4. Consider the following system of PDEs
\frac{\partial A}{\partial t} = D \frac{\partial^2 A}{\partial x^2} + r_1(A + B)[1 - (A + B)/K],
\frac{\partial B}{\partial t} = r_2(A + B)[1 - (A + B)/K],
Here, D is diffusion constant. $r_1, r_2$ and K are parameters.
Set A = Uu, B = Vv, x = LX and t = T$\tau$, where U, V, L and T are
dimensional scales and u, v, X and $\tau$ are dimensionless variables. Show that
equations (3) and (4) can be non-dimensionalised to produce the form
\frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial X^2} + p(u + v)[1 - (u + v)],
\frac{\partial v}{\partial t} = (1 - p)(u + v)[1 - (u + v)],
where U, V, L, T and p should all be defined in terms of given parameters.
