Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Population Dynamics and the Logistic Model**
Consider a population \( P \) that obeys the logistic model. The dynamics of the population are described by the differential equation:
\[
\frac{dP}{dt} = \frac{5}{700} P(7 - P) \quad \text{for} \quad P > 0.
\]
We are asked to determine the intervals where the population is increasing or decreasing and to find a specific value of \( P \) given initial conditions.
**(a) The population is increasing on the interval**
\[
\text{[ ]} \quad \text{(use interval notation)}
\]
**(b) The population is decreasing on the interval**
\[
\text{[ ]} \quad \text{(use interval notation)}
\]
**(c) Assume that \( P(0) = 2 \). Find \( P(85) \).**
\[
P(85) = \text{[ ]}
\]
---
### Explanation
**Differential Equation Description:**
The given logistic model describes how the population \( P \) changes over time \( t \). The rate at which the population changes, \( \frac{dP}{dt} \), is proportional to the current population \( P \) and the factor \( 7 - P \). This implies that the population growth rate slows down as the population approaches the upper limit, \( 7 \).
**Key Points:**
1. **Increasing Population:**
- When \( \frac{dP}{dt} > 0 \), the population is increasing.
- This occurs when the product \( P(7 - P) \) is positive, i.e., \( 0 < P < 7 \).
2. **Decreasing Population:**
- When \( \frac{dP}{dt} < 0 \), the population is decreasing.
- This happens when \( P > 7 \).
3. **Equilibrium Points:**
- \( \frac{dP}{dt} = 0 \) suggests two equilibrium points: \( P = 0 \) and \( P = 7 \).
4. **Calculating \( P(85) \):**
- You will need to solve the differential equation with the given initial condition \( P(0) = 2 \).
Using these guidelines, students can analyze the intervals](https://content.bartleby.com/qna-images/question/897f167f-9a34-4bd4-9e82-b53258962c55/81185268-11f8-48c2-9399-2f305d702383/u0srcy9_processed.jpeg)
Transcribed Image Text:**Population Dynamics and the Logistic Model**
Consider a population \( P \) that obeys the logistic model. The dynamics of the population are described by the differential equation:
\[
\frac{dP}{dt} = \frac{5}{700} P(7 - P) \quad \text{for} \quad P > 0.
\]
We are asked to determine the intervals where the population is increasing or decreasing and to find a specific value of \( P \) given initial conditions.
**(a) The population is increasing on the interval**
\[
\text{[ ]} \quad \text{(use interval notation)}
\]
**(b) The population is decreasing on the interval**
\[
\text{[ ]} \quad \text{(use interval notation)}
\]
**(c) Assume that \( P(0) = 2 \). Find \( P(85) \).**
\[
P(85) = \text{[ ]}
\]
---
### Explanation
**Differential Equation Description:**
The given logistic model describes how the population \( P \) changes over time \( t \). The rate at which the population changes, \( \frac{dP}{dt} \), is proportional to the current population \( P \) and the factor \( 7 - P \). This implies that the population growth rate slows down as the population approaches the upper limit, \( 7 \).
**Key Points:**
1. **Increasing Population:**
- When \( \frac{dP}{dt} > 0 \), the population is increasing.
- This occurs when the product \( P(7 - P) \) is positive, i.e., \( 0 < P < 7 \).
2. **Decreasing Population:**
- When \( \frac{dP}{dt} < 0 \), the population is decreasing.
- This happens when \( P > 7 \).
3. **Equilibrium Points:**
- \( \frac{dP}{dt} = 0 \) suggests two equilibrium points: \( P = 0 \) and \( P = 7 \).
4. **Calculating \( P(85) \):**
- You will need to solve the differential equation with the given initial condition \( P(0) = 2 \).
Using these guidelines, students can analyze the intervals
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