicitly represented and has a mass evolving continuously over time, and where the substrate concentration is represented as a conventional ordinary differential equation. These two components are coupled with the bacterial consumption. Mechanisms acting on the bacteria are explicitly described (growth, division and up-take). Bacteria interact via consumption. We set the exact Monte Carlo simulation algorithm of this model and its mathematical representation as a stochastic process. We prove the convergence of this process to the solution of an integro-differential equation (IDE) when the population size tends to infinity. The IDE is discretized with the help of finite differences, with simulation as well as the IBM are developed in Python with the help of the Gamma-Team (UMR Mistea) [63] .

Finally with O. Ovaskainen (Univ. of Helsinki) we developed an evolution model for the chemostat.

We proposed a new “hybrid” model for the simulation of biofilm growth in a plug flow bioreactor, that combines information from three scales: a microscopic one for the individual bacteria, a mesoscopic or “coarse-grained” one that homogenizes at an intermediate scale the quantities relevant to the attachment/detachment process, and a macroscopic one in terms of substrate concentration. In contrast to existing partial differential equations models, this approach is based on a description of biological mechanisms at the individual scale, thus bringing in a biological justification of the attachment/detachment process responsible for the macroscopic behavior [20] .

We pursue our study of the “atto-fox” question in the classical Rosenzweig-MacArthur model for a resource-consumer relationship: for certain values ​​of parameters the system has a limit cycle such that the smallest value reached by the resource on this cycle is so small that the model validity is jeopardized [65] .