Brief review of previous lecture

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Brief review of previous lecture

• Some history on Pearl and Lotka the logistic
  equation

• THE DEBATE in population ecology regarding
  importance of density-dependent processes in
  determining animal abundances
• (especially Nicholson vs. Andrewartha Birch).

• Classical density dependence vs. density vagueness

• Issues in detecting density dependence from time
  series

• Population regulation vs. population limitation

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Lecture Outline Age-structured Matrix Models

• Incorporating age-specific survival and fecundity
  into population growth models using matrix
  projections.

• Calculating age-specific survival and fecundity
  from a multi-year census.

• Setting up and projecting a Leslie Matrix

• Stable age distributions

• Lambda for age-structured population

• Reproductive value

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• Exponential and logistic-type growth models
  assume that population has no age or size
  structure.

• However, survival and reproduction often differ
  among individuals of different ages.

• Consideration of age-specific vital rates can be
  critical in many areas of wildlife conservation
  translocations and reintroductions, harvesting,
  population viability analysis (PVA), control of
  invasive species.

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General approach
Nt 1 ? Nt
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• Focus on birth-pulse models

• Often assume closed population, but approach can
  be extended to include dispersal

• Model both sexes or only females (e.g., fecundity
  is the number of daughters per adult female)

• Assume no variation among individuals within an
  age class

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Example using hypothetical data for helmeted
honeyeater

• Riparian forests of Victoria, Australia

• Pairs occupy territories within colonies

• Dramatic decline following European settlement
  and habitat clearing

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Data from multi-year census
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Calculating Survival Rates
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• Means of three yearly estimates.
• Use for mean matrix and deterministic projection

• Can use variation among estimates to add
  stochasticity

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Calculating Fecundity

• Fecundity for a given year is number of offspring
  produced in that year that survive to the next
  year divided by number of potential parents in
  that year.

• Age-specific fecundity is average number of
  offspring per individual of age x at time t that
  are counted at time t1.

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Calculating average fecundity

• If fecundity varies among age classes, then need
  to use different approach (e.g., counts of
  fledglings per nest).

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Age-specific fecundities should be estimated as
the average over all individuals in age class,
not just the breeding ones.
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The Leslie Matrix

• Fecundities are elements of the top row.
• Survival rates are elements of the subdiagonal.

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A Composite age class consists of all individuals
of a certain age or older
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Full matrix
New matrix with composite age class
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Matrix Projection

• First, consider the equations for projecting age
  class abundances

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Projection with the Leslie matrix

(Example calculations on board)
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Initial population growth depends on initial age
distribution
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Stable age distribution
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• Repeatedly multiplying an age distribution by a
  Leslie matrix eventually will produce a stable
  age distribution.

(aka dominant right eigenvector of projection
matrix in matrix algebra)

• A stable age distribution for a population that
  is neither increasing nor decreasing is termed a
  stationary age distribution.

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Lambda of a Structured Population

• After a population reaches a stable age
  distribution, it will grow exponentially with
  rate equal to lambda.

• Lambda is termed the dominant eigenvalue of the
  projection matrix.

• Lambda is a long-term, deterministic measure of
  growth rate of a population in a constant
  environment.

• The stable age distribution and lambda are
  independent of the initial age distribution they
  depend on the projection matrix.

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Reproductive Value

• Another useful measure that can be calculated
  from a Leslie matrix is reproductive value (aka
  dominant left eigenvector).

• Reproductive value is the relative contribution
  to future population growth an individual in a
  certain age class is expected to make.

• Reproductive value equals the number of offspring
  an individual of a given age class will produce,
  expressed relative to newborns (i.e., first age
  class always equals 1.0).

• Both survival rates and fecundities affect
  reproductive values.

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Reproductive value for sparrowhawks with
senescence
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Reproductive values for the common frog
Pre-juveniles 1.0 Juveniles
60.8 Adults 271.0
How could reproductive values be useful in
wildlife conservation and management decisions?