A Convection-Diffusion Model for Gang Territoriality

TL;DR

This paper introduces an agent-based convection-diffusion model for gang territoriality, demonstrating how indirect graffiti interactions lead to phase transitions between mixed and segregated states, supported by numerical and analytical stability analyses.

Contribution

It develops a novel agent-based and continuum model for gang territoriality, analyzing phase transitions and stability based on graffiti dynamics and agent interactions.

Findings

Phase transition between mixed and segregated states identified.

System parameters like graffiti rate influence critical behavior.

Continuum equations enable precise phase transition location prediction.

Abstract

We present an agent-based model to simulate gang territorial development motivated by graffiti marking on a two-dimensional discrete lattice. For simplicity, we assume that there are two rival gangs present, and they compete for territory. In this model, agents represent gang members and move according to a biased random walk, adding graffiti with some probability as they move and preferentially avoiding the other gang's graffiti. All agent interactions are indirect, with the interactions occurring through the graffiti field. We show numerically that as parameters vary, a phase transition occurs between a well-mixed state and a well-segregated state. The numerical results show that system mass, decay rate and graffiti rate influence the critical parameter. From the discrete model, we derive a continuum system of convection-diffusion equations for territorial development. Using the continuum equations, we perform a linear stability analysis to determine the stability of the equilibrium solutions and we find that we can determine the precise location of the phase transition in parameter space as a function of the system mass and the graffiti creation and decay rates.