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 problem 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.

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# Learning Objective

Use logistic growth models.

# How-to Guide

1. **Write the Logistic Differential Equation:**  
   Begin with the logistic differential equation:  
   $$
   \frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)
   $$  
   where $$ P(t) $$ is the population at time $$ t $$, $$ r $$ is the intrinsic growth rate, and $$ K $$ is the carrying capacity.

2. **Separate Variables:**  
   Rewrite the differential equation to separate the variables:  
   $$
   \frac{dP}{P\left(1 - \frac{P}{K}\right)} = r\,dt
   $$

3. **Perform Partial Fraction Decomposition:**  
   Decompose the left-hand side integral into partial fractions.  
   Write:  
   $$
   \frac{1}{P\left(1 - \frac{P}{K}\right)} = \frac{1}{P} + \frac{1}{K-P}
   $$
   (Make sure to adjust coefficients appropriately if needed.) The goal is to express the integrand as a sum of terms that are easier to integrate.

4. **Integrate Both Sides:**  
   Integrate the equation with respect to $$ P $$ and $$ t $$:  
   $$
   \int \left(\frac{\,dP}{P} + \frac{\,dP}{K-P}\right) = \int r\,dt
   $$  
   The integrations yield:  
   $$
   \ln |P| - \ln |K-P| = rt + C
   $$  
   where $$ C $$ is the constant of integration.

5. **Solve for $$ P(t) $$:**  
   Combine the logarithms:  
   $$
   \ln \left|\frac{P}{K-P}\right| = rt + C
   $$  
   Exponentiate both sides to eliminate the logarithm:  
   $$
   \left|\frac{P}{K-P}\right| = e^{rt+C} = Ae^{rt}
   $$  
   where $$ A = e^C $$ is a constant. We can then solve for $$ P(t) $$:  
   $$
   \frac{P}{K-P} = Ae^{rt}
   $$
   Isolate $$ P $$ by cross-multiplying:
   $$
   P = Ae^{rt}(K-P)
   $$
   $$
   P + Ae^{rt}P = Ae^{rt}K
   $$
   $$
   P(1 + Ae^{rt}) = Ae^{rt}K
   $$
   $$
   P(t) = \frac{K Ae^{rt}}{1 + Ae^{rt}}
   $$
   Alternatively, you can express the solution using the constant determined by the initial condition:
   $$
   P(t) = \frac{K}{1 + Be^{-rt}}
   $$
   where $$ B $$ is determined from the initial condition.

6. **Determine the Constant Using Initial Conditions:**  
   Suppose the initial condition is $$ P(0) = P_0 $$. Substitute $$ t = 0 $$ into the solution:  
   $$
   P_0 = \frac{K}{1 + B}
   $$
   Solve for $$ B $$:
   $$
   B = \frac{K - P_0}{P_0}
   $$
   The logistic growth model then becomes:  
   $$
   P(t) = \frac{K}{1 + \left(\frac{K-P_0}{P_0}\right)e^{-rt}}
   $$

# Example

Consider a population of bacteria where:

- Intrinsic growth rate: $$ r = 0.1 $$
- Carrying capacity: $$ K = 1000 $$
- Initial population: $$ P_0 = 10 $$

Using the steps above:

1. Write the general solution:
   $$
   P(t) = \frac{K}{1 + \left(\frac{K-P_0}{P_0}\right)e^{-rt}}
   $$

2. Calculate the constant $$ B $$:
   $$
   B = \frac{K - P_0}{P_0} = \frac{1000 - 10}{10} = \frac{990}{10} = 99
   $$

3. Substitute the values into the logistic model:
   $$
   P(t) = \frac{1000}{1 + 99e^{-0.1t}}
   $$

This function $$ P(t) $$ represents the bacteria population at time $$ t $$ based on the logistic growth model.
# Learning Objective Use logistic growth models. # How-to Guide 1. **Start with the Logistic Equation:** Use the formula: $$ \frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right) $$ where $$ P $$ is the population, $$ r $$ is the growth rate, and $$ K $$ is the carrying capacity. 2. **Separate Variables:** Rewrite the equation to separate $$ P $$ and $$ t $$: $$ \frac{dP}{P\left(1 - \frac{P}{K}\right)} = r\,dt $$ 3. **Integrate Both Sides:** Integrate to find the relationship between $$ P $$ and $$ t $$: $$ \ln \left|\frac{P}{K-P}\right| = rt + C $$ where $$ C $$ is the constant of integration. 4. **Solve for $$ P(t) $$:** Rearrange the equation to express $$ P(t) $$ in terms of $$ t $$: $$ P(t) = \frac{K}{1 + Be^{-rt}} $$ where $$ B $$ is determined by initial conditions. 5. **Apply Initial Conditions:** Use the initial population $$ P_0 $$ at $$ t = 0 $$ to find $$ B $$: $$ B = \frac{K - P_0}{P_0} $$ Substitute $$ B $$ back into the equation to get the specific logistic growth model. # Example Suppose a population of bacteria grows with: - Growth rate $$ r = 0.1 $$ - Carrying capacity $$ K = 1000 $$ - Initial population $$ P_0 = 10 $$ Using the logistic model: $$ P(t) = \frac{1000}{1 + 99e^{-0.1t}} $$ This equation shows how the population $$ P(t) $$ changes over time $$ t $$ based on the logistic growth model.

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