For this discussion, you will be able to pick one of the learning objectives we have covered in
Module 6 or Module 7 in this case “Use logistic growth models” and develop a step-by-step list
on how to solve the problern generically. Your post will contain all of the following:

State the learning objective that you are covering.

The How-to Guide you created for solving a generic form of the objective in your own words.

Give an example to illustrate the learning objective.

Question

For this discussion, you will be able to pick one of the learning objectives we have covered in
Module 6 or Module 7 in this case “Use logistic growth models” and develop a step-by-step list
on how to solve the problern generically. Your post will contain all of the following:

State the learning objective that you are covering.
The How-to Guide you created for solving a generic form of the objective in your own words.
Give an example to illustrate the learning objective.

UpStudy AI · Accepted Answer

### Learning Objective
**Use logistic growth models**

### How-to Guide for Solving a Logistic Growth Problem

1. **Identify the Basic Equation**  
   The logistic growth model is given by  
   $$
   \frac{dP}{dt} = r P\left(1-\frac{P}{K}\right),
   $$  
   where:
   - $$ P(t) $$ is the population at time $$ t $$.
   - $$ r $$ is the intrinsic growth rate.
   - $$ K $$ is the carrying capacity.

2. **Write the General Solution**  
   The general solution for this differential equation is  
   $$
   P(t)=\frac{K}{1+A e^{-rt}},
   $$  
   where $$ A $$ is a constant determined by the initial condition.

3. **Determine the Constant $$ A $$ Using the Initial Condition**  
   If the initial condition is given as $$ P(0)=P_0 $$, substitute $$ t=0 $$ into the general solution:
   $$
   P_0 = \frac{K}{1+A e^{-r\cdot 0}} = \frac{K}{1+A}.
   $$
   Solve for $$ A $$:
   $$
   1+A = \frac{K}{P_0} \quad \Longrightarrow \quad A = \frac{K}{P_0} - 1 = \frac{K-P_0}{P_0}.
   $$

4. **Write the Specific Logistic Growth Model**  
   Substitute $$ A $$ back into the general solution:
   $$
   P(t)=\frac{K}{1+\left(\frac{K-P_0}{P_0}\right) e^{-rt}}.
   $$
   This equation can now be used to calculate $$ P(t) $$ for any time $$ t $$.

5. **Analyze or Predict Population Behavior**  
   With the model established, you can:
   - Determine population size at any time $$ t $$.
   - Analyze long-term behavior as $$ t \to \infty $$ (the population approaches the carrying capacity $$ K $$).
   - Assess the effect of changing the parameters $$ r $$ and $$ K $$

### Example

Suppose we have the following parameters:
- Carrying capacity: $$ K = 1000 $$
- Growth rate: $$ r = 0.1 $$ (per unit time)
- Initial population: $$ P(0) = 50 $$

1. **Determine the Value of $$ A $$:**
   $$
   A = \frac{K-P_0}{P_0} = \frac{1000-50}{50} = \frac{950}{50} = 19.
   $$

2. **Write the Specific Logistic Model:**
   $$
   P(t)=\frac{1000}{1+19e^{-0.1t}}.
   $$

3. **Interpret the Model:**
   - At $$ t=0 $$:
     $$
     P(0) = \frac{1000}{1+19e^{0}} = \frac{1000}{20} = 50,
     $$  
     confirming the initial condition.
   - As $$ t \to \infty $$:
     $$
     P(t) \to \frac{1000}{1+19\cdot 0} = 1000,
     $$  
     hence the population approaches the carrying capacity.

This step-by-step process shows how to set up and solve a logistic growth model using an initial condition to determine the constant and then apply the model for prediction and analysis.
**Learning Objective:** Use logistic growth models **How-to Guide:** 1. **Identify the Logistic Equation:** $$ \frac{dP}{dt} = r P\left(1-\frac{P}{K}\right) $$ where: - $$ P(t) $$ = population at time $$ t $$ - $$ r $$ = growth rate - $$ K $$ = carrying capacity 2. **Find the General Solution:** $$ P(t) = \frac{K}{1 + A e^{-rt}} $$ where $$ A $$ is determined by initial conditions. 3. **Determine Constant $$ A $$:** Use the initial condition $$ P(0) = P_0 $$: $$ A = \frac{K - P_0}{P_0} $$ 4. **Write the Specific Model:** Substitute $$ A $$ back into the general solution: $$ P(t) = \frac{K}{1 + \left(\frac{K - P_0}{P_0}\right) e^{-rt}} $$ 5. **Analyze the Model:** - Calculate $$ P(t) $$ for any $$ t $$ - Understand long-term behavior as $$ t \to \infty $$ (population approaches $$ K $$) - Adjust $$ r $$ and $$ K $$ to see their effects **Example:** - $$ K = 1000 $$ - $$ r = 0.1 $$ - $$ P(0) = 50 $$ 1. **Calculate $$ A $$:** $$ A = \frac{1000 - 50}{50} = 19 $$ 2. **Specific Model:** $$ P(t) = \frac{1000}{1 + 19 e^{-0.1t}} $$ 3. **Interpretation:** - At $$ t = 0 $$: $$ P(0) = 50 $$ - As $$ t \to \infty $$: $$ P(t) \to 1000 $$ This guide helps set up and solve logistic growth models using initial conditions to predict population behavior over time.

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