Age-Structured Matrix Models

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Age-Structured Matrix Models
1, 2

• Abdessamad Tridane
• MTBI Summer 2007.

1.Adapted from lecture by Dr. Robert Schooley,
University of Illinois 2. Figures from Akcakaya
et al. 1999. Applied Population Ecology
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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

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• 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), habitat
  protection and enhancement.

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General approach
Nt 1 ? Nt
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• Focus on birth-pulse models

• Assume closed population

• 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

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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

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Data from multi-year census
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Calculating Survival Rates
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• Means of three yearly estimates.
• Use for mean matrix and deterministic projection

• Can use variation among estimates to add
  stochasticity

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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.

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Calculating average fecundity

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

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Age-specific fecundities should be estimated as
the average over all individuals in age class,
not just the breeding ones.
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The Leslie Matrix

• Fecundities are elements of the top row.
• Survival rates are elements of the subdiagonal.

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A Composite age class consists of all individuals
of a certain age or older
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Full matrix
New matrix with composite age class
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Matrix Projection

• First, consider the equations for projecting age
  class abundances

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Projection with the Leslie matrix

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

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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.

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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.

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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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Reproductive value for sparrowhawks with
senescence
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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)

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Pre-breeding census
Fx mxS0
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Post-breeding census
Fx Sxmx
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Stage Structure

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

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Transition Matrices and Loop Diagrams

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

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• 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
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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

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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 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.
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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.

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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.

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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.

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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.
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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.