NEET 2025 Zoology Question 180 Solution

In nature, resource availability is finite and limited. No population can sustain geometric/exponential growth indefinitely:

• • A habitat has finite resources to support a maximum possible number of individuals, beyond which no further growth is possible. This limit is termed the carrying capacity ($K$).
• • A population growing in a habitat with limited resources shows an initial lag phase, followed by phases of acceleration and deceleration, and finally reaches an asymptote as $N \rightarrow K$.
• • A plot of population density ($N$) in relation to time ($t$) yields a sigmoid ($S$-shaped) logistic curve, designated as Verhulst-Pearl Logistic Growth.

Deriving the differential equation from first principles:

• • Unconstrained intrinsic growth rate: $\frac{dN}{dt} = rN$.
• • Correction for carrying capacity (environmental resistance): As $N$ approaches $K$, the fraction of unutilized carrying capacity is $\frac{K - N}{K}$.
• • When $N \ll K$, $\frac{K - N}{K} \approx 1$, and growth is exponential ($rN$).
• • When $N = K$, $\frac{K - N}{K} = 0$, and growth stops ($\frac{dN}{dt} = 0$, reaching the asymptote).
• • Combining these yields: $\frac{dN}{dt} = rN \left( \frac{K - N}{K} \right)$.

The canonical equation for Verhulst-Pearl Logistic Growth is $\frac{dN}{dt} = rN \left( \frac{K - N}{K} \right)$, matching Option (2).