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
-Ratios
-Capture-recapture
-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