Crime Modeling with Truncated Levy Flights and Effects of Police Patrol

In this paper, we developed a truncated Levy flight model to study the crime dynamics. In the discrete case, our model allows criminals to perform long jumps in between committing crimes with a speed light. This is a more realistic extension of a pioneering random walk model by Short et. al and a Levy flight model thereafter in Chaturapruek, et al. We also derive a continuum limit and perform a stability analysis to study the formation of crime hotspots. Our model is more realistic than the Levy Flight Model, and provides an alternative to the Random Walk Model when the criminals can perform long jumps in between committing crimes. In the next step, we introduce patrolling police officers to our new model following that in. We examine the effects of police patrol when the police choose to adopt different strategies, including unbiased random walk, biased random walk, and truncated Levy flight. We evaluate the effectiveness of the police patrol with the number of crime events in a given time frame. With spatially non-uniform initial conditions, we find that the truncated Levy flight to be the best strategy in general.