将偏微分方程\begin{cases} \frac{\partial u}{\partial t}(x,t) = \frac{\partial^2 u}{\partial x^2}(x,t) + u-u^2-u*v/(0.5*v+1)\\ \frac{\partial v}{\partial t}(x,t) = \frac{\partial^2 v}{\partial x^2}(x,t) + 20*u(x,t-tau)*v/(0.5*v(x,t-tau)+1)-v-v^2, \end{cases} $$转化为常微分方程组

对于偏微分方程组，我们可以使用空间离散化方法，如有限差分法，将其转化为常微分方程组。这里以有限差分法为例，将空间区间$[0,L]$等分为$N$个小段，每个小段的长度为$h=L/N$。时间区间$[0,T]$等分为$M$个小段，每个小段的长度为$\Delta t=T/M$。令$u_i^m\approx u(ih,m\Delta t)$，$v_i^m\approx v(ih,m\Delta t)$，则有：

$$\frac{\partial u}{\partial t}(x,t)\approx \frac{u_i^{m+1}-u_i^m}{\Delta t}$$

$$\frac{\partial^2 u}{\partial x^2}(x,t)\approx \frac{u_{i+1}^m-2u_i^m+u_{i-1}^m}{h^2}$$

将上式代入原方程组第一行得：

$$\frac{u_i^{m+1}-u_i^m}{\Delta t}=\frac{u_{i+1}^m-2u_i^m+u_{i-1}^m}{h^2}+u_i^m-u_i^2-\frac{u_i^m v_i^m}{0.5v_i^m+1}$$

整理得：

$$u_i^{m+1}=\frac{\Delta t}{h^2}(u_{i+1}^m-2u_i^m+u_{i-1}^m)+u_i^m-\Delta t(u_i^m)^2-\frac{\Delta t u_i^m v_i^m}{0.5v_i^m+1}+u_i^m$$

$$u_i^{m+1}=\frac{\Delta t}{h^2}(u_{i+1}^m-2u_i^m+u_{i-1}^m)+2u_i^m-\Delta t(u_i^m)^2-\frac{\Delta t u_i^m v_i^m}{0.5v_i^m+1}$$

同理，将空间离散化得：

$$v_i^{m+1}=\frac{\Delta t}{h^2}(v_{i+1}^m-2v_i^m+v_{i-1}^m)+20\Delta t u_i^{m-\tau}\frac{v_i^m}{0.5v_i^m+1}-\Delta tv_i^m-\Delta t(v_i^m)^2$$

将时间离散化后，整理得到：

\begin{cases} u_i^{m+1}=\Delta t(\frac{1}{h^2}(u_{i+1}^m-2u_i^m+u_{i-1}^m)+2u_i^m-\Delta t(u_i^m)^2-\frac{\Delta t u_i^m v_i^m}{0.5v_i^m+1}) \\ v_i^{m+1}=\Delta t(\frac{1}{h^2}(v_{i+1}^m-2v_i^m+v_{i-1}^m)+20u_i^{m-\tau}\frac{v_i^m}{0.5v_i^m+1}-v_i^m-\Delta t(v_i^m)^2) \end{cases}

这样就将偏微分方程组转化为常微分方程组了。