The Verhulst-Pearl Logistic Growth is described by which equation?

The correct answer is Option (1) → \(\frac{dN}{dt}\) = rN(\(\frac{K-N}{K}\))

No population in nature has unlimited resources. A population in a habitat with limited resources grows slowly at first, then faster, then slower, and finally levels off at the carrying capacity K. This sigmoid growth is called Verhulst-Pearl Logistic Growth. In its equation the growth rate rN is multiplied by (K – N)/K, the share of the carrying capacity still unused. As N comes close to K, this share falls towards zero and growth stops.

Option (1) \(\frac{dN}{dt}\) = rN(\(\frac{K-N}{K}\)) — Correct. This is the logistic equation. When N is small, (K – N)/K is close to 1 and growth is almost exponential; when N reaches K, it becomes zero.

Option (2) \(\frac{dN}{dt}\) = rN(\(\frac{K+N}{K}\)) — Incorrect. With (K + N)/K the factor grows larger as N rises, so the population would speed up instead of levelling off at K.

Option (3) \(\frac{dN}{dt}\) = rN(\(\frac{K}{K-N}\)) — Incorrect. With K/(K – N) the factor rises without limit as N approaches K. This is the opposite of the slowing down seen in logistic growth.

Option (4) \(\frac{dN}{dt}\) = rK(\(\frac{K-N}{N}\)) — Incorrect. It multiplies r by K instead of by N and divides by N instead of by K, so growth would be fastest when the population is smallest. This is not the book’s equation.

NCERT Extract:
Growth Models – “… 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 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:” \(\frac{dN}{dt} = rN\left(\frac{K-N}{K}\right)\)

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