The formula for logistic growth is

A population growing in a habitat with limited resources show initially a lag phase, followed by phases of acceleration and deceleration and finally an asymptote, when the population density reaches the carrying capacity. A plot of N in relation to time (t) results in a sigmoid curve. This type of curve is called verhulst pearl logistic curve.
$\frac{\mathrm{dN}}{\mathrm{dt}}= \mathrm{rN}\left[\frac{\mathrm{K}-\mathrm{N}}{\mathrm{K}}\right]$
where, N = Population density at time ‘t’
            r = intrinsic rate of natural increase
           K = carrying capacity