Energy and Human Ambitions on a Finite Planet, 2021a

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Definition 3.2.3 Overshoot is a generic consequence of delaying negative
feedback. Since negative feedback is a “corrective,” stabilizing influence,
delaying its application allows the system to “get away” from the control,
thereby exceeding the target equilibrium state.
This is a pretty easy concept to underst<strong>and</strong>. The logistic curve of Figure
3.7 first accelerates, then briefly coasts before decelerating to arrive
smoothly at a target. Following an example from [1], it is much like
a car starting from rest by accelerating before applying the brakes to
gently come to a stop when the bumper barely kisses a brick wall.
The driver is operating a negative feedback loop: seeing/sensing the
proximity to the wall <strong>and</strong> slowing down accordingly. The closer to the
wall, the slower the driver goes until lightly touching the wall. Now
imagine delaying the feedback to the driver by applying a blindfold <strong>and</strong>
giving voice descriptions of the proximity to the wall, so that decisions
about how much to brake are based on conditions from a delayed
communication process. Obviously, the driver will crash into the wall
if the feedback is delayed, unless slowing down the whole process
dramatically. Likewise, if the negative consequences—signals that we
need to slow down population growth—arrive decades after the act of
producing more humans, we can expect to exceed the “natural” limit,
Q—a condition called overshoot.
By “generic consequence,” we just mean an
outcome that is characteristic of the situation,
independent of details.
[1]: Meadows et al. (1974), The Limits to
Growth: A Report for the Club of Rome’s Project
on the Predicament of Mankind
Another example of feedback delay leading
to overshoot: let’s say you are holding down
the space bar <strong>and</strong> trying to position the
cursor in the middle of the screen. But your
connection is lagging <strong>and</strong> even though you
release the space bar when you see the cursor
reach the middle, it keeps sailing past due
to the delay: overshooting.
Example 3.2.4 We did not detail the mechanisms of negative feedback
operating on the deer population in Example 3.2.3 that act to stabilize
the population at Q, but to illustrate how delayed negative feedback
produces overshoot, consider predation as one of the operating forces.
To put some simple numbers on it, let’s say that steady state can
support one adult (hunting) mountain lion for every 50 deer. Initially,
when the population was 100 deer, this means two predators. When
the deer population reaches Q 840, we might have ∼17 predators.
But it takes time for the predators to react to the growing number of
prey, perhaps taking a few years to produce the requisite number of
hunting adults. Lacking the full complement of predators, the deer
population will sail past the 840 mark until the predator population
rises to establish the ultimate balance. In fact, the predators will likely
also exceed their steady population in a game of catch-up that leads
to oscillations like those seen in Figure 3.8.
We can explore what happens to our logistic curve if the negative
feedback is delayed by various amounts. Figure 3.8 gives a few examples
of overshoot as the delay increases. To avoid significant overshoot, the
delay (τ) needs to be smaller than the natural timescale governing the
problem: 1/r, where r is the rate in Eqs. 3.5 <strong>and</strong> 3.6. In our deer example
using r 0.5, any delay longer than about 2 years causes overshoot. For
more modest growth rates (human populations), relevant delays are in
decades (see Box 3.1).
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.