We will present firstly the derivation of a detailed 3D physical model. This model takes into account the processes involved at the scales of internal structure of the vegetation up to the scale of the flames. It is obtained by successive "homogenisations" using thermodynamics tools and concept of porous media. It is intended to be as "detailed" as possible.

In a second step we will relate this detailed physical model to two dimensional reaction diffusion models. The main idea is that at a certain length scale L (distance from the fire front, size of a developed fire) the height of the vegetation d is such that the ratio d/L = e is a small parameter, and the vegetation appears as a boundary layer. Indeed one can proceed to an asymptotic expansion using matched asymptotic expansion technique. The expansion inside the vegetation appears as the inner expansion, while the expansion outside the vegetation is the outer expansion. In the third section we will derive the inner expansion and consider the outer expansion as a parameter, in order to obtain a hierarchy of two-dimensional models. The simplest model called model I is essentially a reaction diffusion model. It involves only the temperature of the vegetation, the temperature of the surrounding gas is considered as a parameter. This model can be simplified neglecting the heat conductivity of the vegetation. Then from this new model a quasi closed form for the rate of spread can be obtained. The last part is devoted to the simulation of the model number one; it is shown that there is a percolation critical density of vegetation such that under this value the fire will not propagate.