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Brief review of previous lecture

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Issues in detecting density dependence from time series ... Model both sexes or only females (e.g., fecundity is the number of daughters per adult female) ... – PowerPoint PPT presentation

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Title: Brief review of previous lecture


1
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

2
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

3
  • 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.

4
General approach
Nt 1 ? Nt
5
  • 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

6
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

7
Data from multi-year census
8
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9
Calculating Survival Rates
10
  • Means of three yearly estimates.
  • Use for mean matrix and deterministic projection
  • Can use variation among estimates to add
    stochasticity

11
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.

12
Calculating average fecundity
  • If fecundity varies among age classes, then need
    to use different approach (e.g., counts of
    fledglings per nest).

13
Age-specific fecundities should be estimated as
the average over all individuals in age class,
not just the breeding ones.
14
The Leslie Matrix
  • Fecundities are elements of the top row.
  • Survival rates are elements of the subdiagonal.

15
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16
A Composite age class consists of all individuals
of a certain age or older
17
Full matrix
New matrix with composite age class
18
Matrix Projection
  • First, consider the equations for projecting age
    class abundances

19
Projection with the Leslie matrix

(Example calculations on board)
20
Initial population growth depends on initial age
distribution
21
Stable age distribution
22
  • 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.

23
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.

24
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.

25
Reproductive value for sparrowhawks with
senescence
26
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?
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