## Section: New Results

### Interactions within populations

#### Emulsion polymerization process

Participant : Ahmed Noussair.

Roughly speaking, polymerization is the formation of huge molecules (the molecules of the polymer) from smaller ones (the molecules of the monomer). In the case of emulsion polymerization this process takes place in an aqueous medium in the presence of appropriate auxiliary substances. We derive a model of coupled equations describing the emulsion polymerization process with size structure . The nonlinear size growth is modelled with a transport equation with bilinear polymerization operator while the concentrations of monomers , of appropriate emulsifier substance and the supersaturation satisfies a nonlinear system of ordinary differential equations. The inflow boundary condition is a nonlocal condition and depend on the above concentrations and on the surface coverage. We first prove global existence and uniqueness of solution to the nonlinear transport equation then and prove the existence of solutions to the full system. This research is related to the developpement of polymer-based nanotechnology delivery system for antibiotic.

#### Binary interactions in proliferating multispecies cells population

Participant : Ahmed Noussair.

We examine existence and uniqueness of a global solution and some basic mathematical issues associated with the development of a numerical scheme for a model of a binary interactions between of multispecies cells. The system consists of nonlinear transport equations with bilinear operator of type similar to Boltzmann's and with a nonlocal boundary condition. We construct a numerical schemes and we prove the convergence via Bv compactness argument. A Numerical code in Fortran is available.