Traveling Wave Solutions in a Reaction-Diffusion Model for Criminal Activity

Abstract

We study a reaction-diffusion system of partial differential equations, which can be taken to be a basic model for criminal activity. We show that the assumption of a populations natural tendency towards crime significantly changes the long-time behavior of criminal activity patterns. Under the right assumptions on these natural tendencies we first show that there exists traveling wave solutions connecting zones with no criminal activity and zones with high criminal activity, known as hotspots. This corresponds to an invasion of criminal activity onto all space. Second, we study the problem of preventing such invasions by employing a finite number of resources that reduce the payoff committing a crime in a finite region. We make the concept of wave propagation mathematically rigorous in this situation by proving the existence of entire solutions that approach traveling waves as time approaches negative infinity. Furthermore, we characterize the minimum amount of resources necessary to prevent the invasion in the case when prevention is possible. Finally, we apply our theory to what is commonly known as the gap problem in the excitable media literature, proving existing conjectures in the literature.