Energy and Human Ambitions on a Finite Planet, 2021a

3 Population 33
3.2 Logistic Model
Absent human influence, the population of a particular animal species
on the planet might fluctuate on short timescales (year by year) <strong>and</strong>
experience large changes on very long timescales (centuries or longer).
But by-<strong>and</strong>-large nature finds a rough equilibrium. Overpopulation
proves to be temporary, as exhaustion of food resources, increased
predation, <strong>and</strong> in some cases disease (another form of predation, really)
knock back the population. 5 On the other h<strong>and</strong>, a small population
finds it easy to exp<strong>and</strong> into abundant food opportunities, <strong>and</strong> predators
reliant on the species have also scaled back due to lack of prey.
We have just described a form of negative feedback: corrective action to
remedy a maladjusted system back toward equilibrium.
Definition 3.2.1 Negative feedback simply means that a correction is
applied in a direction opposite the recent motion. If a pendulum moves to the
right, a restoring force pushes it back to the left, while moving too far to the
left results in a rightward push. A mass oscillating on a spring demonstrates
similar characteristics, as must all equilibrium phenomena.
We can make a simple model for how a population might evolve in an
environment hosting negative feedback. When a population is small <strong>and</strong>
resources are abundant, the birth rate is proportional to the population.
Example 3.2.1 If a forest has 100 breeding-aged deer, or 50 couples, we
can expect 50 fawns in a year (under the simplifying <strong>and</strong> unimportant
assumption of one fawn per female per year). If the forest has 200
deer, we can expect 100 fawns. The birth rate is simply proportional to
the population capable of giving birth. 6
5: For reference, the SARS-CoV2 p<strong>and</strong>emic
of 2020 barely impacted global population
growth rates. When population grows by
more than 80 million each year, a disease
killing even a few million people barely
registers as a hit to the broader trend.
The word negative may sound like something
we would not want, but its cousin—
positive feedback—leads to disastrous runaway
conditions. An example of positive
feedback is the bacteria example from Chapter
1: having more bacteria only increases
the rate of growth. Exponentials are the
hallmark of positive feedback, while equilibrium
signals negative feedback.
6: ...nonegative feedback yet
If the setup in Example 3.2.1 were the only element to the story, we
would find exponential growth: more offspring means a larger population,
which ultimately reaches breeding age to produce an even larger
population. 7 But as the population grows, negative feedback will begin
to play a role. We will denote the population as P, <strong>and</strong> its rate of change
as We might say that the growth rate, or Ṗ,is
7: We have just described a state of positive
feedback: more begets more.
Ṗ.8
8: Ṗ is a time derivative (note the dot on
top), defined as Ṗ dP/dt. But don’t panic
if calculus is not your thing: what we describe
here is still totally
Ṗ rP, (3.3)
underst<strong>and</strong>able.
where r represents the birth rate in proportion to the population (e.g.,
0.04 if 4% of the population will give birth in a year). 9 This equation 9: In terms of the growth rate we used
just re-iterates the simple idea that the rate of population growth is before, p,asinEq. 3.1, r ln(1 + p).Sofor
instance, if growing at 2%, p 0.02 <strong>and</strong> r
dependent on (proportional to) the present population. The solution to
also is 0.02 (r ≈ p for small values of p).
this differential equation is an exponential:
P P 0 e r(t−t 0) , (3.4)
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.