# Math Insight

### Bacteria growth model exercises

The bacteria model page explained how to develop a discrete dynamical system model based on experimental measurements of population density. In the following exercises, you can explore similar models and datasets.

#### Exercise 1

Following the procedure outlined in the bacteria model page, compute $B_1$, $B_2$, and $B_3$ for the following discrete dynamical systems. \begin{align*} {\rm{a.}} \hspace{2mm} B_0 & = 4 & \hspace{5mm} B_{t+1} - B_t & = 0.5 \times B_t \\ {\rm{b.}} \hspace{2mm} B_0 & = 4 & \hspace{5mm} B_{t+1} - B_t & = 0.1 \times B_t \\ {\rm{c.}} \hspace{2mm} B_0 & = 0.2 & \hspace{5mm} B_{t+1} - B_t & = 0.05 \times B_t \\ {\rm{d.}} \hspace{2mm} B_0 & = 0.2 & \hspace{5mm} B_{t+1} - B_t & = 1 \times B_t \\ {\rm{e.}} \hspace{2mm} B_0 & = 100 & \hspace{5mm} B_{t+1} - B_t & = 0.4 \times B_t \\ {\rm{f.}} \hspace{2mm} B_0 & = 100 & \hspace{5mm} B_{t+1} - B_t & = 0.01 \times B_t \end{align*}

#### Exercise 2

Write a solution equation for the dynamical systems of Exercise 1 similar to equation (5) of the bacteria page.

#### Exercise 3

When developing the linear model, observe that we plotted $B_{t+1} - B_t$ versus $B_t$. The points are $(B_0, B_1-B_0)$, $(B_1, B_2-B_1)$, etc. The second coordinate, $B_{t+1} - B_t$ is the population increase during time period $t$, given that the population at the beginning of the time period is $B_t$. Explain why the point (0,0) would be a point of this graph.

#### Exercise 4

Below are tables with four sets of data. For each data set, follow the steps from the bacteria growth page to fit a dynamical system to the data. The dynamical system should be of the form $B_0 = \text{value from the table}, \hspace{1cm}\goodbreak B_{t+1} - B_t = r \times B_t.$ Find the number $r$ so that the values $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ and $B_6$ computed from the dynamical system are close to the corresponding numbers in the table. Once you have calculated $r$, compute the numbers $B_1$ to $B_6$ from the dynamical system and compare your computed values with the original data.

(a)
$t$$B_t 01.99 12.68 23.63 34.89 46.63 58.93 612.10 (b) t$$B_t$
00.015
10.021
20.031
30.040
40.055
50.075
60.106
(c)
$t$$B_t 022.1 123.4 226.1 327.5 430.5 534.4 636.6 (d) t$$B_t$
0287
1331
2375
3450
4534
5619
6718

To aid you in calculating the number $r$, you can use the below applet, which is similar to original bacteria applet but uses $P_t$ to indicate population size at time $t$. Enter the data and find a line close to it. Be sure that the line goes through the origin. For better control over the applet and to be able to print or save your results, we suggest downloading the applet to your computer. On the applet information page, you can find a link to the applet file and instructions for installing Geogebra.

Fitting a linear model to population change as a function of population size. In the table of the right panel, enter up to six values for the population size $P_t$ at time $t$. (In the right panel, $P_t$ is displayed as P_t.) The applet will automatically calculate the changes $P_{t+1}-P_t$ and plot the points $(P_t,P_{t+1}-P_t)$, which show the population change versus the population size at the beginning of the interval. By moving the points $A$ and $B$ (represented by the red diamonds) with the mouse, you can try to fit a line through the points and read off its slope. You can zoom in and out with the buttons label by + and -. The arrows allow you finer control over the zoom in each direction.

#### Exercise 5

The bacterium V. natriegens was also grown in a growth medium with pH of 7.85. Data for that experiment is shown in the following table. Repeat the analysis demonstrated in the bacteria growth page for this data. Compare your computed relative growth rate of V. natriegens at pH 7.85 with the computed relative growth rate at pH 6.25, which we calculated to be 2/3.

Measurements of V. natriegens density at pH 7.85.
Time (min)Population Density
0 0.028
160.047
320.082
480.141
640.240
800.381

#### Exercise 6

What discrete dynamical system would describe the growth of an Escherichia coli population in a nutrient medium that had 250,000 E. coli cells per milliliter at the start of an experiment and one-fourth of the cells divided every 30 minutes?