When does the logistic growth curve reach its asymptote?

When the population size ('N') equals the carrying capacity ('K')

When K =N in a logististic growth curve, it is asymptote.

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,i.e. when the population density (N ) reaches the carrying capacity (K)

Population growth curve is logistic, when responses are limiting the growth, Here, K is carrying capacity and N is population density.