  We study the asymptotic speed of a random front for solutions $u_t(x)$ to
stochastic reaction-diffusion equations of the form \[
\partial_tu=\farc{1}{2}\partial_x^2u+f(u)+\sigma\sqrt{u(1-u)}\dot{W}(t,x),~t\ge
0,~x\in\Rm, \] arising in population genetics. Here, $f$ is a continuous
function with $f(0)=f(1)=0$, and such that~$|f(u)|\le K|u(1-u)|^\gamma$
with~$\gamma\ge 1/2$, and $\dot{W}(t,x)$ is a space-time Gaussian white noise.
We assume that the initial condition $u_0(x)$ satisfies $0\le u_0(x)\le 1$ for
all $x\in\Rm$, $u_0(x)=1$ for~$x<L_0$ and $ u_0(x)=0$ for~$x>R_0$. We show that
when $\sigma>0$, for each $t>0$ there exist~$R(u_t)<+\infty$
and~$L(u_t)<-\infty$ such that $u_t(x)=0$ for $x>R(u_t)$ and $u_t(x)=1$
for~$x<L(u_t)$ even if $f$ is not Lipschitz. We also show that for all
$\sigma>0$ there exists a finite deterministic speed~$V(\sigma)\in\Rm$ so
that~$R(u_t)/t\to V(\sigma)$ as $t\to+\infty$, almost surely. This is in
dramatic contrast with the deterministic case $\sigma=0$ for nonlinearities of
the type $f(u)=u^m(1-u)$ with $0<m<1$ when solutions converge to $1$ uniformly
on $\Rm$ as $t\to+\infty$. Finally, we prove that when $\gamma>1/2$ there
exists $c_f\in\Rm$, so that~$\sigma^2V(\sigma)\to c_f$ as~$\sigma\to+\infty$
and give a characterization of $c_f$. The last result complements a lower bound
obtained by Conlon and Doering \cite{cd05} for the special case of
$f(u)=u(1-u)$ where a duality argument is available.
