Forest fire spreading: A nonlinear stochastic model continuous in space and time

Forest fire spreading is a complex phenomenon characterized by a stochastic behavior. Nowadays, the enormous quantity of georeferenced data and the availability of powerful techniques for their analysis can provide a very careful picture of forest fires opening the way to more realistic models. We propose a stochastic spreading model continuous in space and time that has the potentiality to use such data in their full power. The state of the forest fire is described by the subprobability densities of the green trees and of the trees on fire that can be estimated thanks to data coming from satellites and earth detectors. The fire dynamics is encoded into a kernel function that can take into account wind conditions, land slope, spotting phenomena, and so on, bringing to a system of integrodifferential equations whose solutions provide the evolution in time of the subprobability densities. That makes the model complementary to models based on cellular automata that furnish single instantiations of the stochastic phenomenon. Moreover, stochastic models based on cellular automata can be derived from the present model by space and time discretization. Existence and uniqueness of the solutions is proved by using Banach's fixed-point theorem. The asymptotic behavior of the model is analyzed as well. By specifying a particular structure for the kernel, we obtain numerical simulations of the fire spreading under different conditions. For example, in the case of a forest fire evolving toward a river, the simulations show that the probability density of the trees on fire is different from zero beyond the river due to the spotting phenomenon. The kernel could be slightly modified to include firefighters interventions and weather changes.