Communications in Mathematical Sciences

Volume 19 (2021)

Number 8

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

Pages: 2139 – 2175



Alethea B. T. Barbaro (Delft Institute of Applied Mathematics, Delft University of Technology, Delft, Netherlands)

Nancy Rodriguez (Department of Applied Mathematics, University of Colorado, Boulder, Co., U.S.A.)

Havva Yoldas (Camille Jordan Institute, Claude Bernard University of Lyon 1, Villeurbanne, France)

Nicola Zamponi (Institute for Analysis and Scientific Computing, Technische Universität Wien, Austria)


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 masses of the densities are smaller than a critical value. This result is complemented with numerical experiments.


partial differential equations, cross-diffusion, gang dynamics, entropy method, weak stability, linear stability, equilibrium solutions, long-time behaviour, numerical simulations

2010 Mathematics Subject Classification

35A01, 35A23, 35K55, 35K57, 35K65

A. Barbaro was supported by the NSF through grant No. DMS-1319462. N. Rodríguez was partially funded by the NSF DMS-1516778. H. Yoldaş was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 639638). N. Zamponi acknowledges support from the Alexander von Humboldt foundation. The authors gratefully acknowledge the American Institute of Mathematics (AIM), where this project began.

Received 11 September 2020

Accepted 6 May 2021

Published 7 October 2021