# American Institute of Mathematical Sciences

2007, 4(2): 187-203. doi: 10.3934/mbe.2007.4.187

## An optimal adaptive time-stepping scheme for solving reaction-diffusion-chemotaxis systems

 1 Department of Mathematics, Michigan State University, East Lansing, MI 48864, United States 2 Department of Applied Mathematics, Providence University, Taichung, Taiwan

Received  July 2005 Revised  December 2006 Published  February 2007

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 reactiondiffusion- 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.
Citation: Chichia Chiu, Jui-Ling Yu. An optimal adaptive time-stepping scheme for solving reaction-diffusion-chemotaxis systems. Mathematical Biosciences & Engineering, 2007, 4 (2) : 187-203. doi: 10.3934/mbe.2007.4.187
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