论文标题
分析城市竞争对手帮派互动的交叉扩散模型
Analysis of a cross-diffusion model for rival gangs interaction in a city
论文作者
论文摘要
我们研究了两个物种的交叉扩散模型，该模型的灵感来自于二维离散晶格上基于代理模型的对流扩散方程系统。已提出后一种模型通过使用涂鸦标记来模拟帮派领土发展。我们找到了系统的两个能量功能，使我们能够证明弱稳定性结果并确定平衡溶液。我们表明，在从弱稳定性结果中获得的弱解的自然定义下，该系统不允许隔离溶液。此外，在密度质量的乘积小于临界值的情况下，我们介绍了解决方案的长期行为的结果。该结果与数值实验相辅相成。
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 product of the masses of the densities are smaller than a critical value. This result is complemented with numerical experiments.
