Read the following text and answer the following question on the basis of the same :
Growth of population with time shows specific and predictable patters. Two types of growth patter of populaiton are exponetial and logistic growth . When resource in the habitat are unlimited each species has the ability to relize fully its innate potential to grow in number . Then the population gorws in exponential fashion . When the resource are limited growth curve shows an inital slow rate and then it accelerates and finally slows giving the gorwth curve which is sigmoid .
The equations correctly represents Verhulst-Pearl logistic growth is:

Options

A. `dN//dt = rN (K-N)//K`
B. `dN//dt = rN//K`
C. `dN//dt = N(K-N)//K`
D. `dN//dt = K(K-N)//N`

Explanation

To answer the question regarding the equation that correctly represents Verhulst-Pearl logistic growth, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Population Growth Patterns**:
- There are two main types of population growth: exponential and logistic.
- Exponential growth occurs when resources are unlimited, allowing populations to grow rapidly without constraints.

2. **Identifying Logistic Growth**:
- Logistic growth occurs when resources are limited. In this case, the growth starts slow, accelerates, and then slows down, forming a sigmoid (S-shaped) curve.

3. **Formulating the Logistic Growth Equation**:
- The logistic growth equation incorporates the carrying capacity (K), which represents the maximum sustainable population size in the environment.
- The general form of the logistic growth equation is:
\[ \frac{dN}{dt} = rN \left(\frac{K - N}{K}\right) \]
- Here, \( \frac{dN}{dt} \) is the rate of change of the population size, \( r \) is the intrinsic growth rate, \( N \) is the current population size, and \( K \) is the carrying capacity.

4. **Identifying the Correct Equation**:
- Based on the information provided, we need to identify which of the given options matches the logistic growth equation.
- The correct representation should include the terms \( rN \) and \( \frac{K - N}{K} \).

5. **Conclusion**:
- After analyzing the options, the equation that matches the logistic growth model is the one that follows the structure outlined above.

### Final Answer:
The equation that correctly represents Verhulst-Pearl logistic growth is:
\[ \frac{dN}{dt} = rN \left(\frac{K - N}{K}\right) \]