Analysis of a cross-diffusion model for rival gangs interaction in a city

TL;DR

This paper analyzes a two-species cross-diffusion model inspired by gang territorial behavior, establishing stability, equilibrium, and long-term dynamics, supported by numerical experiments.

Contribution

It introduces energy functionals for the model, proves weak stability, and characterizes equilibrium and long-term behavior, advancing understanding of gang interaction modeling.

Findings

Weak stability of the system established

No segregated solutions under weak solution framework

Long-term behavior depends on the product of densities' masses

Abstract

We study a two-species cross-diffusion model that is inspired by a system of convection-diffusion equations derived from an agent-based model on a two-dimensional discrete lattice. The latter model has been proposed to simulate gang territorial development through the use of graffiti markings. We find two energy functionals for the system that allow us to prove a weak-stability result and identify equilibrium solutions. We show that under the natural definition of weak solutions, obtained from the weak-stability result, the system does not allow segregated solutions. Moreover, we present a result on the long-term behavior of solutions in the case when the product of the masses of the densities are smaller than a critical value. This result is complemented with numerical experiments.