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

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 mass of the densities are less than one. This result is complemented with numerical experiments.