Reaction–diffusion dynamics of protesting activity on networks: spreading speeds, Allee effects, and wave pinning

Abstract Protesting activity often displays spatio-temporal patterns, previously modeled with reaction–diffusion systems. With the global rise of social media, it is important to understand how online networks shape these dynamics. In this work, we extend the reaction–diffusion model of Berestycki et al. (Netw Heterog Media 10:443–475, 2015) to regular network structures, capturing the interplay between activity levels and social tension. Depending on parameters, the system exhibits Fisher–KPP, weak Allee, or strong Allee growth. We analyze spreading speeds in the Fisher–KPP and weak Allee regimes, where a phase transition occurs between pushed and pulled waves. Numerical and asymptotic analyses characterize this transition and its limiting behavior. In the strong Allee (bistable) regime, we identify conditions under which small diffusion leads to pinned waves.