Population Regulation For all these questions about population growth and models describing it, the common observation in nature is that most populations of plants and animals seem to remain fairly constant in size from year to year… An important question in ecology is what mechanisms
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For all these questions about population growth and models
describing it, the common observation in nature is that
most populations of plants and animals seem to remain
fairly constant in size from year to year…
An important question in ecology is what mechanisms
“regulate” or “control” population size?
If those populations seem to be in equilibrium, are they
near K? What keeps a population near K?
The most straightforward chain of argument goes…
Resources Become Limiting
Competition Among Individuals for Resources
Competition has Effects on Birth and Death Rates
Does this “regulate” or just limit population size?
What is regulation?
The ‘engineering-style’ definition:
The amplitude of any perturbation to a variable at its
set point will be decreased by regulation to restore the
variable to its set point.
A practical example:
The thermostat in your house is set to a specific
temperature (“the set point”). Should temperature in your
house increase, the thermostat turns on your central air
conditioning to bring the house temperature back down (or
vice versa when it becomes to cold and the furnace is
The idea that populations are “regulated” was highly
There were two opposed schools…
David Lack (e.g.1954) argued that population size was
regulated by food, predators, and disease, i.e. by biotic
Andrewartha & Birch, at around the same time, claimed
that numbers were determined by factors extrinsic to the
population acting on it. For example, r is strongly affected
The data used by Andrewartha & Birch came from studies
of thrips (Thrips imaginis) growing in roses. They were able
to predict population size from past size and weather in the
previous fall and current spring fairly well. They found
little evidence of density-dependence.
Here are observed and predicted numbers at peak for a
number of years, using only previous numbers and weather
They also showed that r was strongly affected by
temperature and moisture, key variables in climatic pattern.
cool & dry0.01
cool & moist0.03
warm & dry0.01
moderate & moist0.1
Does this mean that thrips are “regulated” entirely in a
density-independent way? The possibility caused a long
controversy. For this case, it was largely settled in 1961 by
What would indicate density-dependence?
1) What happens to per capita
growth rate as the population
approaches K? Per capita growth
rate declines. So should the
change per unit time in lnN.
Here’s what Smith found in
Andrewartha & Birch’s data…
(Oct. to Nov. is growth to peak)
2) Smith’s 2nd argument came from variance in population
size and growth. A basic principle in statistics is that the
variance of a sum of two independent variables should
be the sum of the variances of the individual variables, i.e.
Var (X + Y) = Var (X) + Var (Y)
Look at population growth…
ln N(t + 1) = ln N(t) + ln N(t)
but in the data the variance
is much smaller as the mean
N(t) (or ln N(t)) approaches
its annual peak. Again, here’s
the figure showing that...
For the negative relationship between numbers and the
change in numbers as the peak density is approached to
be as strong as this, there must have been a strongly
negative covariance between the variables (ln N(t) and
Δln N(t). That is exactly what would be expected in
logistic growth. Growth rate (or Δln N(t)) should
decrease as population size increases.
The argument developed, in part, because of the kinds of
species studied by supporters of the two points of view...
Supporters of Andrewartha & Birch studied insects, for
which growth and mortality are strongly affected by
Supporters of Lack generally were working with vertebrate
species, where behaviour (territory defense) and interactions
(competition and predation) often apparently limit population
Actually, this long lasting argument should never have
occurred. Darwin expressed clearly how both abiotic
(density-independent) and biotic (density-dependent) factors
interact to determine population size and growth…
“Climate plays an important part in determining the
average number of species, and periodical seasons of
extreme cold or drought seem to be the most effective
of all checks…The action of climate seems at first to be
quite independent of the struggle for existence; but in
so far as climate chiefly acts in reducing food, it brings
on the most severe struggle between the individuals,
whether of the same or distinct species, which subsist
on the same kind of food.”
There are many other examples of apparently density-independent dynamics in populations (many are studies of insects!).
In a grain weevil, for example, the intrinsic rate of increase (r) varies 10-fold with minor changes in humidity and temperature in environmental chambers.
