  We study the transient dynamics of single species reaction diffusion systems
whose reaction terms $f(u)$ vary nonlinearly near $u\approx 0$, specifically as
$f(u)\approx u^{2}$ and $f(u)\approx u^{3}$. We consider three cases, calculate
their traveling wave fronts and speeds \emph{analytically} and solve the
equations numerically with different initial conditions to study the approach
to the asymptotic front shape and speed. Observed time evolution is found to be
quite sensitive to initial conditions and to display in some cases nonmonotonic
behavior. Our analysis is centered on cases with $f'(0)=0$, and uncovers
findings qualitatively as well quantitatively different from the more familiar
reaction diffusion equations with $f'(0)>0$. These differences are ascribable
to the disparity in time scales between the evolution of the front interior and
the front tail.
