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.

翻译：对于帮派领地模型 \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*} 其中 $u$ 和 $v$ 表示两个敌对帮派的密度，它们喷涂涂鸦（分别以密度 $z$ 和 $w$ 表示）并部分远离对方帮派的涂鸦，我们构造了全局有界经典解。通过利用定量全局估计，我们证明：若 $\|u_0\|_{L^\infty(\Omega)}$ 和 $\|v_0\|_{L^\infty(\Omega)}$ 充分小，则这些解收敛到齐次稳态。此外，我们进行了数值实验，表明对于不同参数选择，系统可能表现为扩散主导或对流主导：在前一种情况下，解收敛到常数稳态；而在后一种情况下，观察到诸如分离等非平凡渐近行为。为进行这些实验，我们应用了保持正性的非线性有限元通量校正传输方法（FEM-FCT）。随后，我们利用不动点迭代同时处理系统中的非线性项和所提出的非线性格式。