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.

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 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.

UpStudy AI · Accepted Answer

# 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.

Solución de inteligencia artificial de Upstudy

Responder

Learning Objective

How-to Guide

Example

Solución

Learning Objective

How-to Guide

Example

Bonus Knowledge

preguntas relacionadas

Latest Other Questions