Population Regulation For all these questions about population growth and models

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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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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

“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…

Increased N

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

turned on).

The idea that populations are “regulated” was highly

controversial…

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

factors.

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

by weather.

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

factors...

They also showed that r was strongly affected by

temperature and moisture, key variables in climatic pattern.

Climater

cool & dry 0.01

cool & moist 0.03

warm & dry 0.01

moderate & moist 0.1

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

F.E. Smith…

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

weather.

Supporters of Lack generally were working with vertebrate

species, where behaviour (territory defense) and interactions

(competition and predation) often apparently limit population

size.

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.

When density-dependence

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...

Intense

competition

Resources limited

Plentiful resources

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

density-dependent...

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

that way.

Where population growth is density-independent, there is

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

size.

Organisms showing density-dependent “regulation” of

• population size…
• appear to have a carrying capacity K
• are limited by resources
• However, even these species may show changes in
• population size near K.
• What we observe in some species could be described as
• “loose” regulation, and the population is not necessarily
• kept close to K. One cause may be environmental variation
• altering the effective K. We’ll return to this idea at the end.

In many species, under real environmental conditions

• there is no clear regulation of population size because at
• the densities occurring in nature birth and death rates are
• effectively density-independent.
• In these species, patterns of population change are
• opportunistic. Populations grow rapidly (exponentially)
• when conditions are good.
• Exponential growth is followed by large crashes in
• numbers when conditions worsen.
• This is the pattern usually seen in insects and weedy plants
• with annual life cycles.

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.

weren’t counted directly, but the size of the colony is directly

proportional to the number of craters that surround nest

entrances.

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

The “trajectory” of plant self-thinning is well established.

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...

Finally,…

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

population size.

References:

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.

Ecol. 17:200-222.

Lack, D. (1954) The Natural Regulation of Animal Numbers. Oxford

Univ. Press, New York, N.Y.

Smith, F.E. (1961) Ecology, 42:403-7.