Which one of the following equations represents the Verhulst-Pearl Logistic Growth of population?

The correct answer is: dN/dt = rN ( (K − N) / K )

Step-by-Step Explanation:

The Verhulst-Pearl Logistic Growth model describes how a population grows when resources (such as food and space) are limited. Unlike exponential growth, which assumes unlimited resources, logistic growth accounts for the carrying capacity of the environment.

Let us break down the mathematical formulation of this model step-by-step:
1. Exponential Growth Component:
If resources were unlimited, the rate of change of population size ($N$) over time ($t$) would be directly proportional to the current population size. This is expressed as:
$\frac{dN}{dt}=rN$
where $r$ is the intrinsic rate of natural increase (birth rate minus death rate).

2. Introducing Environmental Resistance (Carrying Capacity):
In nature, habitats have a limit to the population size they can support, known as the carrying capacity ($K$). As the population density ($N$) approaches the carrying capacity ($K$), resource competition increases, slowing down the rate of growth.

To reflect this limitation, we multiply the exponential growth rate by a feedback term that represents the fraction of carrying capacity still available for growth:
$\text{Available Resource Fraction}=\frac{K-N}{K}$

3. Combining the Components:
By multiplying the exponential growth term by the environmental resistance term, we obtain the Verhulst-Pearl Logistic Growth equation:
$\frac{dN}{dt}=rN\left(\frac{K-N}{K}\right)$

Let's analyze the behavior of this equation under different population sizes to confirm its validity:
- When $N$ is very small (near 0), the term $\frac{K-N}{K}$ is close to 1, meaning the population grows exponentially ($rN$).
- As $N$ approaches $K$, the term $\frac{K-N}{K}$ approaches 0, causing the growth rate ($\frac{dN}{dt}$) to slow down and eventually stop once the population reaches carrying capacity ($N=K$).