Which one of the following equations represents Verhulst Pearl Logist…

The correct answer is Option (3) → \( \displaystyle \frac{dN}{dt} = rN\left(\frac{K-N}{K}\right) \)

The factor \( \left(\frac{K-N}{K}\right) \) is the part of the carrying capacity still unused. When N is small, this factor is close to 1 and the population grows almost exponentially. When N reaches K, the factor becomes 0 and growth stops.

Option (1) \( \displaystyle \frac{dN}{dt} = rN\left(\frac{r-N}{K}\right) \) — Incorrect. It subtracts the population density from the growth rate r. A rate and a number of individuals cannot be subtracted, and the term does not fall to zero at the carrying capacity.

Option (2) \( \displaystyle \frac{dN}{dt} = rN\left(\frac{K-r}{K}\right) \) — Incorrect. This bracket has no N in it, so it stays the same however big the population becomes. Growth would never slow down as the population nears K.

Option (3) \( \displaystyle \frac{dN}{dt} = rN\left(\frac{K-N}{K}\right) \) — Correct. This is the logistic equation. The term (K − N)/K slows growth as N comes close to K and stops it at K.

Option (4) \( \displaystyle \frac{dN}{dt} = rK\left(\frac{K-N}{K}\right) \) — Incorrect. The growth rate depends on the number of individuals now present (N), not on K. Writing rK in place of rN would give the fastest growth when the population is smallest, which does not happen.

Note: A similar question was also asked in CUET (UG) 2026 on 29-05-2026 (Slot 1) and on 21-05-2026 (Slot 1), and in CUET (UG) 2024 on 29-05-2024 (Slot 1).

NCERT Extract:
Population Growth – …

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