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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 densitiesof two rivaling gangs which spray graffiti (with densities $z$ and $w$,respectively) and partially move away from the other gang's graffiti, weconstruct global, bounded classical solutions. By making use of quantitativeglobal estimates, we prove that these solutions converge to homogeneous steadystates if $\|u_0\|_{L^\infty(\Omega)}$ and $\|v_0\|_{L^\infty(\Omega)}$ aresufficiently small. Moreover, we perform numerical experiments which show thatfor different choices of parameters, the system may become diffusion- orconvection-dominated, where in the former case the solutions converge towardconstant steady states while in the later case nontrivial asymptotic behaviorsuch as segregation is observed. In order to perform these experiments, weapply a nonlinear finite element flux-corrected transport method (FEM-FCT)which is positivity-preserving. Then, we treat the nonlinearities in both thesystem and the proposed nonlinear scheme simultaneously using fixed-pointiteration.

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