We have studied the discrete logistic growth model and seen the difficulties computing and analyzing populations using this model. Most biologists use the continuous version of the logistic growth model for their studies of populations. This model is used very extensively and can be written with the following formula

where P0 is the initial population, M is the carrying capacity of the population, and r is the Malthusian growth rate (early exponential growth rate) of the culture. Below is a table with data from Gause [1] on a growing culture of the yeast Schizosaccharomyces kephir (a contaminant culture of brewers yeast)

Time (hr)

0

14

33

57

102

126

Volume

1.27

1.7

2.73

4.87

5.8

5.83

 

a. Use Excel to find the best values of parameters P0, M, and r, then write these values clearly in your lab report. (The Help page gives details on using Excel's Solver for finding these parameters.) Include the sum of squares error. Also, write the complete formula with the best parameter fit in your report. If the value of r gives the Malthusian growth rate for low populations, then use this to determine the doubling time for this culture of yeast. As noted above, M is the carrying capacity of the population. Give a brief biological interpretation of this parameter and describe what your value of M says about what happens to this experimental culture of yeast. Create a graph showing both the data and the logistic growth function, p(t).

b. The growth rate for a culture can be found by taking the derivative of the population function. Differentiate the logistic growth function p(t) with the parameters found in Part a. and write this formula in your lab report. Create a graph of the derivative of the logistic growth function, p'(t).

c. The turning point of the population or the mid-log phase for this culture of yeast is where the growth of the culture is at a maximum. (This is also the point of inflection for the original logistic growth function, p(t).) Find when the logistic growth function reaches the turning point by finding the maximum of the derivative of the logistic growth function, p'(t). Write the time of the turning point, the maximum growth that you find, and the population (volume) of the culture at this time.

[1] G. F. Gause, Struggle for Existence, Hafner, New York, 1934.