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Abstract

This paper investigates the conditions for the stability and emergence of patterns in a new three-component reaction-diffusion system. The system describes the coexistence and interaction of water reservoirs, vegetation, and bushfire activity in a given ecosystem. We perform a detailed stability analysis to determine the parameter space where an unstable homogeneous equilibrium becomes stable with respect to spatially nonuniform perturbations. We also use diffusion to generate traveling trains in the form of periodic orbits of the linearized system. These orbits are remnants of an unstable equilibrium in the absence of diffusion and arise from a nonsingular eigenvalue crossing of the imaginary axis, while a third eigenvalue remains real and negative, thereby ensuring linear stability for monocromatic waves. These phenomena differ from ``classical'' Turing and Hopf bifurcations, as the model does not involve distinct ``activators'' and ``inhibitors'', and the effects observed are not the byproduct of diffusion with necessarily differing speeds. Also, differently from the classical Turing pattern, the role of diffusion in this context is to stabilize, rather than destabilize, homogeneous equilibria. We also consider the case of plant competition, showing a suitable form of Turing instability for slow-frequency oscillations in a small rainfall regime.