Verhulst-Pearl logistic growth equation can be represented by -

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

In logistic growth, growth slows as the population (N) nears the carrying capacity (K). The factor (K − N)/K is the share of K still unused: near 1 when N is small, and 0 when N reaches K.

Option (1) \( dt/dN=rN(K-N/K) \) — Incorrect. The growth rate is dN/dt, change in N per unit time; dt/dN is upside down.

Option (2) \( dN/dt=rK(K-N/K) \) — Incorrect. Growth depends on the number now present, N, not on K.

Option (3) \( dN/dt = rN(K-N/K) \) — Correct. The Verhulst-Pearl logistic growth is written dN/dt = rN((K − N)/K), where N is the population density, r the intrinsic rate of natural increase and K the carrying capacity. As N nears K, growth slows, giving an S-shaped curve.

Option (4) \( dN/dt=rN(N-K/N) \) — Incorrect. (N − K) is negative below K, so the population would shrink while there is still room to grow.

NCERT Extract:
Population Growth – “… 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
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