Structured cell growth models in a chemostat

In this paper, we want to describe the growth of the cell during its cycle; we construct some structured ODE models in low dimension. The structuration in ODE models takes account of the different stages in the cell evolution. This kind of modelisation allows richer mathematical behavior than classical unstructured models. As oscillatory behaviors can be observed in biological experiences [3], we construct systems where limit cycle can appear. To prove the existence of closed orbit or not for our models, we use the properties of competitive systems, and the Bendixson criterion.