**Modeling Bacterla Growth**

Bacteria have beneficial uses, such as in the production of foods, beverages, and medicines. On the other hand, some bacteria are important indicators of poor environmental quality. Engineers in the food, environmental, and chemical industries often must understand and be able to model the growth of bacteria. The following table gives data on the growth of a certain bacteria population with time. Fit an equation to this data.

**• Solution**

There is no simple equation that describes bacterial growth under a variety of conditions, so we do not have a predetermined mathematical function to use. The exponential growth

law, y = be'” or its equivalent form y = b( 10 )’11 I, sometimes fits the data, and we will see if it works here. There are other functions that could be tried, but for brevity, here we will try three polynomial fits (linear, quadratic, and cubic), and an exponential fit. We will

examine their residuals to determine which best fits the data. The script file is given below. Note that we can write the exponential form as y = b(lo)m, = 10 m,+a, where b = 10″. The coefficients a and m will be obtained with the poly fit function. % Time data

x = [0:19]; % Population data y = [6,13,23,33,54,83,118,156,210,282, 350,440,557,685,815,990,1170,1350,1575,1830];

% Linear fit

pl = poly fit(x,y,1);

% Quadratic fit

p 2 = poly fit(x,y,2);

% Cubic fit

p3 = polyfit(x,y,3);

% Exponential fit

p4 = polyfit(x,log10(y) ,1);

% Residuals

res1 polyval(p1,x)-y;

res2 polyval(p2,x)-y;

res3 polyval(p3,x)-y;

res4 10.Apolyval(p4,x)-y;

Youcan then plot the residuals as shown in Figure 5.6-7. Note that there is a definite pattern in the residuals of the linear fit. This indicates that the linear function cannot match the curvature of the data The residuals of the quadratic fit are much smaller, but there is still a pattern, with a random component. This indicates that the quadratic function also cannot match the curvature of the data. The residuals of the cubic fit are even smaller, with no strong pattern and a large random component. This indicates that a polynomial degree higher than three wiIl not be able to match the data curvature any better than the cubic. The residuals for the exponential are the largest of all, and indicate a poor fit. Note also

how the residuals systematically increase with t, indicating that the exponential cannot describe the data’s behavior after a certain time.

Thus the cubic is the best fit of the four models considered. Its coefficient of determination is r 2 = 0.9999. The model is

y = 0.1916(3 + 1.208212 + 3.6071 + 7.7307

where y is the bacteria population in ppm and ( is time in minutes.