Brief review of previous lecture

1
Brief review of previous lecture

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

• Creating and projecting a Leslie matrix for an
  age-structured population

• Initial, stable, and stationary age distributions

• Lambda is a long-term, deterministic measure of
  growth rate of an age-structured population.

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

2
Lecture Outline Age- and Stage-structured Models

• Timing of sampling estimating fecundity for
  matrix

• Stage-structured models

• Determining stages
• Stage-transition matrices and loop diagrams

• Adding density dependence and stochasticity

• Sensitivity analysis

• Values to wildlife research and management
• Sensitivities and elasticities
• Case study for cheetahs

3
Timing of sampling the Leslie matrix

• Fecundity values for the matrix are products of
  survival rates and fertility rates, and
  calculation depends on timing of census relative
  to breeding season.

• Sx are survival rates (Mills uses Px)
• Mx are fertility rates (mean number of offspring
  per individual of age x)

4
Pre-breeding census
Fx mxS0
5
Post-breeding census
Fx Sxmx
6
Stage Structure

• Age is not always the best indicator of
  demographic change.

7
(No Transcript)
8
Transition Matrices and Loop Diagrams

• Lets start with a Leslie matrix for an
  age-structured model
• (helmeted honeyeater)

9

• A common type of stage-structured model

• Individuals can remain in current stage during
  time step or transition to next stage

• No stage skipping or reversals

Lefkovitch matrix
10
(No Transcript)
11
(No Transcript)
12
Density Dependence

• Adding density dependence to structured models is
  more complicated than for non-structured models
  because many variables are potentially density
  dependent (age-specific survival and fecundity)
  and not just the growth rate.

• 1. Which vital rates are density-dependent?
• 2. How do those rates change with density?
• 3. Which classes contribute to the
  density-dependence? (For instance, is juvenile
  survival influenced by total density or by
  juvenile density?)

• Additional problem we rarely have long-term
  demographic data to detect and estimate type of
  density dependence

13
Several approaches
1. Assume that total abundance affects all
elements of stage matrix proportionately (method
used in RAMAS Ecolab).
2. For territorial species, use territory size to
estimate upper limit for number of breeders and
model with Ceiling Model. This makes transition
from pre-reproductive to reproductive classes
density-dependent.
3. Choose one (or a few) vital rates for which
data exist and model these rates with specific
density-dependent functions (e.g., Ricker,
Beverton-Holt). Assume other rates are
density-independent.
14
Adding Demographic Stochasticity

• We use same approach as for models without age
  structure
• determine whether each individual survives or
  reproduces using statistical distributions
  such as binomial or Poisson.

• But now we must track fate of individuals
  separately within age classes.

15
Adding Environmental Stochasticity

• We estimate temporal variations in vital rates
  from past observations and use these to predict
  future population sizes.

• At each time step, before doing the matrix
  multiplication, we randomly sample the matrix
  elements (or vital rates) from statistical
  distributions with appropriate means and standard
  deviations.

16
Additional Considerations

• Estimates of environmental stochasticity may
  include sampling variation. Ideally, the sampling
  variation should be stripped off so that pure
  process variance is used in projections.

• Are vital rates correlated with each other?
  RAMAS Ecolab assumes a positive correlation. For
  instance, in a bad year all survival rates and
  fecundities are below average.

17
Sensitivity Analysis
How sensitive is population growth rate to
different matrix elements?
How sensitive is population growth rate to vital
rates used to calculate the matrix elements?
18
Sensitivity Analysis
Adult survival
19
Sensitivity Values

• Convenient to have single value that summarizes
  sensitivity of lambda to each vital rate.

Manual Perturbation Method

• Compute ? with current value of vital rate (v)

2. Compute ? with v ?, where ? is some small
value (say 0.01) , while holding all other rates
constant.
20

• This approach for calculating sensitivities can
  be applied to deterministic and to stochastic
  models (using average lambda).

• It also can be used to ask how sensitive is
  extinction risk to each matrix element or vital
  rate. We just replace lambda in above formula
  with something like Probability of declining to
  50 individuals.

• Sensitivity of a matrix element depends on the
  reproductive value of that age class, and the
  proportion of individuals in that class at the
  Stable Age Distribution.

21
Elasticity Values

• Sensitivity values reflect absolute changes in
  vital rates, which can be a problem for
  comparisons.

