A reduced PDE model for wildfire spread with probabilistic interpretation

Recently, a stochastic forest fire model was proposed in which the forest state is represented by sub-probability densities of firing, green, and burnt trees, whose evolution is governed by a master equation formulated as a system of integro-differential equations. Within this framework, the present paper restricts attention to the case of short-range fire propagation and derives a reduced model whose dynamics is described by a reaction-diffusion-convection equation for the firing-tree sub-probability density, coupled with an ordinary differential equation for the green-tree sub-probability density. The reduced system is interpreted as a short-range approximation of the original integro-differential model, while retaining the same probabilistic meaning of the evolving densities. Numerical simulations show that it reproduces the main qualitative features of the full model in the parameter regimes considered, while quantitative comparisons based on Hellinger and total variation distances indicate that the two dynamics remain close over simulation times corresponding to real times of the order of a few tens of hours. At the same time, the reduced formulation yields a substantial reduction in computational cost, making it particularly suitable for repeated simulations and for the early stages of wildfire propagation, where stochastic effects are relevant and fire spotting can still be neglected. When used together with the original integro-differential model, the reduced model contributes to a stochastic modelling framework that complements approaches based on cellular automata. Finally, some mathematical properties of the reduced model, including nonnegativity of solutions and long-time asymptotic behaviour, are also discussed.