Thi Nhu Thao Nguyen
(Université de Bourgogne-Franche-Comté)

Small rodents, as voles, are considered harmful because they can cause a lot of damages in crops or forests when their population becomes large. But at the same time, voles are a source of food for many predators like foxes or some birds of prey. The large variations in their populations, therefore, have consequences in the diet of these predators. It is therefore important, to propose and study macroscopic models to describe the spatial dynamics of these voles in Eastern France, in particular
in the department of Doubs, based on the results monitored from the Chrono-Environment laboratory.

We will describe the evolution of the density of the voles population, in which models will be based on partial differential equations.

Firstly, we proposed a model of a voles population spatial dynamics via transport equations on a graph. At this scale, space is not taken into account, but the model took into account the Allee effect which stipulates that if the population of voles in a colony is below a threshold, the reproduction rate may decrease until the disappearance of the colony. We study the dynamics in several colonies, taking into account the movement of young voles from one colony to another.

Next, we proposed a model which, this time, takes into account the space variable, thus making it possible to describe the dynamics of voles on a scale that can be the size of a region (the direction of the dispersion of voles depends on the topography). We proved the existence and stability of entropy weak solutions for the macroscopic PDE model. In addition, we developed a digital diagramof the spatial model.

Finally, I also wish to introduce the last task which I am in the process of completing, that is considering the existence of a nonlinear system consisting of a hyperbolic equation for predators coupled with a parabolic for prey.