Energy and Human Ambitions on a Finite Planet, 2021a

3 Population 35
1.0
population, P, as fraction of maximum, Q
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
inflection point (linear phase)
population, P
halfway to Q at t = t 0
0.0
−12 −10 −8 −6 −4 −2 0 2 4 6 8 10 12
time relative to midpoint (t ¡ t 0 )
Figure 3.7: Logistic population curve (blue),
sometimes called an S-curve, as given in Eq.
3.6, in this case plotting for r 0.5 to match
examples in the text. The red curve is the
exponential that would result without any
negative feedback.
<strong>and</strong> suppress the rate of growth until it stops growing altogether when
P reaches Q.
Example 3.2.3 Continuing the deer scenario, let’s say the forest can
ultimately support 840 adults, 12 <strong>and</strong> keep r 0.5 as the uninhibited 12: . . . tuned for a convenient match to the
growth rate. Using these numbers, Eq. 3.6 yields 100 adults at t t numbers we have used in the foregoing
0 −4
examples
years (effectively the initial state in Example 3.2.1). One year later, at
t t 0 − 3, Eq. 3.6 yields 153—very close to the nominal addition of 50
members. But now four years in (t t 0 ), we have 420 instead of the
13
739 we got under unrestricted exponential growth in Example 3.2.2.
13: Not coincidentally, P Q/2 at the halfway
point, t t 0 .
The logistic curve is the dream scenario: no drama. The population simply
approaches its ultimate value smoothly, in a tidy manner. We might
imagine or hope that human population follows a similar path. Maybe
the fact that we’ve hit a linear phase—consistently adding one billion
people every 12 years, lately—is a sign that we are at the inflection, <strong>and</strong>
will start rolling over toward a stable endpoint. If so, we know from the
logistic curve that the linear part is halfway to the final population.
Three consecutive 12-year intervals appear
in Table 3.2. If the middle one is the midpoint
of a logistic linear phase—in 2011 at
7 billion people—it would suggest an ultimate
population of 14 billion.
3.2.1 Overshoot
But not so fast. We left out a crucial piece: feedback delay. The math
that leads to the logistic curve assumes that the negative feedback 14 acts
instantaneously in determining population rates.
14: . . . based on remaining resources, Q −P,
at the moment in Eq. 3.5
Consider that human decisions to procreate are based on present conditions:
food, opportunities, stability, etc. But humans live for many
decades, <strong>and</strong> do not impose their full toll on the system until many years
after birth, effectively delaying the negative feedback. The logistic curve
<strong>and</strong> equation that guided it had no delay built in.
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.