For $N$ cells in a one-dimensional ring, imagine a dynamical system given by, for each cell $1\leq i\leq N$
$$
\begin{align}
\frac{dx_i}{dt}&=f(x_i,y_{i-1},y_{i+1})\\
\frac{dy_i}{dt}&=g(x_i,y_{i})
\end{align}
$$
for any non-linear $f$ and $g$. A reaction-diffusion system (in 1 dimension) is of the form
$$
\begin{align}
\frac{\partial u}{\partial t}&=K_u\partial_{xx}u(t,x)+F(u,v)\\
\frac{\partial v}{\partial t}&=K_v\partial_{xx}v(t,x)+G(u,v)
\end{align}
$$
for constants $K_u,K_v$ and non-linear functions $F,G$.
Is it possible to write the first system as a reaction-diffusion system? I suspect it is since I can think of the first system as a discretization of the one-dimensional ring (see here), but I am lacking the right notation. Any ideas?
Best Answer
To stay analogous to your example, lets rename the variables:
\begin{align}\frac{du_{i}}{dt} & =f(u_{i},v_{i-1},v_{i+1})\\ \frac{dv_{i}}{dt} & =g(u_{i},v_{i}) \end{align}
next, lets go to the continuum limit:
$u_{i}\rightarrow u(i\Delta x) $and similarly for $v_{i}$
so
\begin{align} \frac{du(t,x)}{dt} & =f(u(t,x),v(t,x-\Delta),v(t,x+\Delta))\\ \frac{dv(t,x)}{dt} & =g(u(t,x),v(t,x)) \end{align}
now expand to second order in $\Delta$
\begin{align} \frac{du(t,x)}{dt} & \approx f(u(t,x),v(t,x)-\Delta\partial_{x}v+\frac{1}{2}\Delta^{2}\partial_{x}^{2}v,v(t,x)+\Delta\partial_{x}v+\frac{1}{2}\Delta^{2}\partial_{x}^{2}v)\\ \frac{dv(t,x)}{dt} & =g(u(t,x),v(t,x)) \end{align}
A consistent expansion of $f$ will provide you with the reaction-diffusion form, possibly with a convection ($\partial_{x}$) term.