Instructions for printer-friendly copy.

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## Population Growth and Regulation

Given ideal circumstances, populations
have the capacity to increase dramatically in size.

Population growth rate depends on both
intrinsic and extrinsic factors.

Population ecologists use mathematical models
to describe and predict theoretical population growth.

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x ## Density and Distribution

A population's density is the number of
individuals per unit area or volume.

A population's dispersion is the pattern of
spacing among individuals.

Spacing can be

• random
• clumped
• uniform

These patterns can aid prediction of
intraspecific interactions. x

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x ## Random Distribution

If members of a population exhibit
random dispersion, it means that there are no strong intraspecific factors controlling their distance
from each other.

Relative locations are determined
by chance.
Example: wind-dispersed plants

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x ## Clumped Distribution

Clumped dispersion is a relatively
common pattern. It may indicate

• local abundance of food or other resource
• mating swarms (temporary)
• social congregation (natural selection)
• a pack of wolves will hunt more successfully than a single wolf
• in a herd or flock, an individual has a better chance of escape

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x ## Uniform Distribution

Uniform dispersion is a relatively uncommon pattern.

It can result from negative intraspecific interactions such as x

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x ## Population Dynamics

A population's gain occurs via births and immigration.
A population's loss occurs via deaths and emigration.

• If population gain = loss, then population size will not change.

• If population gain > loss, then population size will increase.

• If population loss < gain, then population size will decrease.

Demography is the study of changes
in the vital statistics of populations over time.

(An actuary is a demographics expert who determines
how likely you are to be dead in the next year,

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## The Cohort

A cohort is a subset of a population consisting of
all individuals born in the same year.

Populations of ling-lived species have more cohorts
and more interactions among individuals of different ages.

Populations consisting of multiple cohorts
tend to be more resistant to extinction
than those consisting of only one or very few cohorts.

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x ## Life Tables

A life table is a representation of a population's age-specific survival,
based on the survivorship of a given cohort.

Click on the photo of Belding's Ground Squirrels (Urocitellus beldingi)
to view an example of a life table.

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x ## Survivorship Curves

Life table data can be plotted to yield an age-specific survivorship curve.
Three idealized curves represent common survivorship patterns.

• Type I
• low risk of juvenile death
• likelihood of death increases with advancing age

• Type II
• constant risk of death at all life stages

• Type III
• high risk of juvenile death
• long adult life expectancy (for its species) x

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## Population Growth

Populations are not always stable in size.

Births, deaths, immigration, and emigration do not necessarily generate a state of equilibrium.

Three different mathematical models can be used to describe the growth of populations.

• arithmetic growth
• exponential (geometric) growth
• logistic growth

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x ## Arithmetic Growth

The Arithmetic Model describes a population growth rate
that is unaffected by population size.

A population undergoing arithmetic growth
increases in size by the same increment over each time interval.

This is described by the equation

Nt+1 = Nt + B - D

where

• Nt = population size at time t
• B = number of births from time (t) to time (t+1)
• D = number of deaths from time (t) to time (t+1)
• (B - D) = C = net change in population size x

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Arithmetic growth generally does not occur in natural populations.

## Arithmetic Growth

Because immigration (I) is numerically equivalent to births
and emigration (E) is numerically equivalent to deaths,
the model can be modified to include them:

Nt+1 = Nt + (B + I) - (D + E)

Although the model assumes that deaths and births are
independent of population size, in reality, as population size increases...

• absolute number of births and deaths increases
• birth and death rates per individual remain the same

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x ## Exponential (=Geometric) Growth

The Exponential Model ...describes the growth of a
constantly reproducing population uninhibited by biotic or abiotic constraints.

A population undergoing exponential, or geometric growth
increases rapidly in size, with at its maximum intrinsic rate of growth.
This can be described by the equation:

Nt+1 = Nt + [(b - d) x Nt]

where

• Nt = population size at time t
• b = average # of births per female (fecundity)
• d = average # of deaths per capita (mortality)
• (b - d) = rate of change/individual/generation (net reproductive rate (r))
• (b - d) x Nt = net change in # for a given population size

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x  ## Exponential Growth: Biotic Potential

A population's maximum intrinsinc rate of increase (rmax)
(also known as its biotic potential) is the number of births
minus the number of deaths per generation time.

In this model...

