Instructions for printer-friendly copy.

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(photo by J.C. Stahl)

    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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    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.

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    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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    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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    Uniform Distribution

    Uniform dispersion is a relatively uncommon pattern.

    It can result from negative intraspecific interactions such as

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    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,
    and adjusts your life insurance rate accordingly.)

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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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    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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    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)

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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.

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    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

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(photo by Yuri Arcurs, iStock)

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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    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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    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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    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.

    Start with a population of 10 individuals:

    • 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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    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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    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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    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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(click for source)

    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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(click for source)

    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

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

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    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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    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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    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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(source: National Geographic)

    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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    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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    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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    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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(click for source)


(click for source)

    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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(click for source)

    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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    Predictions

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

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    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?

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    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.

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    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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    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.

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    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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    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.

    <-- But here's what actually happened on the set.