The Effect of Wind on the Propagation of an Idealized Forest Fire

A reaction-diffusion model for the propagation of an idealized forest fire is revisited to include the effect of wind on the speed of propagation. We study the existence of one-dimensional travelling wave solutions. When the wind velocity is zero or when the wind blows from the burning region, the existence and uniqueness of the travelling wave is proved. In the case when the wind blows into the burning region, we show that there exist at least two travelling wave solutions for small wind speed, and there is no travelling wave solutions for large wind speed. The analysis relies on comparison principles whereby the questions of existence and uniqueness are addressed via the construction of appropriate lower and upper solutions for the travelling waves. The theoretical results are supplemented with numerical examples for each case of wind velocity.