Stochastic Reaction-Diffusion Models with Travelling-Wave Solutions

A wide range of physical, chemical, and biological systems exhibit a dynamical behaviour characterised by the emergence of travelling waves that propagate from a stable phase into an unstable one. At the deterministic level, such behaviour is typically described by the Fisher–Kolmogorov–Petrovsky–Piscunov (FKPP) equation, but finite-size effects or external noise can significantly alter the dynamics. Finite-size effects arise from small particle numbers, leading to density cut-offs and internal fluctuations, while external noise originates from random environmental perturbations. Although finite-size effects have been widely investigated in the literature, existing models – usually formulated as extensions of the FKPP equation – are often phenomenological or based on theoretically inconsistent derivations. In this Thesis, we employ a well-established stochastic procedure to derive a family of stochastic differential equations from a class of interacting-particle models, encompassing hard-core and point-like particles. This framework allows us to determine corrections to the FKPP equation, compare them with the literature, establish a criterion for the relevance of stochastic fluctuations, and analyse the impact of internal and external noise on the wave propagation speed.