A study of general reaction-advection-diffusion equations describing dynamics between target, partaker, and guardian

Abstract: This paper introduces a reaction-advection-diffusion system that models interactions among three actors: a target, a partaker, and a guardian. The framework is versatile, capturing phenomena ranging from the emergence and movement of crime hotspots in urban areas to shifts in public attitudes during critical events as individuals and control units move through space. We prove local and global existence of solutions under realistic assumptions and showcase the model through two applications: protest dynamics driven by new demonstrators joining and escalating hostility in enclosed environments such as classrooms or offices. Numerical simulations highlight the resulting complex spatial patterns and temporal dynamics.