On solutions of spatial pattern formation in nonlinear reaction-diffusion equations

1993 Fiscal Year Final Research Report Summary

• Project/Area Number: 04804008
• Research Category: Grant-in-Aid for General Scientific Research (C)
• Allocation Type: Single-year Grants
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Doshisha University
• Principal Investigator: KAWASAKI Kohkichi Doshisha University, Science and Engineering Research Institute, Professor, 理工学研究所, 教授 (10150799)
• Project Period (FY): 1992 – 1993
• Keywords: reaction-diffusion equation / nonlinear equation / traveling wave / spatial pattern / chemotaxis

Research Abstract

There are many chemotactic phenomena where livings gather by response to gradient of some chemical substances, for example, bacterial colonies and aggregations of slime molds. These phenomena can be described by nonlinear reaction-diffusion equations, that is, nonlinear partial deferential equations of population density u(x, t) and concentration of chemical substance c(x, t). This research aims to obtain the condition of existence of solutions and the speed of traveling waves which are expandign with periodical tailes.
By the numerical simulations, we have obtained the parameter values in which some spatial patterns with periodical spots appear. Also we have obtained dependency on parameter values for speed and wave-length of traveling waves. The equations have essentially five parameters and the appearance of spatial spots depends on strength of chemotaxis and decomposition rate of chemotactic substance. The spatial patterns obtained by numerical simulations give good agreement with the patterns that had been obtained by the experiments of bacterial culture. The numerical calculations are performed by the alternating-dierction implicit method on large 2-dimensional lattice points (800x800).