Question

Which of the following equations depicts Verhulst-Pearl logistic population growth ?

• A. $\frac{dN}{dt} = rN \left(\frac{K-N}{N}\right)$
• B. $\frac{dN}{dt} = rN \left(\frac{K-N}{K}\right)$
• C. $\frac{dN}{dt} = rN \left(\frac{K}{K-N}\right)$
• D. $\frac{dN}{dt} = rN \left(\frac{K+N}{K}\right)$

Hint:

Logic Tip: The term $\left(\frac{K-N}{K}\right)$ represents the "unutilized capacity" of the environment. When $N = K$ (population hits carrying capacity), the term becomes zero, meaning population growth ($\frac{dN}{dt}$) completely stops!

Answer 1

The Correct Option is B

Step 1 : Understanding the Question:
The question asks to identify the correct differential equation for logistic population growth, also known as the Verhulst-Pearl equation. This model describes population dynamics in an environment with limited resources.
Step 2 : Key Formulas and Approach:
1. $N$: Population density at time $t$.
2. $r$: Intrinsic rate of natural increase.
3. $K$: Carrying capacity (maximum population supportable).
4. The growth rate decreases as the population size ($N$) approaches the carrying capacity ($K$).
Step 3 : Detailed Explanation:

Conceptual Basis: When resources (food/space) are limited, a population cannot grow exponentially ($dN/dt = rN$) indefinitely. There is a limit ($K$) beyond which no more individuals can be supported.

Environmental Resistance: The factor $(K - N) / K$ represents the "unutilized capacity" of the environment. As $N$ gets closer to $K$, this factor becomes smaller, slowing down the growth rate.

Mathematical Form: The full equation is $dN/dt = rN [(K - N) / K]$. This produces a sigmoid or S-shaped growth curve.

Evaluating Options: Option (B) correctly shows the relationship where growth is proportional to current population and the remaining fractional capacity of the habitat.

Step 4 : Final Answer:
The correct mathematical representation of Verhulst-Pearl logistic growth is found in the second choice, option (B).