A population P obeys the logistic model. It satisfies the equation 5 dP P(7-P) for P >0. dt 700 (a) The population is increasing on the interval (b) The population is decreasing on the interval (c) Assume that P(0) = 2. Find P(85). P(85) (use interval notation) ***** (use interval notation) *****************

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Chapter2: Second-order Linear Odes
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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
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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