Which of the following equation is correct about Verhulst-Pearl Logis…

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

The factor \( \frac{(K - N)}{K} \) 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) \( \displaystyle dN/dt = (h - d)\frac{(K - N)}{K} \) — Incorrect. The population N is missing in front of the fraction. Growth depends on how many individuals there are to reproduce, so the rate must be multiplied by N.

Option (2) \( \displaystyle dN/dt = rN\frac{(N - K)}{K} \) — Incorrect. The bracket is turned round. While N is below K, (N − K) is negative, so this equation would make a small population shrink instead of grow.

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

Option (4) \( \displaystyle dN/dt = (h - d)\frac{(N - K)}{K} \) — Incorrect. It has both faults: N is missing in front, and the bracket is turned round.

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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 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”