# Question

The population of a culture of yeast cells is studied in the laboratory to see the effects of limited resources (food, space) on population growth. At 2-h intervals, the size of the population (measured as total mass of yeast cells) is recorded.

(a) Make a graph of the yeast population as a function of elapsed time. Draw a best-fit smooth curve.

(b) Notice from the graph of part (a) that after a long time, the population asymptotically approaches a maximum known as the carrying capacity. From the graph, estimate the carrying capacity for this population.

(c) When the population is much smaller than the carrying capacity, the growth is expected to be exponential: m(t) = m0ert, where m is the population at any time t, m 0 is the initial population, r is the intrinsic growth rate (i.e., the growth rate in the absence of limits), and e is the base of natural logarithms. To obtain a straight line graph from this exponential relationship, we can plot the natural logarithm of m/ m0:

ln m / m0 = In en = rt

Make a graph of ln (m/m0) versus t from t = 0 to t = 6.0 h, and use it to estimate the intrinsic growth rate r for the yeast population.

Time (h) Mass (g)

0.0 ........................................... 3.2

2.0 ........................................... 5.9

4.0 ........................................... 10.8

6.0 ........................................... 19.1

8.0 ........................................... 31.2

10.0 .......................................... 46.5

12.0 .......................................... 62.0

14.0 .......................................... 74.9

16.0 .......................................... 83.7

18.0 .......................................... 89.3

20.0 .......................................... 92.5

22.0 .......................................... 94.0

24.0 .......................................... 95.1

(a) Make a graph of the yeast population as a function of elapsed time. Draw a best-fit smooth curve.

(b) Notice from the graph of part (a) that after a long time, the population asymptotically approaches a maximum known as the carrying capacity. From the graph, estimate the carrying capacity for this population.

(c) When the population is much smaller than the carrying capacity, the growth is expected to be exponential: m(t) = m0ert, where m is the population at any time t, m 0 is the initial population, r is the intrinsic growth rate (i.e., the growth rate in the absence of limits), and e is the base of natural logarithms. To obtain a straight line graph from this exponential relationship, we can plot the natural logarithm of m/ m0:

ln m / m0 = In en = rt

Make a graph of ln (m/m0) versus t from t = 0 to t = 6.0 h, and use it to estimate the intrinsic growth rate r for the yeast population.

Time (h) Mass (g)

0.0 ........................................... 3.2

2.0 ........................................... 5.9

4.0 ........................................... 10.8

6.0 ........................................... 19.1

8.0 ........................................... 31.2

10.0 .......................................... 46.5

12.0 .......................................... 62.0

14.0 .......................................... 74.9

16.0 .......................................... 83.7

18.0 .......................................... 89.3

20.0 .......................................... 92.5

22.0 .......................................... 94.0

24.0 .......................................... 95.1

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