suffices that there is an *upper limit* $\bar{\mathcal K}$, in which case $0 \leq p / \bar{\mathcal K} \leq
1$. This scaled version will be used in (\[eq:fisherKPP\]) below. Fisher and KPP were particularly interested in the traveling wave case, $p(t,\mathbf{x}) = f(\mathbf{x}_{0} + \mathbf{v} t)$. Notice what happens here if $f$ exists: $$\mathbf{v} \cdot \nabla f = \lambda
\left(1-\frac{f}{\mathcal{K}}\right) f + c\,\nabla^2 f,$$ which in 1-D becomes a second order ordinary differential equation $$v \frac{{\rm d} f}{{\rm d} x} = \lambda
\left(1-\frac{f}{\mathcal{K}}\right) f + c \frac{{\rm d}^2
  f}{{\rm d} x^2}.$$ If $v$ were known, this ODE could be solved using `pvp4c` from `MatLab`, for example. By the rescalings show in Table \[tab:scalings\], for constant $\mathcal{K}$ the Fisher/KPP equation (\[eq:fisherKPP0\]) will be used in the form $$\frac{\partial u}{\partial t} = (1-u)u + \frac{1}{2}\,\nabla^2 u,
\label{eq:fisherKPP}$$ where the only sensible solutions have $0 \leq u \leq 1$. The initial distribution $u(0,\mathbf{x}) = u_{0}(\mathbf{x})$ must be defined for all $\mathbf{x}$.

  --------------------------------------------------------------------
   In (\[eq:fisherKPP\])             In (\[eq:fisherKPP0\])
  ----------------------- --------------------------------------------
             $             $\sqrt{\frac{\lambda}{2 c}} \, \mathbf{x}$
        \mathbf{x}$       

            $t$                           $\lambda t$

            $u$                         $p/\mathcal{K}$
  --------------------------------------------------------------------

  : Variables in left column are scaled versions of those in right column.\[tab:scalings\]

In Murray ([@murray2002], eq. (11.17)) our growth coefficient $\lambda$ is 