When does the logistic growth curve reach its asymptote?

The correct answer is Option (2) → When the population size ('N') equals the carrying capacity ('K')

A habitat with limited resources can support only a certain maximum number of a species, its carrying capacity (K). A population growing there shows logistic (sigmoid) growth, and the curve finally levels off at an asymptote.

Option (1) When the intrinsic rate of natural increase (r) reaches its maximum — Incorrect. r is a constant of the species in the logistic equation; it does not peak at the asymptote. Growth stops there because N has reached K.

Option (2) When the population size ('N') equals the carrying capacity ('K') — Correct. The curve levels off when N = K. The factor (K – N)/K then becomes zero, so dN/dt is zero and the population stops growing.

Option (3) When the population size ('N') is greater than the carrying capacity ('K') — Incorrect. The habitat cannot support a population above K, so the curve does not level off there. It flattens at K.

Option (4) When the population size ('N') is less than the carrying capacity ('K') — Incorrect. While N is below K, resources are still left and the population keeps growing. The curve has not yet levelled off.

NCERT Extract:
Population Growth – “… In nature, a given habitat has enough resources to support a maximum possible number, beyond which no further growth is possible. Let us call this limit as nature’s carrying capacity (K) for that species in that habitat. 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 population growth is called Verhulst-Pearl Logistic Growth (Figure 11.3) and is described by the following equation:
\( \displaystyle\frac{dN}{dt} = rN\left(\frac{K-N}{K}\right) \)
Where N = Population density at time t
r = Intrinsic rate of natural increase
K = Carrying capacity”

Figure 11.3 Population growth curve a when responses are not limiting the growth, plot is exponential, b when responses are limiting the growth, plot is logistic, K is carrying capacity