Logistic growth is represented by which equations

Options

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

Answer

**Step-by-Step Text Solution:**

1. **Understanding Growth Models**:
- There are two main types of population growth models: exponential growth and logistic growth.
- Exponential growth occurs when resources are unlimited, leading to continuous population increase. This is often represented by a J-shaped curve.
- Logistic growth occurs when resources are limited, leading to a more realistic model of population dynamics, represented by an S-shaped or sigmoid curve.

2. **Phases of Logistic Growth**:
- The logistic growth model consists of three main phases:
- **Lag Phase**: Initial slow growth as the population adapts to the environment.
- **Log Phase (Exponential Phase)**: Rapid population increase as resources are utilized efficiently.
- **Stationary Phase**: Growth slows down and stabilizes as the population reaches the carrying capacity of the environment.

3. **Logistic Growth Equation**:
- The equation that represents logistic growth is given by:
\[ \frac{dn}{dt} = r \cdot n \left(1 - \frac{n}{K}\right) \]
- Where:
- \(\frac{dn}{dt}\) = rate of change of the population over time
- \(r\) = intrinsic rate of natural increase
- \(n\) = population density at time \(t\)
- \(K\) = carrying capacity of the environment

4. **Identifying the Correct Equation**:
- Among the options provided, the correct representation of logistic growth is the one that matches the equation mentioned above.
- It is important to note that population density is typically represented with a capital \(N\) rather than a small \(n\).

5. **Conclusion**:
- The correct equation for logistic growth is:
\[ \frac{dn}{dt} = r \cdot n \left(1 - \frac{n}{K}\right) \]
- This equation reflects the limitations imposed by resource availability on population growth.

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