Identify the correct Verhulst-Pearl logistic growth equation.

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

When growth is limited by resources, a population grows in a sigmoid curve. This is Verhulst-Pearl logistic growth. In the equation, N is the population density at time t, r is the intrinsic rate of natural increase and K is the carrying capacity. The factor \( \displaystyle\frac{K-N}{K} \) shows how much room is left. We can test any form with two cases: N very small, and N equal to K.

(1) \( \displaystyle\frac{dN}{dt} = rN\left(\frac{K-N}{K}\right) \) — Correct. For a small N, the bracket is close to 1, so the growth is almost rN, as in exponential growth.

    \( \displaystyle N=K:\quad \frac{dN}{dt} = rN\left(\frac{K-K}{K}\right) = 0 \)

So growth stops at the carrying capacity.

(2) \( \displaystyle\frac{dN}{dt} = rN\left(\frac{K-N}{N}\right) \) — Incorrect. Dividing by N cancels the N outside. The growth rate becomes \( r(K-N) \), which does not fall to rN for a small population.

(3) \( \displaystyle\frac{dN}{dt} = rN\left(\frac{N-K}{K}\right) \) — Incorrect. The sign is reversed. For N below K the bracket is negative, so the population would shrink while it is still below the carrying capacity.

(4) \( \displaystyle\frac{dN}{dt} = rN\left(\frac{N-K}{N}\right) \) — Incorrect. It has both faults: the sign is reversed and the denominator is N.

Note: A similar question was also asked in CUET (UG) 2026 on 21-05-2026 (Slot 1), in CUET (UG) 2025 on 30-05-2025 (Slot 1), in CUET (UG) 2024 on 29-05-2024 (Slot 1), and in CUET (UG) 2023 on 25-05-2023 (Slot 3).

NCERT Extract:
Population Growth – “… 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
Since resources for growth for most animal populations are finite and become limiting sooner or later, the logistic growth model is considered a more realistic one.”

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