Here we obtain the equation for the per capita growth rate of the yeast population given at the bottom of Page 423 of the textbook.
The yeast population size was measured every hour and is given in Table 1 of Section 7.1 (these data can also be downloaded from the link on the left). A rough approximation for the per capita growth rate, as a function of population size, can be obtained from these data as follows.
(1) Calculate the difference in population size between each successive point in time. If N(t) and N(t+1) are successive measurements at hour t and t+1, then we calculate N(t+1)-N(t) for all times from t=0 to t=35.
(2) Divide the results from part (1) by the time that has elapsed between successive measurements (namely 1 hour). This gives a rough approximation for the slope dN/dt as we have seen in Section 3.1 of the textbook.
(3) Divide the results from part (2) by N(t) to obtain a rough approximation for the per capita rate of change,
at each time from t=0 to t=35.
(4) Plot the results from part (3) against the value of N(t) at each time from t=0 to t=35.
(5) Fit a linear relationship to the scatter plot from part (4). This will give a function for the per capita growth rate of the form
where b is the y-intercept and m is the slope.
The following animation displays the scatter plot resulting from part (4) above and it allows you to explore different choices for the slope m and intercept b of the linear fit.
In Exercise 11.3.25 of the textbook we conduct a formal statistical analysis of the data to show that b=0.55 and m=-0.0026.