In a paper published last November in Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, authors Henri Berestycki, Nancy Rodríguez, and Lenya Ryzhik show that the assumption of a population's natural tendency towards crime significantly changes long-term criminal activity patterns. The authors use a reaction-diffusion system to study criminal activity.
A reaction-diffusion model of a system describes how components of a system interact (in this case population zones and criminals) and predict how one or more of the components (in this case criminals) move in space and time. Reaction-diffusion models have traveling wave solutions where the waveform is a function of a single variable and the wave travels at a constant speed. The authors show that traveling wave solutions of such a model connect zones with no criminal activity with zones of high criminal activity (or hotspots). This corresponds to an invasion of criminal activity into all space.
The paper studies the problem of preventing such invasions by employing a finite number of resources that reduce payoff for committing a crime, as well as characterizes the minimum amount of resources necessary to prevent invasion of criminal activity.
While it is natural to expect that in a population with a natural tendency toward crime—based on motivation, opportunity, and running average of crime in an area— a criminal hot or warm spot might form, the authors find an interesting observation in populations with indifference or zero tendency toward criminal activity. In cases where there is indifference, criminal activity begins to dominate with a high enough payoff.
Lack of crime here is a steady state, but when it becomes unstable, criminal activity begins to dominate. In the case of populations with a natural tendency to be peaceful or crime-free, there is an interesting interplay between natural tendency and payoff for committing a crime. Here there are three steady states: complete lack of crime, small amount of crime, or a hotspot. Since small amount of crime is usually unstable, the population tends toward either zero crime or a hotspot in the long run.
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Traveling Wave Solutions in a Reaction-Diffusion Model for Criminal Activity, Multiscale Modeling & Simulation, 11(4), 1097-1126. epubs.siam.org/doi/abs/10.1137/12089884X