Biology Reference
In-Depth Information
IV. THE LOGISTIC MODEL
The exercises from the last section raise some important questions. In
particular, the solutions of both the discrete and the continuous models
are unbounded functions, and therefore describe unlimited growth. In any
given environment, however, the factors that support growth—for
example, the availability of food or nesting sites—are limited. Any
environmental degradation, such as air or water pollution, may also
limit population growth. These limiting factors determine the carrying
capacity of an environment—the maximum number of organisms the
environmental system can support. This is the upper limit on a
sustainable population.
E XERCISE 1-4
What factors do you think would determine the carrying capacity for
human populations?
To illustrate that populations do not grow without limit, Figure 1-4
shows the growth of the same yeast culture from Table 1-4 throughout
the entire 18-hour data collection period (Carlson [1913]). Figure 1-4 also
contains the solution curve of our continuous model with r
0.49. As
anticipated, our model exhibits unlimited growth, while the actual yeast
culture appears to approach a maximum population size. One might
suppose that the decrease in growth rate is caused by depletion of the
yeast's food supply—namely, sugar. However, analysis of the medium
showed that sugar was still available (Richards [1928]). Rather, the
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0
0
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18
Time
FIGURE 1-4.
Predicted and actual values for yeast population. Comparison between the solution curve of the
model dP/dt ¼ rP(t) with r ¼ 0.49 (dashed line) and the 18-hour yeast growth data (solid line), after
Carlson (1913).
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