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

We study a reaction-diffusion system of partial differential equations, which can be taken to be a basic model for criminal activity, first introduced in [Berestycki and Nadal, J. Appl. Math., 21 (2010), pp. 371--399]. We show that the assumption of a population's 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 for 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. Furt...

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