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

We revisit a hyperbolic reaction–diffusion wildfire model with relaxation effects and extend it by incorporating an advective transport term that accounts for wind-driven fire spread. After a planar two-dimensional reformulation and systematic nondimensionalization, 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 principal focus of the study isto assess the influence of the effective wind velocity on the propagation dynamics of the fire-affected tree fraction. In particular, we examine how wind-induced advection modifies front morphology, internal thickness and local stability properties of travelling combustion fronts. To derive analytical solutions to the exteded forest fire model, we apply the Simple Equations Method (SEsM) in its (1,1) variant, using a Riccati-type ordinary differential equation as the simple equation. This approach yields several physically relevant families of real-valued exact travelling-wave solutions of the extended hyperbolic model. The obtained solutions describe transition fronts connecting fire-unaffected and fully fire-affected states or vise versa. Numerical simulations are performed to illustrate and validate the analytical solutions, demonstrating how the internal front thickness and profile morphology depend on the specific Riccati solution variant and on the magnitude of the effective wind velocity. A phase-plane stability and bifurcation analysis of the reduced travelling-wave system is carried out. The equilibrium states corresponding to fire-free and fully burned regimes are classified as nodes, foci, or saddles depending on the relaxation and reaction parameterd as well as the traveing wave speed and the effective wind velocity. Hopf bifurcation thresholds with respect to the effective wind velocity parameter are identified, revealing transitions between monotone front propagation and oscillatory regimes. The existence, admissibility, and qualitative structure of travelling wave fronts are interpreted in terms of invariant manifolds and heteroclinic connections between equilibrium points. Finally, a regime map in the parameter plane spanned by the effective wind velocity and the travelling-wave speed is constructed. This diagram delineates regions of qualitatively different propagation behavior, including monotone advancing fronts, oscillatory fronts, and regimes where travelling-wave solutions cease to exist. The regime map provides a compact dynamical characterization of wind-assisted wildfire spread in hyperbolic reaction–diffusion systems with relaxation.