Abstract: In this talk, we first recall classical models for animal displacement and we point the necessity of a new model for fish motion, the socalled "Persistent turning walker" (PTW) model. Using a multiscale approach, we explain how to derive a diffusive equation starting from the PTW model. Some numerical simulations will illustrate the theory.
In a second part, we investigate a minimal model introduced by Vicsek et al. describing alignment rule between particles. This model is used for example to describe fish interaction (combined with an attractionrepulsion rule). Despite its simplicity a lot of questions are opened about it and in particular what is the dynamic at a larger scale. Our first work is to derive a continuous timedependent equation starting from this algorithm. Then, using again a multiscale approach, we explain how to derive a macroscopic model from this individual dynamic. After some explanations about the specificity of the hyperbolic equation obtained, we introduce the numerical challenges for resolving this equation. Finally, we compare numerical simulations of the model both at the particle and macroscopic level.
