When does the growth rate of a population following the logistic model equal zero ? The logistic model is given as dN/dt = rN(1-N/K):

Options

A. when N/K equals zero
B. when N/K is exactly one.
C. when death rate is greater than birth rate.
D. when N nears the carrying capacity of the habitat.

Answer

To determine when the growth rate of a population following the logistic model equals zero, we start with the logistic growth equation:

\[ \frac{dN}{dt} = rN \left(1 - \frac{N}{K}\right) \]

Where:
- \( \frac{dN}{dt} \) is the rate of change of the population size over time.
- \( N \) is the current population size.
- \( r \) is the intrinsic rate of natural increase.
- \( K \) is the carrying capacity of the environment.

### Step-by-Step Solution:

1. **Understanding the Equation**: The growth rate \( \frac{dN}{dt} \) will equal zero when the right-hand side of the equation is zero. Thus, we need to solve the equation:
\[ rN \left(1 - \frac{N}{K}\right) = 0 \]

2. **Setting the Factors to Zero**: The product on the left side will be zero if either factor is zero:
- \( rN = 0 \)
- \( 1 - \frac{N}{K} = 0 \)

3. **Analyzing Each Factor**:
- For \( rN = 0 \): This implies either \( r = 0 \) (no growth) or \( N = 0 \) (no population).
- For \( 1 - \frac{N}{K} = 0 \): Rearranging gives us:
\[ \frac{N}{K} = 1 \quad \Rightarrow \quad N = K \]
This indicates that the population size \( N \) has reached the carrying capacity \( K \).

4. **Conclusion**: The growth rate \( \frac{dN}{dt} \) equals zero when:
- The population size \( N \) is equal to the carrying capacity \( K \) (i.e., \( N = K \)).
- Alternatively, it can also be zero if there is no population present (i.e., \( N = 0 \)).

Thus, the growth rate of a population following the logistic model equals zero when \( N = K \).

### Final Answer:
The growth rate of a population following the logistic model equals zero when \( N = K \).

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