Wouldn’t we expect insect growth, then, to be sensitive to environmental variation (drought, heavy rainfall, extreme cold or heat) in the real world?
In the end, what we consider to be the critical factor depends on the organism we are studying.
occurs, and affects r, those
effects are manifest through
changes in birth rate and
death rate. Here is what the
separate relationships look like
in abstract form...
for each individual
The result of changes in both birth and death rates is an
equilibrium population size, K. At size K, birth and death
rates are equal, i.e.
b = d
Population size K is called a stable point.
Now let’s compare the birth rate that would be observed
when population “regulation” is density-independent versus
Birth rate is not a function of density when “regulation” is
density-independent. In fact, this shouldn’t be called “regulation”.
And the death rate under each situation...
Similarly, when “regulation” is density-independent there
is no relationship between death rate and population size or
density (and no ‘regulation).
Density-independent birth and death rates tell us that
crowding is not important in populations “regulated” in
Where population growth is density-independent, there is
no tendency for the population to return to an equilibrium
value, or K.
In fact, K is not defined under those conditions. There is no
unique density where b = d, and therefore no equilibrium.
So, what might a graph of population size over time look
like for a density-independent population? ...
Is there an equilibrium evident? No.
Is there a pattern evident? Not with respect to population
In some species, population regulation is apparent, and
population size fluctuates around K because birth and death
rates are density-dependent.
The logistic model you have seen and used is one (and
literally the simplest) model of density-dependence. The
growth rate (effective r) decreases as N increases, due to a
decrease in the birth rate and/or an increase in the death rate.
The effective r decreases to 0 at population size K.
remember dN/dt = rN(K – N/K)
effective r = r(K – N/K)
Although many other species fit this pattern, it was first
described and widely fits data for vertebrate species.
Here are a couple of examples to show you that the model
does apply to other populations, as well:
An experiment where aphids were introduced onto individual
pea plants, and population growth on those pea plants was
followed. (Interval between ticks on the x-axis is 2 days.)
These are willows in England after myxamatosis essentially
wiped out the population of rabbits that had eaten most
seedlings. Thus, here it is not a new population, but one that
is starting from small numbers due to removal of predation.
Finally, logistic growth in an ant colony in Brazil. The ants
weren’t counted directly, but the size of the colony is directly
proportional to the number of craters that surround nest
In plants, population regulation incorporates a “second
level”. In a densely planted population, there is
mortality, but the surviving
individuals grow. Individual
plant weight increases as
density decreases. The process
is called self-thinning. This
figure demonstrates it for
horseweed (Erigeron canadensis)
The “trajectory” of plant self-thinning is well established.
The slope of a plot of log individual plant weight against
log plant density is -3/2. Therefore, self-thinning is called
the -3/2 power law. Since plants all follow the same law,
plots for much different species follow parallel lines...
In still other species, regulation by density-dependent
birth and death rates is present, but the regulation is “loose”.
When this occurs, population size may depart substantially
from K (or K may vary substantially).
Such loose regulation occurs when birth and death rates
have a range of possible values at any population size. Here
we cannot establish a single-valued function relating either/
or birth and death rate to density.
This is called “density-vague” regulation.
Here is what you might see as a population trajectory. With
density-vague regulation, it may reflect continuous variation
in K… or it may reflect density-vague responses in birth
and death rates.
How can such variation occur in a regulated population?
Here is an abstract view of the ranges of birth and death
rates possible plotted over a range of population sizes…
K might be any value
in the indicated range,
i.e. anywhere within
the ~diamond shaped
box, depending on
Andrewartha, H.G. and L.C. Birch (1954) The Distribution and
Abundance of Animals. Univ. Chicago Press, Chicago
Davidson, J. and H.G. Andrewartha (1948) The influence of rainfall,
evaporation and atmospheric temperature on fluctuations in the size
of a natural population of Thrips imaginis (Thysanoptera). J. Anim.
Lack, D. (1954) The Natural Regulation of Animal Numbers. Oxford
Univ. Press, New York, N.Y.
Smith, F.E. (1961) Ecology, 42:403-7.