Population Viability Analysis II: Demographic models

I.  Projection matrix models

1.  Advantages
    - Assess vital rates of particular class
        - May be more accurate for a species that has undergone change in vital rate
        - Particularly well suited to understand effects of different management scenarios
2.  Primary disadvantage is that demographic models require more parameters

3.  Requirements
    - Ideally, at least 2-3 years of demographic data
    - Age-, size-, or stage-structured data

4.  Use three types of parameters
    1.  Survival
    2.  State
    3.  Number of offspring

II.  Constructing a Matrix Model

Step 1: Conduct a demographic study
    -individuals representative of entire population
    -censuses at regular intervals
    -ideally, enough data to estimate variability

Step 2: Is population best classified by age, size or life stage?
    -Can be a mix
    -Usually track one sex (females)
    -States highly correlated with all vital rates
    -Balance between accuracy and practicality
    -Setting class boundaries balances information and numbers

Step 3: Estimate vital rates
    -Comment on vital rates v. matrix elements
    -Rates depend on timing of data collection
    -Statistical models take advantage of additional information

Step 4: Construct a projection matrix

Step 5. Construct an initial population vector

Step 6: Project the matrix
    - matrix multiplication
    - How to calculate λ
        The dominant eigenvalue of the matrix
        Aw = λ1w
    - The stable age distribution
        the right eigenvector of the matrix
        Aw = λ1w
    - Reproductive value
        the relative contribution of each class to future population growth
        the left eigenvector of the matrix, v’A = λ1v’

Step 7:  Sensitivity analysis

A.  Any approach to estimate the effects of a change in a vital rate or matrix element on any measure of  population viability

        1. Easy approach: change a vital rate slightly, and observe changes in λ

        2.  More complicated approach:
            a.  Solve sensitivity analytically, using stable age distribution and reproductive value
                    Sij = δλ1/δaij = Σ (viwj/vkwk)

            b.  More complicated for vital rates, especially those that affect more than one matrix element

B.  Elasticity analysis

        1. Often a fairer measure of the relative importance of changing a vital rate on viability

        2. Accounts for the proportional change in λ relative to the proportional change in a vital rate

III.  Incorporating variability into matrix models

1.  Different matrices represent different environmental conditions
    a.  Have computer select a matrix for each model time step
        b.  Can incorporate environmental correlations
        c.  Long-term population growth rate is predicted by the most likely log population growth rate
        – determine population size over a series of years
        – take the arithmetic mean of log (N(t+1)/N(t)) generated from all pairs of adjacent years (= log λs)
        – determine CDF

2.  Construct a matrix with random variables
    a.  Estimate parameters
        – Ideally, mean, variance, and correlation for each parameter

    b. Incorporate correlations in vital rates

        c. Incorporate density dependence
        – need to determine presence and functional form for many parameters
        – Many classes that may exert effects on any given class – which ones?
        – Assume a maximum number of individuals in one or more classes
    – Model specific vital rates with density dependent functions

    d.  Simulate population growth to determine CDF