Since you say that the curve initially accelerates,
I visualize the curve as an "S" curve.
Maybe like this:
http://en.wikipedia.org/wiki/Generalised_logistic_curve
Here is the formula in Excel:
y=k_a+k_c/((1+k_t*EXP(-k_b*(x-2*k_m)))^(1/k_t))
where
y= population
x= time
k_a lower asymptote
k_c upper asymptote minus k_a
k_m time of maximum growth
k_b growth rate
k_t asymmetry of max growth time

A simplified formula might look like this:
y=a/(b+EXP(-c*x))