Chiu Chichia, Yu Jui-Ling
Department of Mathematics, Michigan State University, East Lansing, MI 48864, USA.
Math Biosci Eng. 2007 Apr;4(2):187-203. doi: 10.3934/mbe.2007.4.187.
Reaction-diffusion-chemotaxis systems have proven to be fairly accurate mathematical models for many pattern formation problems in chemistry and biology. These systems are important for computer simulations of patterns, parameter estimations as well as analysis of the biological systems. To solve reaction-diffusion-chemotaxis systems, efficient and reliable numerical algorithms are essential for pattern generations. In this paper, a general reaction-diffusion-chemotaxis system is considered for specific numerical issues of pattern simulations. We propose a fully explicit discretization combined with a variable optimal time step strategy for solving the reaction-diffusion-chemotaxis system. Theorems about stability and convergence of the algorithm are given to show that the algorithm is highly stable and efficient. Numerical experiment results on a model problem are given for comparison with other numerical methods. Simulations on two real biological experiments will also be shown.
反应-扩散-趋化系统已被证明是用于化学和生物学中许多模式形成问题的相当精确的数学模型。这些系统对于模式的计算机模拟、参数估计以及生物系统分析都很重要。为了解决反应-扩散-趋化系统，高效且可靠的数值算法对于模式生成至关重要。在本文中，针对模式模拟的特定数值问题，考虑了一个一般的反应-扩散-趋化系统。我们提出了一种完全显式离散化方法，并结合可变最优时间步长策略来求解反应-扩散-趋化系统。给出了关于该算法稳定性和收敛性的定理，以表明该算法具有高度的稳定性和效率。给出了一个模型问题的数值实验结果，以便与其他数值方法进行比较。还将展示两个实际生物实验的模拟结果。
