Age-structured models (continued): Estimating l from Leslie matrix models

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Age-structured models (continued)Estimating l
from Leslie matrix models

• Fish 458, Lecture 4

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The facts on l

• Finite rate of population increase
• ler rln(l), therefore NtNlt
• A dimensionless number (no units)
• Associated with a particular time step
• (Ex l 1.2/yr not the same as l 0.1/mo)
• l gt1 pop. llt1 pop

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Matrix Population Models Definitions

• Matrix- any rectangular array of symbols. When
  used to describe population change, they are
  called population projection matrices.
• Scalar- a number a 1 X 1 matrix
• State variables- age or stage classes that define
  a matrix.
• State vector- non-matrix representation of
  individuals in age/stage classes.
• Projection interval- unit of time define by
  age/stage class width.

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Basic Matrix Multiplication
4x1 3x2 2x3 0 2x1 - 2x2 5x3 6 x1 -
x2 - 3x3 1
0 6 1
x1 x2 x3
4 3 2 2 2 5 1 1 3

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What does this remind you of?

• n(t 1) An(t)
• Where
• A is the transition/projection matrix
• n(t) is the state vector
• n(t 1) is the population at time t 1
• This is the basic equation of a matrix population
  model.

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Eigenvectors Eigenvalues
When matrix multiplication equals scalar
multiplication
Aw ?w
vA ?v
v,w Eigenvector ? Eigenvalue

• Rate of Population Growth (?) Dominant
  Eigenvalue
• Stable age distribution (w) Right Eigenvector
• Reproductive values (v) Left Eigenvector

Note Eigen is German for self.
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Example Eigenvalue
A x y
A x y

• -6
• 2 -5

• -6
• 2 -5

4 1
6 3
-3 -3
1 1


No obvious relationship between x and y
Obvious relationship between x and y
x is multiplied by -3
Thus, A acts like a scalar multiplier. How is
this similar to ??
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Characteristic equations

• From eigenvalues, we understand that Ax ?x
• We want to solve for ?, so
• Ax - ?x 0 (singularity)
• or
• (A- ?I)x 0
• I represents an identity matrix that converts ?
  into a matrix on the same order as A.
• Finding the determinant of (A- ?I) will allow one
  to solve for ?. The equation used to solve for ?
  is called the Characteristic Equation

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Solution of the Projection Equation n(t1)
An(t)

• ?4 - P1F2 ?2 - P1P2F3 ? - P1P2P3F4 0
• or alternatively (divide by ?4)
• 1 P1F2 ?-2 P1P2F3 ?-3 P1P2P3F4 ?-4

This equation is just the matrix form of Eulers
equation 1 S lxmxe-rx
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Constructing an age-structured (Leslie) matrix
model

• Build a life table
• Birth-flow vs. birth pulse
• Pre-breeding vs. post-breeding census
• Survivorship
• Fertility
• Build a transition matrix

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Birth-Flow vs. Birth-Pulse

• Birth-Flow (e.g humans)Pattern of reproduction
  assuming continuous births. There must be
  approximations to l(x) and m(x) modeled as
  continuous, but entries in the projection matrix
  are discrete coefficients.
• Birth-Pulse (many mammals, birds, fish)Maternity
  function and age distribution are discontinuous,
  matrix projection matrix very appropriate.

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Pre-breeding vs. Post-breeding Censuses
Pre-breeding (P?1) Populations are accounted for
just before they breed.
Post-breeding (P?0) Populations are accounted for
just after they breed
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Calculating Survivorship and Fertility Rates for
Pre- and Post-Breeding Censuses
Different approaches, yet both ways produce a ?
of 1.221.
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The Transition/Population Projection Matrix
4 age class life cycle graph
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Example Shortfin Mako (Isurus oxyrinchus)
Software of choice PopTools
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Mako Shark Data
Mortality M1-6 0.17 M7-w 0.15 Fecundity
12.5 pups/female Age at female maturity 7
years Reproductive cycle every other 2 years
Photo Ron White
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Essential Characters of Population Models

• Asymptotic analysis A model that describes the
  long-term behavior of a population.
• Ergodicity A model whose asymptotic analyses
  are independent of initial conditions.
• Transient analysis The short-term behavior of
  a population useful in perturbation analysis.
• Perturbation (Sensitivity) analysis The extent
  to which the population is sensitive to changes
  in the model.

Caswell 2001, pg. 18
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Uncertainty and hypothesis testing

• Characterizing uncertainty
• Series approximation (delta method)
• Bootstrapping and Jackknifing
• Monte Carlo methods
• Hypothesis testing
• Loglinear analysis of transition matrices
• Randomization/permutation tests

Caswell 2001, Ch. 12
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References
Caswell, H. 2001. Matrix Population Models
Construction, Analysis, and Interpretation.
Sunderland, MA, Sinauer Associates. 722
pp. Ebert, T. A. 1999. Plant and Animal
Populations Methods in Demography. San Diego,
CA, Academic Press. 312 pp. Leslie, P. H. 1945.
On the use of matrices in certain population
mathematics. Biometrika 33 183-212. Mollet, H.
F. and G. M. Cailliet. 2002. Comparative
population demography of elasmobranch using life
history tables, Leslie matrixes and stage-based
models. Marine and Freshwater Research 53
503-516. PopTools http//www.cse.csiro.au/popto
ols/