On Wind Effects in a Hyperbolic Advection–Reaction–Diffusion Forest Fire Model: Analytical Solutions, Stability, and Bifurcation Analysis

We revisit a hyperbolic wildfire model based on reaction–diffusion dynamics with relaxation effects and extend it by incorporating an advection transport term that accounts for wind-driven fire spread. After a planar two-dimensional reformulation and non-dimensionalization of the model, the analysis is restricted to the minimal ignition regime characterized by the presence of a logistic reaction term governing the evolution of the fire-affected tree fraction. The focus of the study is to assess the influence of the effective wind velocity on the propagation dynamics of the fire-affected tree fraction. For this purpose, analytical solutions of the extended wildfire model are derived by applying the Simple Equations Method (SEsM) in its (1,1) variant using a Riccati-type ordinary differential equation as a simple equation. The obtained families of exact solutions describe physically relevant transition fronts connecting fire-unaffected and fully fire-affected states, or vice versa. Numerical simulations of the derived analytical solutions are performed to demonstrate how the internal front thickness and the profile morphology depend on the specific variant of the Riccati-type solution and on the magnitude of the effective wind velocity. A phase-plane stability and bifurcation analysis of the reduced traveling wave system is carried out. Hopf bifurcation thresholds with respect to the effective wind velocity parameter are identified, revealing transitions between monotone front propagation and oscillatory regimes. A regime map is constructed in the parameter plane spanned by the effective wind velocity and the traveling wave speed. This regime diagram delineates regions of qualitatively different propagation behavior, including monotone advancing fronts, possible oscillatory regimes, and regimes in which traveling wave fronts cease to exist.