Which of the following equations correctly represents Verhulst-Pearl logistic growth ?

Options

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

Explanation

To determine which equation correctly represents the Verhulst-Pearl logistic growth model, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Logistic Growth**:
- Logistic growth occurs when a population grows in an environment with limited resources. This means that the population will grow rapidly at first but will slow down as it approaches the carrying capacity of the environment.

2. **Identify Key Components**:
- The carrying capacity (K) is the maximum population size that the environment can sustain.
- The intrinsic rate of natural increase (R) is the difference between the birth rate and the death rate of the population.

3. **Recognize the Equation Structure**:
- The general form of the logistic growth equation is:
\[ \frac{dn}{dt} = Rn \left(1 - \frac{n}{K}\right) \]
- Here, \( \frac{dn}{dt} \) represents the change in population size over time, \( n \) is the current population size, \( R \) is the intrinsic growth rate, and \( K \) is the carrying capacity.

4. **Analyze the Given Options**:
- Look for an equation that includes both the intrinsic growth rate (R) and the carrying capacity (K) in the form mentioned above.

5. **Select the Correct Option**:
- Based on the explanation provided, the correct equation representing Verhulst-Pearl logistic growth is:
\[ \frac{dn}{dt} = Rn \left(1 - \frac{n}{K}\right) \]
- This equation incorporates the limiting factor of carrying capacity and the growth rate.

### Final Answer:
The equation that correctly represents Verhulst-Pearl logistic growth is:
\[ \frac{dn}{dt} = Rn \left(1 - \frac{n}{K}\right) \]