Population Viability Analysis II: Demographic models

I.  Projection matrix models

- 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
-Ratios
-Capture-recapture

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