Demographic techniques

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Demographic techniques

• Vital Statistics
• Chapter 10

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Life Tables

• Life tables summarize demographic information
  (typically for females) in a convenient format,
  including
• age (x)
• number alive
• survivorship (lx) lx s0s1s2s3 ... sx-1
• mortality rate (mx)
• probability of survival between x and x1 (sx)
• fecundity (bx)

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Survivorship Schedule l(x)

• S(x) is the number of survivors born to a
  particular cohort of age x
• S(x) converted to l(x) as the proportion of the
  original cohort that survives to age x

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Survival Probabilty g(x)

• Survival probability is the probability of
  surviving from age x to age x1
• It is calculated as g(x)l(x1)/l(x)

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Types of Survivorship Curves

• Type I Survivorship Curve
• High survivorship during young and intermediate
  ages, low during old age (humans)
• Type II Survivorship Curve
• Intermediate in all age classes, straight line
  when plotted logarithmically (birds)
• Type III Survivorship Curve
• High at young ages, lower at older ages (insects)

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Data for Life Tables

• Cohort life tables are based on data collected
  from a group of individuals born at the same time
  and followed throughout their lives
• difficult to apply to mobile and/or long-lived
  animals
• Static life tables consider survival of
  individuals of known age during a single time
  interval
• Age at death observed
• Age structure directly observed
• Both require some means of determining ages of
  individual
• Both assume populations are in equilibrium and
  environment does not change

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Cohort Life table of a grass
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Static Life Table of a sheep
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The Intrinsic Rate of Increase 1

• The Malthusian parameter (rm) or intrinsic rate
  of increase is the exponential rate of increase
  (r) assumed by a population with a stable age
  distribution.
• rm is approximated (ra) by performing several
  computations on a life table, starting with
  computation of R0, the net repro-ductive rate,
  (Slxbx) across all age classes.

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The Intrinsic Rate of Increase 2

• The net reproductive rate, R0, is the expected
  total number of offspring of an individual over
  the course of her life span.
• R0 1 represents the replacement rate
• R0 lt 1 represents a declining population
• R0 gt 1 represents an increasing population
• The generation time for the population is
  calculated as T Sxlxbx/Slxbx

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The Intrinsic Rate of Increase 3

• Computation of ra is based on R0 and T as
  follows
• ra logeR0/T
• Clearly, the intrinsic rate of natural increase
  depends on both the net reproductive rate and the
  generation time
• large values of R0 and small values of T lead to
  the most rapid population growth

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Stable Age Structure

• An interesting feature of life tables is that
  projections into the future of populations with
  age structure will show a stable age structure
• This will occur after some time period during
  which the age structure will fluctuate
• Will eventually settle down

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Population Growth 1

• Exponential growth results in a continuously
  accelerating curve of increase (or continuously
  decelerating curve of decrease).
• The rate at which individuals are added to the
  population is
• dN/dt rN
• This equation encompasses two principles
• the exponential growth rate (r) expresses
  population increase on a per individual basis
• the rate of increase (dN/dt) varies in direct
  proportion to N

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Exponential Population Growth

• A population exhibiting exponential growth has a
  smooth curve of population increase as a function
  of time.
• The equation describing such growth is
• N(t) N(0)ert
• where N(t) number of individuals after t time
  units
• N(0) initial population size
• r exponential growth rate
• e base of the natural logarithms (about 2.72)

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Life Table (1)
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Projecting life tables through time
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Age distribution and total size
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Age structure and Growth rate
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Calculation of Exponential rate of increase
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Instantaneous and finite rate of increase are
related.

• Instantaneous rate and finite rate growth
  equations describe the same data equally well.
• These models are related by
• ? er
• and
• loge ? r

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Environmental conditions and intrinsic rates of
increase.

• The intrinsic rate of increase depends on how
  individuals perform in that populations
  environment.
• Individuals from the same population subjected to
  different conditions can establish the reaction
  norm for intrinsic rate of increase across a
  range of conditions
• these vary within and between species

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Summary

• When birth and death rates vary by age,
  predicting future population growth requires
  knowledge of age-specific survival and fecundity.
• Life tables summarize demographic data.
• Analyses of life table data permit determination
  of population growth rates and stable age
  distributions.

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Summary

• Populations have potential for explosive growth
  but this is dependent on the following
  assumptions
• Stable age structure
• Constant age-specific mortality and fertility
  rates
• r can be thought as the theoretical constant to
  compare actual growth rates in nature

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• Growth rate (r ) generally not associated with
  abundance
• But can recover faster from disturbances
• r will increase if
• Reproduce sooner
• Reproduce more often
• Reproduce more productively (more offspring)
• Can offset poorer survival

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Other terms

• Reproductive Value v(x)
• is the relative number of offspring than remain
  to be born to individuals of a given age
• v (x)

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Reproductive Value Continued

• Reproductive value of newborns is always set at 1
• Reproductive value of older age classes is
  relative to that of newborns
• Reproductive value reflects survivorship to its
  current age, its survivorship and reproduction
  during future ages, and the magnitude of r.
• Typically, v(x) peaks near age of first
  reproduction, then drops off at older ages.
• Natural Selection operates most strongly on
  individuals with high reproductive value
  predators will impact populations if they focus
  on individuals with high reproductive value

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Age structures

• Stable age structure
• Population growing exponentially with fixed
  life-table parameters
• Stationary age structure
• Fertility balances mortality
• No population growth
• Takes form of the lx distribution

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Evolution of demographic traits

• Big-bang (semelparous) vs repeated (iteroparous)
  reproduction
• Trade off involves gains in reproductive success
  by repeated breeding
• Big-bang favored when benefits occur only at high
  levels of reproductive effort
• Repeated if benefits can occur at low levels of
  effort
• Environmental variability will tend to select for
  repeated breeding