• Population growth rate is proportional to population size.
• Population growth rate is independent of population density.
In a growing population, change in population size is not constant
because there are more individuals reproducing in each successive generation.

All organisms have the capacity to grow in this fashion.

A species' maximum intrinsic rate of increase (rmax)
is also known as its biotic potential.

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x ## Exponential Growth: An Example

Let's say in Species X:
• The average female has 5 babies per generation.
• Of the five offspring, an average of 2 die per generation.
• Intrinsic rate of increase (r) is 5 - 2 = 3.
• Population increase per generation is calculated by
multiplying each generation's population size by r.

• Generation 1 = 10
N of next generation will be 10 x 3, or 30. (r = 31)
• Generation 2 = 30.
N of next generation will be (10 x 3) x 3, or 90. (r = 32)
• Generation 3 = 90.
N of next generation will be [(10 x 3) 3] x 3, or 270. (r = 33)
• Generation 4 = 270.
N of next generation will be {[(10 x 3) x 3] x 3} x 3, or 810. (r = 34)

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x ## Exponential vs. Geometric Growth

Populations undergoing exponential or geometric growth
both exhibit a J-shaped growth curve.

But there is a difference between them.

• The exponential growth model applies to species that
reproduce throughout the year.

• The geometric growth model applies to species that
breed only over a limited time during the year.

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x ## Exponential Growth: Constant Reproduction

Exponential growth also can be expressed as: dN/dt = rN

where

• t = time interval
• N = initial popuation size
• r = intrinsic growth rate (birth rate - death rate)
• dN/dt = change in population size over a time interval

Remember...

• Population growth rate is proportional to population size.
• Population growth rate is independent of population density.

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x ## Geometric Growth: Intermittent Reproduction

Geometric growth can be expressed as: Nt = N0λ

where

 N0 = current population size Nt = future population size λ = ratio of population size at one time interval (e.g., year) to population size at the preceding time interval

• If births > deaths, then λ > 1.0
• If births < deaths, then λ < 1.0
• (λ cannot be negative, because population size cannot be negative.)

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## Doubling Time

Doubling time is the time required for an exponentially
growing population to double in size.

As r increases, doubling time decreases.

Biotic potential (r) varies considerably among species.

• Most small-bodied organisms grow quickly, both in body and population size.
• Large-bodied organisms grow more slowly, both in body and population size.

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## The Rule of 70

The doubling time (in years) for a population undergoing exponential growth
can be calculated with the Rule of 70.

70/r

• Growth rate must be expressed as a whole number.
• Growth rate of 5% must be entered as 5, not 0.05

For example:

A population with a 5% growth rate will double its size in 70/5, or 14 years!

(The alternative formula shown at the left also can be used.)

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## Population-limiting Factors

A population cannot grow exponentially forever.

Common results of high population density include

• reduced fecundity (why?)
• increased mortality (why?)
• increased rate of disease (why?)
• possibly increased per capita predation (due to altered predator "search image")
• increased contact with toxic waste produced by the population itself

A limiting factor is any environmental condition that prevents a population from attaining unlimited growth at r.

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x ## Density-dependent factors

Density-dependent factors increase in intensity
as population density increases.

Better adapted individuals will be more successful
under density-dependent pressures.

Density dependence may be negative or positive.

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x  ## Negative Density-dependence

A factor that causes a population's growth rate to decrease as population size increases is a negative density dependent factor.

Common negative factors include limiting resources such as:

 food physical space nesting sites sunlight (in a shady forest) water (in a xeric environment)

Examples:

• Fruit flies raised at different densities, same amount of food.
• Common Terns colonizing new nesting islands
• Plants experimentally grown at high densities
• smaller
• less fecund
• have lower survivorship
...than those grown at lower densities.

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x ## Positive Density-dependence

A factor that causes a population's growth rate to increase
as population size increases is a positive density dependent factor.

Typically occurs only at very low population density.

Moderate population increase can increase population growth rate:

• increases chances of finding a mate (or being pollinated)
• reduces inbreeding suppression
• can improve a male-skewed sex ratio

Example:

• Cowslip plants were pollen-limited at low pop'n density,
but showed improved seed production with slight population increase.

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## Density-independent factors

Density-independent factors do not affect population growth rate as population density increases.

• temperature
• rainfall
• sunlight (in an open prairie)
• natural disasters

Such factors affect all individuals in the population,
regardless of population density.

