The logistic growth curve is the curve which shows a decrease in the growth rate when the population reaches its carrying capacity. It is represented by equation dN/dt = rN(K-N/K) where, N is the population density at a given time, r = intrinsic rate of natural increase, K= carrying capacity.
Asymptote is a straight line associated with a curve. A population that survives in a habitat with limited resources shows a lag phase during the initial stage as the organism needs to adjust during the initial stage and the population is less than the carrying capacity.
After, the initial stage it shows accelerated growth as the population increases and utilizes the resources. The resources get utilized at the later stage and thus the growth deaccelerates and further, it turns to an asymptote when the population density becomes equivalent to carrying capacity. So, in the case of an asymptote, K=N.