Energy and Human Ambitions on a Finite Planet, 2021a

3 Population 37
1.8
population as fraction of steady-state max, Q
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
¿ = 0.0/r
¿ = 0.5/r
¿ = 1.0/r
¿ = 1.5/r
¿ = 2.0/r
−10 −5 0 5 10 15 20
time relative to nominal midpoint
Figure 3.8: Feedback delay generally results
in overshoot <strong>and</strong> oscillation, shown for various
delay values, τ. The black curve (τ 0)
is the nominal no-delay logistic curve. As
the delay increases, the severity of overshoot
increases. Delays are explored in increments
of 0.5 times the characteristic timescale of
1/r (using r 0.5 here to match previous examples,
so that a delay of τ 1.5/r equates
to 3 time units on the graph, for instance).
The delay durations are also indicated by
bar lengths in the legend.
Eventually all the curves in Figure 3.8 converge to the steady state
value of 1.0, 15 but human population involves complexities not captured
in this bare-bones mathematical model. 16 All the same, the generic
phenomenon of overshooting when negative feedback is delayed is a
robust attribute, even if the oscillation <strong>and</strong> eventual settling does not
capture the future of human population well.
15: . . . meaning that population P arrives
at Q
16: For instance, a dramatic overshoot <strong>and</strong>
collapse could be disruptive enough to take
out our current infrastructure for fossil-fuelaided
agriculture so that the Q value essentially
resets to some lower value.
12
10
<strong>Human</strong> Population (Gppl)
8
6
4
2
0
1800 1850 1900 1950 2000 2050 2100 2150 2200
year
Figure 3.9: <strong>Human</strong> population data points
(blue) <strong>and</strong> a logistic curve (red) that represents
the best fit to data points from 1950
onward. The resulting logistic function has
Q ≈ 12 Gppl, r 0.028, <strong>and</strong> a midpoint at
the year 1997. The actual data sequence has
a sudden bend at 1950 (Green Revolution?)
that prevents a suitable fit to a larger span of
data. In other words, the actual data do not
follow a single logistic function very well,
which is to be expected when conditions
change suddenly (energy <strong>and</strong> technology,
in this case) [14, 15].
Box 3.1: Will <strong>Human</strong> Population Overshoot?
Are humans in danger of population overshoot? What is our r value?
It is tempting to take r 0.01 corresponding to the present 1%
growth rate. This would imply that any delay shorter than 100 years
will not produce significant overshoot, which seems reassuring. But
if human population is following a logistic curve rather than an
exponential, resource availability is already exerting a moderating
influence, now appearing to be in the linear “cruise” phase roughly
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.