Periodic major disturbances

 major storms volcanic eruptions fires droughts floods tsunamis
...and other natural disasters can cause extreme population fluctuations, regardless of population density.

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x ## Logistic Growth

The Logistic Model describes the growth of a population
whose increase is inhibited by biotic and/or abiotic factors.

A population undergoing logistic growth exhibits
exponential growth eventually stabilized by environmental resistance.

A population's carrying capacity (K) is the maximum number
of individuals its environment can sustain indefinitely.

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x ## The Logistic Equation

This is described by the equation:

dN/dt = [(rmax)(N)] x [K - N/K]

where

• dN = incremental change in population size
• dt = time interval
• rmax = maximum population growth rate
• N = population size
• K = carrying capacity

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x ## Stability Follows the Crash

As a population reaches environmental carrying capacity,
it will sometimes overshoot K, and then
suffer a population "crash" before recovering.

This pattern may repeat multiple times,
with overshoot amplitude decreasing with each iteration.

Eventually, the population stabilizes at carrying capacity,
and remains so as long as the environment is stable.

Most natural populations exhibit logistic growth.

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## Life History Strategies: r-selected Species

At low population density, natural selection should favor
strategies that bring the population close to rmax.
• early sexual maturity
• short generation time
• increased fecundity

Species exhibiting these strategies are known as
r-selected or opportunistic.

Opportunistic species are most successful in
• unstable habitats
• newly colonized habitats (resources are not limiting)

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## Life History Strategies: K-selected Species

At high population density (at or close to K),
natural selection should favor
• ability to reproduce with few resources
• ability to compete well for limited resources
• ability to use resources very efficiently

S;pecies these strategies are known as
K-selected or equilibrial.

Equilibrial species establish in stable habitats
where population size remains relatively constant
as it approaches K.

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x ## Predictions

The logistic growth model predicts population growth
at both very high and very low population densities. x

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x ## Human Population Parameters

For its first 200,000 years of existence, our species has exhibited
population growth characteristics typical of a K-selected species.

Since the mid-1800s, however, human population growth
began to take an upswing.

From 1870 to 2000, in countries like the U.S.

• Mortality in children under 10 fell from 25% to less than 0.5%.
• Male survival from age 10 - 65 increased from ~40% to 87%.
• Life expectancy at age 65 increased from ~10 years to ~19 years.

To what might you attribute this change? x

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x ## Human Population Growth Rate

The U.S. population has continually increased since its inception.

In 1910, the growth rate began to decline.
Based on these data, Pearl and Reed (1920)
predicted U.S. population size would reach
a maximum of 197 million.

 But by 2018, the U.S. population was 327.2 million. Global human population has shown a similar trend, as shown in the figure to the left. x

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x ## Population Age Structure

The age structure of a population can be illustrated with an age structure pyramid.
The shape of the pyramid indicates whether population size is
• stable (highest frequency: reproductive adults)
• increasing (highest frequency: juveniles, young adults)
• decreasing (highest frequency: post-reproductive adults)

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x ## Birth and Death

• Infant mortality is the number of
infant deaths per 1000 live births.

• Life expectancy is the predicted
average survival of an individual at any given age.

• Infant mortality and life expectancy
are inversely correlated.

• Energy consumption and life expectancy
per capita are positively correlated. x

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x ## Fluctuating Population Size

Populations fluctuate in size naturally over time.

Climatic and seasonal factors can severely decrease populations.
Survivors

• benefit from increased resource availability
• may have increased reproductive output

A population near the maximum density its habitat can support
may be more strongly affected by small environmental changes.

These can magnify the effects of density-dependent factors,
causing significant fluctuations in population size.

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x ## Boom and Bust Cycles

Many species undergo seasonal "boom and bust" cycles
occurring with seasonal climate changes.

The effects of density-dependent factors are not always immediately apparent.
In many cases, there is a "time lag" between a factor's effect
and the resulting change in population density.

When climate changes affect population density of a prey species,
the numbers of its predators generally follow after a time lag.

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## Suicidal Lemmings: The Mythology

A very old meme is that of the suicidal lemming.

When lemming populations increase to a critical density, so it's said,
they disperse away from their birthplace.

Many are so desperate to escape crowding
that they jump off cliffs into the ocean and drown.

This "common knowledge" was popularized in the Disney documentary
White Wilderness, which was shown to tens of thousands of
elementary school kids in the 1960s.