Identify the equation representing logistic growth of a population.

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

In logistic growth the population first grows fast and then levels off at the carrying capacity (K). The term (K − N)/K is close to 1 when N is small and becomes 0 when N reaches K, so growth stops there; the curve is S-shaped.

Option (1) \(\frac{dN}{dt} = rN\left(\frac{K - N}{K}\right)\) — Correct. The Verhulst-Pearl logistic growth is written dN/dt = rN((K − N)/K), where r is the intrinsic rate of natural increase and K the carrying capacity. Growth slows as N nears K, giving an S-shaped curve.

Option (2) \(\frac{dN}{dt} = rN\) — Incorrect. This is exponential growth, with unlimited resources; it gives a J-shaped curve.

Option (3) \(\frac{dN}{dt} = N\left(\frac{N - K}{K}\right)\) — Incorrect. It has no r, and (N − K) is negative below K, so the population would shrink.

Option (4) \(\frac{dt}{dN} = rN\) — Incorrect. The rate is upside down (dt/dN) and there is no limit K.

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