In this paper, we study pattern formations in an aggregation and diffusion cell migration model with Dirichlet boundary condition. The formal continuum limit of the model is a nonlinear parabolic equation with a diffusivity which can become negative if the cell density is small and spatial oscillations and aggregation occur in the numerical simulations. In the classical diffusion migration model with positive diffusivity and non-birth term, species will vanish eventually with Dirichlet boundary. However, because of the aggregation mechanism under small cell density, the total species density is conservative in the discrete aggregation diffusion model. Also, the discrete system converges to a unique positive steady-state with the initial density lying in the diffusion domain. Furthermore, the aggregation mechanism in the model induces rich asymptotic dynamical behaviors or patterns even with 5 discrete space points which gives a theoretical explanation that the interaction between aggregation and diffusion induces patterns in biology. In the corresponding continuous backward forward parabolic equation, the existence of the solution, maximum principle, the asymptotic behavior of the solution is also investigated.

翻译：在本文中,我们用Drichlet边界条件的集合和传播细胞迁移模型研究模式的形成。模型的正规连续限制是非线性抛物线等方程式,如果细胞密度小,且空间振动和聚合在数值模拟中发生,则该等方程式会变成负差。在具有积极差异性和非出生术语的典型扩散迁移模型中,物种最终会随着Drichlet边界而消失。然而,由于小细胞密度下的集合机制,总物种密度在离散集合扩散模型中比较保守。此外,离散系统会聚集到一个独特的正态稳定状态,其初始密度在扩散域中。此外,模型中的聚合机制会诱发大量无症状的动态行为或模式,即使有5个离散的空间点,这些点也从理论上解释集聚与扩散之间的相互作用会诱发生物学模式。在相应的连续向后向偏移方方方程式中,解决方案的存在、最大原则、解决方案的无症状行为也会得到调查。