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

生物通过对某些化学物质梯度的响应而聚集的趋化现象有很多，如细菌的菌落和黏菌的聚集。这些现象可以用非线性反应扩散方程来描述，即种群密度u（x, t）和化学物质浓度c（x, t）的非线性偏相关方程。本研究的目的是得到带周期尾展开的行波解的存在条件和速度。通过数值模拟，我们得到了一些具有周期性斑点的空间模式的参数值。我们还得到了行波的速度和波长与参数值的依赖关系。该方程基本上有五个参数，空间斑点的出现取决于趋化性的强弱和趋化物质的分解速度。数值模拟得到的空间格局与细菌培养实验得到的格局吻合较好。在大的二维点阵点（800x800）上，采用交向隐式方法进行了数值计算。

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).