In the equation,`dN//dt = rN`

Options

A. r' is called the 'intrinsic'rate of natural increase'.
B. r' is the parameter chosen for assessing impacts of any biotic factor an population growth.
C. r' is the parameter chosen for assessing impacts abiotic
D. All of these

Answer

To solve the equation \( \frac{dN}{dt} = rN \), we will analyze the components of the equation and understand its implications in the context of population growth.

### Step-by-Step Solution:

1. **Understanding the Equation**:
The equation \( \frac{dN}{dt} = rN \) represents the rate of change of population size \( N \) over time \( t \). Here, \( r \) is a constant that represents the intrinsic rate of natural increase of the population.

**Hint**: Identify what each symbol in the equation represents. \( N \) is the population size, \( t \) is time, and \( r \) is the growth rate.

2. **Rearranging the Equation**:
We can rearrange the equation to separate the variables:
\[ \frac{dN}{N} = r \, dt \]

**Hint**: To solve differential equations, separating variables can simplify the integration process.

3. **Integrating Both Sides**:
Next, we integrate both sides:
\[ \int \frac{dN}{N} = \int r \, dt \]
This gives us:
\[ \ln |N| = rt + C \]
where \( C \) is the constant of integration.

**Hint**: Remember that integrating \( \frac{1}{N} \) gives \( \ln |N| \).

4. **Exponentiating to Solve for \( N \)**:
To solve for \( N \), we exponentiate both sides:
\[ |N| = e^{rt + C} \]
This can be rewritten as:
\[ N = e^C e^{rt} \]
Let \( K = e^C \), then:
\[ N = K e^{rt} \]

**Hint**: Recognize that \( K \) is a constant that can be determined by initial conditions.

5. **Interpreting the Result**:
The final equation \( N = K e^{rt} \) indicates that the population grows exponentially over time, where \( K \) is the initial population size when \( t = 0 \).

**Hint**: The exponential growth model assumes unlimited resources and ideal conditions for growth.

### Conclusion:
The equation \( \frac{dN}{dt} = rN \) describes exponential growth in a population, where \( r \) is the intrinsic growth rate. This model is a simplification and assumes constant growth without limiting factors.