# Math Insight

### Overview of: Project on developing a logistic model to describe bacteria growth

#### Project summary

By using the logistic model that incorporates the idea of a carrying capacity, fit a discrete dynamical system model to data of both the initial and the longer-term growth of a bacteria colony and compare the results of the model to the data.

#### Requirements for project report

You must work in a group of 3 or 4. Your group's project report must be no longer than three pages typed (10-12 point font, 1 inch margins), with original computer-generated graphs. Please turn in your group's report via Gradescope by the date and time it is due.

Your report needs to contain the following sections:

1. Introduction: Give a short description of the bacteria growth experiment. Introduce the idea of a carrying capacity and why that might be relevant to under the results of this experiment. (No math in this section.)

2. The exponential growth model: The exponential growth model, $$P_{t+1}-P_t = r P_t,$$ predicts a certain pattern for the points $(P_t,P_{t+1}-P_t)$. Explain what this prediction is and how it arises from this equation. Test this prediction with the data. Does the data fit the pattern predicted by the model? If not, explain how the plot of the points $(P_t, P_{t+1}-P_t)$ informs you about the growth rate of the bacteria. Include a plot of the points $(P_t, P_{t+1}-P_t)$.

3. Fitting the logistic model: Explain how you fit the logistic model $$P_{t+1} - P_t = r P_t \left(1 - \frac{P_t}{M} \right)$$ to the bacteria data using a plot of the relative population change $(P_{t+1}-P_t)/P_t$ versus population size $P_t$. Explain what the logistic model predicts about the points $(P_t,(P_{t+1}-P_t)/P_t)$ and how this prediction arises from the logistic model. Show a plot of these points along with a fit of a line through those points. Give the resulting equation of the line, including its slope and $y$-intercept. Write the resulting equations for the low density growth rate $r$ and the carrying capacity $M$, and solve those equations to obtain estimates of $r$ and $M$. The final result of this section is the resulting logistic model, complete with values for the parameters and initial condition from the first data point.

4. The initial validity of the exponential growth model: Explain how the exponential growth model is valid for the initial growth in two ways. First, show that the estimate of $r$ obtained from the logistic model is close to the $2/3$ obtained from the exponential fit to the initial data. Explain why those parameters should be close, i.e., why the logistic model is nearly the same as the exponential growth when $P_t$ is small. Second, explain what the exponential growth model predicts about the points $(P_t,(P_{t+1}-P_t)/P_t)$, and show that the first few data points are close to this prediction.

5. Model comparison: Compare the values computed from your solution with the observed data. Display your results with a graph.

6. Conclusion: Summarize your result. In your summary, evaluate how well the model fit the data in order to conclude how well the logistic model captures the bacteria growth.

Introduction: 2 points
Exponential growth model: 5 points
Fitting the logistic growth: 7 points
Initial validity of exponential growth model: 7 points
Model comparison: 7 points
Conclusion: 2 points

You will be graded on both your analysis and your explanations. Points will be deducted for groups beyond the acceptable 3-4 person range and exceeding the page limit. Late submissions will not be accepted.

#### Points and due date summary

Total points: 30
Assigned: Oct. 23, 2020, 2:30 p.m.
Due: Nov. 6, 2020, 11:59 p.m.