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

有许多趋化现象，其中生命通过一些化学物质的梯度反应而聚集，例如细菌菌落和黏菌聚集物。这些现象可以用非线性反应扩散方程来描述，即人口密度u(x，t)和化学物质浓度c(x，t)的非线性偏微分方程组。本研究的目的是得到周期尾部扩展的行波的解的存在条件和传播速度，通过数值模拟得到出现周期斑点的空间图案的参数值。我们还得到了行波的速度和波长与参数值的依赖关系。方程本质上有五个参数，空间斑点的出现取决于趋化作用的强度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有很好的一致性。数值计算采用交替迭代隐式方法，对大的二维格点(800×800)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).