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

For the gang territoriality model \begin{align*} \begin{cases} u_t = D_u \Delta u + \chi_u \nabla \cdot (u \nabla w), \\ v_t = D_v \Delta v + \chi_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 $\|u_0\|_{L^\infty(\Omega)}$ and $\|v_0\|_{L^\infty(\Omega)}$ 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.