Change in population size equation with prolonged exponential phase can be converted into logistic growth equation by multiplying it with

Options

A. K/N
B. `K-N`/K
C. `K`/K-N
D. 1/N-K

Answer

To solve the question regarding how to convert the population size equation from a prolonged exponential phase to a logistic growth equation, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Exponential Growth Phase**:
- In the exponential growth phase, the population grows without any limitations. The equation representing this phase is:
\[ \frac{dn}{dt} = r \cdot n \]
- Here, \(n\) is the population size, \(r\) is the intrinsic growth rate, and \(\frac{dn}{dt}\) is the change in population size over time.

2. **Understand the Logistic Growth Phase**:
- The logistic growth phase considers the carrying capacity of the environment, which limits population growth. The equation for logistic growth is:
\[ \frac{dn}{dt} = r \cdot n \left( \frac{K - n}{K} \right) \]
- In this equation, \(K\) represents the carrying capacity of the environment.

3. **Convert Exponential to Logistic Growth**:
- To convert the exponential growth equation into the logistic growth equation, we need to account for the carrying capacity. This is done by multiplying the exponential growth equation by a factor that represents the effect of the carrying capacity.
- The factor to multiply by is:
\[ \frac{K - n}{K} \]
- Therefore, the modified equation becomes:
\[ \frac{dn}{dt} = r \cdot n \cdot \frac{K - n}{K} \]

4. **Conclusion**:
- Thus, the equation for converting the prolonged exponential phase into the logistic growth equation is achieved by multiplying the exponential growth equation by the factor \(\frac{K - n}{K}\).

### Final Answer:
The change in population size equation with prolonged exponential phase can be converted into the logistic growth equation by multiplying it with:
\[ \frac{K - n}{K} \]