Reproductive numbers and traveling fronts in models of protest dynamics with police management

Protests often spread through social networks and across cities, a phenomenon that can be modeled using reaction–diffusion systems. Building on the model in [10] , which links protest activity and social tension, we study an extended model that also includes police protest management strategies. We focus on traveling wave solutions and examine how protest activity, social tension, and police density interact to shape the spread of protests [36] . Assuming the existence of traveling fronts in parameter regimes where the system is not monotone, we derive stability conditions for these traveling fronts, extending earlier results that assumed monotonicity. Through numerical simulations, we estimate how fast protests spread and identify threshold conditions, similar to epidemiological reproduction numbers, that determine whether protests grow or fade.