Verhulst - Pearl Logistic growth equation is :

Options

A. `(dN)/(dt)=rN`
B. `(dN)/(dt)=rN[K-(N)/k]`
C. `N_t=N_0e^(rt)`
D. (b - d)

Answer

### Step-by-Step Solution:

1. **Understanding Population Growth Models**:
- There are two primary models of population growth: Exponential Growth and Logistic Growth.
- Exponential growth occurs when resources are unlimited, leading to a J-shaped curve.
- Logistic growth occurs when resources are limited, resulting in an S-shaped curve.

2. **Identifying the Logistic Growth Equation**:
- The logistic growth model accounts for the carrying capacity of the environment, denoted by 'k'.
- The carrying capacity is the maximum population size that an environment can sustain.

3. **Deriving the Logistic Growth Equation**:
- The logistic growth equation is derived from the change in population density over time (dn/dt).
- The equation is given by:
\[ \frac{dn}{dt} = r \cdot n \left( \frac{k - n}{k} \right) \]
- Here:
- \( n \) = population density
- \( r \) = intrinsic growth rate
- \( k \) = carrying capacity

4. **Interpreting the Equation**:
- The term \( (k - n)/k \) represents the fraction of the carrying capacity that is still available for growth.
- As the population \( n \) approaches the carrying capacity \( k \), the growth rate \( dn/dt \) decreases, leading to stabilization of the population.

5. **Conclusion**:
- The correct logistic growth equation is:
\[ \frac{dn}{dt} = r \cdot n \left( \frac{k - n}{k} \right) \]
- This reflects how population growth slows as resources become limited.