Botkin Environmental Science Earth as Living Planet 8th txtbk

3.2 System Responses: Some Important Kinds of Flows 51
Population (millions)
800.0
700.0
600.0
500.0
400.0
300.0
Theoretical U.S. Population Growth
Starting at 2.5 million in 1776, as an exponential with 2.5%
increase a year and as a linear curve with 250,000 added each year.
Exponential Growth
Linear Growth
FIGURE 3.12 Curves 1 and 2: linear and
positive exponential. This graph shows
theoretical growth of the population of the
United States, starting with the 2.5 million
people estimated to have been here in 1776
and growing as an exponential and a linear
curve. Even though the linear curve adds
250,000 people a year—10% of the 1776
population—it greatly lags the exponential by
the beginning of the 20th century, reaching
fewer than 100 million people today, while the
exponential would have exceeded our current
population.
200.0
100.0
1776 1796 1816 1836
1856 1876 1896 1916 1936 1956 1976 1996
Year
1000
Number of individuals
900
800
700
600
500
400
300
Age dependent
mortality
Age independent
mortality
Sooty shearwater
Maximum age = 30
Initial mortality rate = .07
Annual increase in mortality = .01
200
100
(a)
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Age (years)
FIGURE 3.13 Negative exponential. Example: the decline in a population of a species of birds when there
are no births and the mortality rate is 7% per year. The upper curve is a pure negative exponential. (Source:
D.B. Botkin and R.S. Miller, 1974, Mortality rates and survival of birds, American Nat. 108: 181–192.)
(b)
Figure 3.12 shows both a linear relation and a positive
exponential relation. A linear relation is of the form
y = a +bx, where a is the y intercept (in this case, o) and b
is the slope of the line (change in y to change in x, where
y is the vertical axis and x the horizontal). The form of
the positive exponential curve is y = ax b , where a is the
y intercept (in this case o) and b is the slope. However, b
is a positive exponent (power). (See A Closer Look 3.2,
Exponential Growth.)
Figure 3.13 shows two examples of negative exponential
relations. Figure 3.14 is the logistic curve, which
often has the shape of a lazy S; the logistic carrying k is
the population eventually reached or approached, based
on environmental factors. The saturation (Michaelis-
Menton) curve (Figure 3.15) shows initial fast change,
followed by a leveling off at saturation. At the point of
saturation, the net CO 2 fixed (for soybean)is at a lightintensity
value of about 3,000. As light intensity increases
above about 3,000, net fixed CO 2 is nearly constant
(that is, fixed CO 2 saturates at light intensity of 3,000
and does not change if intensity increases).
Number of individuals
400
300
200
100
K=375
1 2 3 4 5 6
FIGURE 3.14 (a) The logistic curve. Growth of a population of a microorganism
in a laboratory test tube under constant conditions with a
constant supply of food. (From G.F. Gause, The Struggle for Existence.)
The logistic carrying capacity is k. If you take a population of such
bacteria into a laboratory and grow them under constant conditions,
you might get the population to change according to the curve above,
as Gause did in the 1930s with other microorganisms. (Source: D.B.
Botkin, Discordant Harmonies: A New Ecology for the 21st Century
[New York: Oxford University Press, 1990].)