For example, a change of 0.15 in adult fecundity
is small if the current value is 0.89, whereas a
change of 0.15 would more than double juvenile
survival if its current value is 0.14.

• To make meaningful comparisons between different
  sensitivity values, usually it is necessary to
  rescale them so that each represents the
  proportional change in lambda due to proportional
  change in the vital rate.

• For manual perturbations using EcoLab, we usually
  will change parameters by some small percent
  instead of an amount to scale the sensitivity
  analyses.

22
Evaluating management options
1. Sensitivity analysis
2. How much can each vital rate be changed with
management?

• How much does the rate vary naturally?
• B. Will the rate respond to available management
  actions?

3. What is the relative financial cost of each
management action?
4. Each management action might affect 1 vital
rate. Hence, best to consider overall effects of
management instead of effect on single
parameters. Use simulations to evaluate
different management scenarios.
23
(No Transcript)
24
Example Cheetah conservation1

• Lack of genetic variation initially considered
  main threat.

• More recent ideas that ecological factors more
  important, especially cub survival (Serengeti).

1Crooks et al. 1998. New insights on cheetah
conservation through demographic modeling.
Conservation Biology 12889-895.
25
Leslie Matrix for cheetahs

• time interval of 6 months plus composite adult
  stage

• Lambda was 0.956 based on mean matrix projection

• Conducted sensitivity analysis of vital rates

26
(No Transcript)
27

• Previous results for deterministic model using
  mean matrix

• Also conducted simulations with environmental
  stochasticity

• Adult survivorship explained most of the
  variation in lambda

28
Adult survivorship (r2 0.75)
Lambda
Newborn young cub survivorship (r2 0.025)
Lambda
Survivorship
29
REFERENCES AGE- AND STAGE-STRUCTURED
MODELS Biek, R., WC Funk, BA Maxwell, and LS
Mills. 2002. What is missing in amphibian decline
research insights from ecological sensitivity
analysis. Conservation Biology 16728-734. Caswel
l, H. 2001. Matrix population models.
Sinauer. Crooks, KR, MA Sanjayan, and DF Doak.
1998. New insights on cheetah conservation
through demographic modeling. Conservation
Biology 12889-895. Crowder, LB, DT Crouse, SS
Heppell, and TH Martin. 1995. Predicting the
impact of turtle excluder devices on loggerhead
sea turtle populations. Ecological Applications
43437-445. De Kroon, H, A Plaisier, J Van
Groenendael, and H. Caswell. 1986. Elasticity
the relative contribution of demographic
parameters to population growth rate. Ecology
671427-1431. Dobson, FS, and MK Oli. 2001. The
demographic basis of population regulation in
Columbian ground squirrels. American Naturalist
158236-247. Fujiwara, M., and H. Caswell. 2001.
Demography of the endangered North Atlantic right
whale. Nature 414537-541. Leslie, PH. 1945. On
the use of matrices in population mathematics.
Biometrika 33183-212. Maguire LA, GF Wilhere, Q
Dong. 1995. Population viability analysis for
red-cockaded woodpeckers in the Georgia piedmont.
J. Wildlife Management 59533-542.
30
Mills, LS, DF Doak, and MJ Wisdom. 1999.
Reliability of conservation actions based on
elasticity analysis of matrix models.
Conservation Biology 13815-829. Morris, WF, and
DF Doak. 2002. Quantitative Conservation Biology
Theory and Practice of Population Viability
Analysis. Sinauer. Oli, MK, and KB Armitage.
2004. Yellow-bellied marmot population dynamics
demographic mechanisms of growth and decline.
Ecology 852446-2455. Reid, JM, EM Bignal, S
Bignal, DI McCracken, and P. Monaghan. 2004.
Identifying the demographic determinants of
population growth rate a case study of
red-billed choughs Pyrrhocorax pyrrhocorax. J.
Animal Ecology 73777-788. Sandercock, BK, K
Martin, and SJ Hannon. 2005. Demographic
consequences of age-structure in extreme
environments population models for arctic and
alpine ptarmigan. Oecologia 14613-24. Wisdom,
MJ, and LS Mills. 1997. Sensitivity analysis to
guide population recovery prairie-chickens as an
example. J. Wildlife Management 61302-312.