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A cross-diffusion system modelling rivaling gangs: global existence of bounded solutions and FCT stabilization for numerical simulation

Mario Fuest, Shahin Heydari

TL;DR

This work analyzes a two-species cross-diffusion model for rival gang territoriality with graffiti-based signals, proving global existence and boundedness of classical solutions under saturating production, and showing convergence to homogeneous steady states for small data. The authors transform the system to a regularized form, derive extensive a priori estimates (including L∞ and gradient bounds), and establish global existence for the original variables. They also prove that smooth heterogeneous steady states cannot exist under analytic production functions, while providing a robust FEM-FCT numerical framework to simulate the model with positivity preservation. Numerical experiments reveal diffusion-dominated regimes that homogenize and convection-dominated regimes that exhibit various segregation patterns, including complete separation under certain parameters, validating the theoretical results and highlighting the method's ability to capture complex territorial dynamics. Overall, the study advances understanding of when rival groups mix or segregate and offers a practical, stable computational approach for simulating cross-diffusion territorial models.

Abstract

For the gang territoriality model \begin{align*} \begin{cases} u_t = D_u Δu + χ_u \nabla \cdot (u \nabla w), \\ v_t = D_v Δv + χ_v \nabla \cdot (v \nabla z), \\ w_t = -w + \frac{v}{1+v}, \\ z_t = -z + \frac{u}{1+u}, \end{cases} \end{align*} where $u$ and $v$ denote the densities of two rivaling gangs which spray graffiti (with densities $z$ and $w$, respectively) and partially move away from the other gang's graffiti, we construct global, bounded classical solutions. By making use of quantitative global estimates, we prove that these solutions converge to homogeneous steady states if $\|u_0\|_{L^\infty(Ω)}$ and $\|v_0\|_{L^\infty(Ω)}$ are sufficiently small. Moreover, we perform numerical experiments which show that for different choices of parameters, the system may become diffusion- or convection-dominated, where in the former case the solutions converge toward constant steady states while in the later case nontrivial asymptotic behavior such as segregation is observed. In order to perform these experiments, we apply a nonlinear finite element flux-corrected transport method (FEM-FCT) which is positivity-preserving. Then, we treat the nonlinearities in both the system and the proposed nonlinear scheme simultaneously using fixed-point iteration.

A cross-diffusion system modelling rivaling gangs: global existence of bounded solutions and FCT stabilization for numerical simulation

TL;DR

This work analyzes a two-species cross-diffusion model for rival gang territoriality with graffiti-based signals, proving global existence and boundedness of classical solutions under saturating production, and showing convergence to homogeneous steady states for small data. The authors transform the system to a regularized form, derive extensive a priori estimates (including L∞ and gradient bounds), and establish global existence for the original variables. They also prove that smooth heterogeneous steady states cannot exist under analytic production functions, while providing a robust FEM-FCT numerical framework to simulate the model with positivity preservation. Numerical experiments reveal diffusion-dominated regimes that homogenize and convection-dominated regimes that exhibit various segregation patterns, including complete separation under certain parameters, validating the theoretical results and highlighting the method's ability to capture complex territorial dynamics. Overall, the study advances understanding of when rival groups mix or segregate and offers a practical, stable computational approach for simulating cross-diffusion territorial models.

Abstract

For the gang territoriality model \begin{align*} \begin{cases} u_t = D_u Δu + χ_u \nabla \cdot (u \nabla w), \\ v_t = D_v Δv + χ_v \nabla \cdot (v \nabla z), \\ w_t = -w + \frac{v}{1+v}, \\ z_t = -z + \frac{u}{1+u}, \end{cases} \end{align*} where and denote the densities of two rivaling gangs which spray graffiti (with densities and , respectively) and partially move away from the other gang's graffiti, we construct global, bounded classical solutions. By making use of quantitative global estimates, we prove that these solutions converge to homogeneous steady states if and are sufficiently small. Moreover, we perform numerical experiments which show that for different choices of parameters, the system may become diffusion- or convection-dominated, where in the former case the solutions converge toward constant steady states while in the later case nontrivial asymptotic behavior such as segregation is observed. In order to perform these experiments, we apply a nonlinear finite element flux-corrected transport method (FEM-FCT) which is positivity-preserving. Then, we treat the nonlinearities in both the system and the proposed nonlinear scheme simultaneously using fixed-point iteration.
Paper Structure (21 sections, 13 theorems, 53 equations, 20 figures, 1 algorithm)

This paper contains 21 sections, 13 theorems, 53 equations, 20 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main analytical results.
  3. Numerical simulations.
  4. Plan of the paper.
  5. Local existence of classical solutions to a transformed system
  6. Global existence and boundedness
  7. L infty estimates
  8. Gradient estimates
  9. Solutions are global in time: proof of Theorem \ref{['th:main']} and Corollary \ref{['cor:ex_small_data']}
  10. Smooth steady states
  11. Convergence to homogeneous steady states for small data: proof of Theorem \ref{['th:conv']}
  12. Lack of smooth heterogeneous steady states
  13. Numerical method
  14. Galerkin approximation
  15. Positivity-preserving flux-corrected transport scheme
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $D_u, D_v, \chi_u, \chi_v > 0$, $f, g$ be as in eq:intro:f_g, $\alpha \in (0, 1)$ and $w_0, z_0 \in C^{2+\alpha}(\overline \Omega; [0, \infty))$. Then there exists $C > 0$ such that for all $M > 0$ and all nonnegative $u_0, v_0 \in C^{2+\alpha}(\overline \Omega)$ with there exists a unique, global, nonnegative classical solution $(u, v, w, z)$ of prob, which satisfies the estimates

Figures (20)

  • Figure 1: Numerical solutions for the model problem (\ref{['prob']}) obtained using the standard Galerkin method, at different time instant when $D_u=D_v=\chi_u=\chi_v =0.25$. For the choice of colors, see the beginning of Subsection \ref{['sep-study']}.
  • Figure 2: The size of gang populations $u$,$v$ and the amount of their corresponding graffiti $z$ and $w$ over time along the line $y=x$ at different time instant $t=0, 400, 500, 600, 700, 1000$ when $D_u=D_v=\chi_u=\chi_v =0.25.$
  • Figure 3: Close up of populations densities along the line $y=x$ at different time instant $t= 400, 700, 1000$ when $D_u=D_v=\chi_u=\chi_v =0.25.$
  • Figure 4: Numerical solutions for the model problem (\ref{['prob']}) obtained using the standard Galerkin method, at different time instant $t=0, 50, 75, 100, 200, 1000$ when $D_u=D_v=3.0$ and $\chi_u=\chi_v =0.25$. For the choice of colors, see the beginning of Subsection \ref{['sep-study']}.
  • Figure 5: The amount of gang densities $u$,$v$ and their corresponding graffiti $z$ and $w$ over time along the line $y=x$ at different time instant $t=0, 50, 75, 100, 200, 1000$ when $D_u=D_v=3.0$ and $\chi_u=\chi_v =0.25.$
  • ...and 15 more figures

Theorems & Definitions (15)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Lemma 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 5 more