论文标题
与聚集和扩散的细胞迁移模型中的模式形成
Pattern formation in a cell migration model with aggregation and diffusion
论文作者
论文摘要
在本文中，我们研究了具有Dirichlet边界条件的聚集和扩散细胞迁移模型的模式形成。该模型的形式连续限量是一个非线性抛物线方程，具有扩散率，如果细胞密度较小并且空间振荡并且在数值模拟中发生聚集可能会变为负。在具有正扩散率和非出生项的经典扩散迁移模型中，物种最终将随着迪利奇（Dirichlet）边界而消失。但是，由于小细胞密度下的聚集机制，在离散聚集扩散模型中，总物种密度是保守的。同样，离散系统会收敛到独特的正稳态，而初始密度则位于扩散域中。此外，即使具有5个离散空间点，模型中的聚集机制也会诱导丰富的渐近动力学行为或模式，这给出了一个理论上的解释，即聚集与扩散之间的相互作用会引起生物学的模式。在相应的连续向前向前抛物线方程中，还研究了溶液的最大原理的存在，也研究了溶液的渐近行为。
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.
