# The Fisher-KPP equation on the lattice Z

## Location

The issue is the large time behaviour of a discrete version of the Fisher-KPP equation. While the model has an interest on its own, our long term motivation is to understand the large time dynamics of epidemiological models on graphs.

The main result is that a front develops, and propagates at an asymptotically linear rate, corrected by a logarithmic term. This correction was first detected in the homogeneous equation by Bramson at the beginning of the 80's, for a particular type of nonlinearity. Bramson's result exploits a remarkable link between the PDE and the Branching Brownian Motion.

Our proof is based on a method devised with J. Nolen and L. Ryzhik. What we had in mind was to provide a deterministic proof of Bramson's result, in order to study situations where the link with the Branching Brownian Motion is less obvious. We will present the general strategy, as well as some ingredients specific to the discrete setting.

Joint work with C. Besse, G. Faye and M. Zhang.