  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.
