A convection–diffusion model for gang territoriality

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.

A Multispecies Cross-Diffusion Model for Territorial Development

“…More recently, advances have been made in the field of movement ecology with models of territorial development motivated by scent marking, many exhibiting cross-diffusion. Some, such as those in [9], are similar to the continuum systems derived in [10] and in Sections 1.1-3.2 of the current work, though there are several subtle but important differences. The interested reader is referred to [9,11,12], which provide an excellent overview of the current state of this ecological literature and important new results for crossdiffusion models in spatial population dynamics.…”

“…Our paper is based on the work of [10], wherein the authors performed a bottom-up approach similar to the Moorcroft model [6], producing a discrete system to describe gang territorial development and formally deriving from it the following system of convectiondiffusion equations:…”

“…A modified version of the convection-diffusion system in [10] was analyzed in [33], where they proved a weak stability result and identified equilibrium solutions; interestingly, though, they did not find segregated equilibrium solutions in this modified system. In this paper, we generalize the model and results of [10] to consider any finite number K of groups. This is important, because it gives the model much more flexibility.…”

“…As in the two-gang model, we observe that our model undergoes a phase transition, and we numerically examine the phase transition and how it is affected by changing our spatial discretization. We then follow [10] in deriving a continuum system. We perform a twodimensional linear stability analysis to identify the critical parameter, showing that above the bifurcation value of β all wave numbers are unstable, in agreement with the observed behavior of the discrete model.…”

“…Furthermore, we find the critical parameters for the transition between well-mixed and segregated territories for all three models we introduce for two groups, with a proposed formula for K groups. This paper's outline is as follows: In Section 1.1, we introduce our extension of the original two-gang agent-based model [10]. The rest of the article is based upon this extension.…”