What is the Leslie Matrix?

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What is the Leslie Matrix?

• Method for representing dynamics of age or size
  structured populations
• Combines population processes (births and deaths)
  into a single model
• Generally applied to populations with annual
  breeding cycle
• By convention, use only female part of population

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Setting up the Leslie Matrix

• Concept of population vector
• Births
• Deaths

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Population Vector
N0 N1 N2 N3 . Ns
s1 rows by 1 column (s1) x 1 Where,
smaximum age
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Births
Newborns (Number of age 1 females) times
(Fecundity of age 1 females) plus
(Number of age 2 females) times (Fecundity
of age 2 females) plus
.. Note fecundity here is defined as number of
female offspring Also, the term
newborns may be flexibly defined (e.g., as
eggs, newly hatched fry, fry that
survive past yolk sac stage, etc.
N0 N1F1 N2F2 N3F3 .FsNs
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Mortality
Number at age in next year (Number at previous
age in prior year) times
(Survival from previous
age to current age)
Na,t Na-1,t-1Sa Another way of putting this
is, for age 1 for example N1,t N0,t-1S0-1
N1,t-1 (0) N2,t-1 (0) N3,t-1 (0)
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Leslie Matrix

(s1) x 1 (s1) x
(s1)
(s1) x 1
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Leslie Matrix

s x 1 s x s

s x 1
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Leslie Matrix

Nt1 A Nt
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Projection with the Leslie Matrix
Nt1 ANt Nt2 AANt Nt3 AAANt Nt4
AAAANt Ntn AnNt
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Properties of this Model

• Age composition initially has an effect on
  population growth rate, but this disappears over
  time (ergodicity)
• Over time, population generally approaches a
  stable age distribution
• Population projection generally shows exponential
  growth

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Properties of this ModelGraphical Illustration
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Properties of this ModelGraphical Illustration
Lambda Nt1 / Nt Thus, Nt1 ? Nt
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Properties of this ModelGraphical Illustration
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Projection with the Leslie Matrix
Given that the population dynamics are ergodic,
we really dont even need to worry about the
initial starting population vector. We can base
our analysis on the matrix A itself Ntn
AnNt
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Projection with the Leslie Matrix
Given the matrix A, we can compute its
eigenvalues and eigenvectors, which correspond to
population growth rate, stable age distribution,
and reproductive value
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Projection with the Leslie MatrixEigenvalues
Whats an eigenvalue? Cant really give you a
plain English definition (heaven knows Ive
searched for one!) Mathematically, these are the
roots of the characteristic equation (there are
s1 eigenvalues for the Leslie matrix),
which basically means that these give us a single
equation for the population growth over time
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Projection with the Leslie MatrixCharacteristic
Equation
1 F1?-1 P1F2 ?-2 P1P2F3 ?-3 P1P2P3F4 ?-4
Note that this is a polynomial, and thus can
be solved to get several roots of the equation
(some of which may be imaginary, that is have
v-1 as part of their solution) The root (?) that
has the largest absolute value is the dominant
eigenvalue and will determine population growth
in the long run. The other eigenvalues will
determine transient dynamics of the population.
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Projection with the Leslie MatrixEigenvectors
Associated with the dominant eigenvalue is two
sets of eigenvectors The right eigenvectors
comprise the stable age distribution The left
eigenvectors comprise the reproductive value (We
wont worry how to compute this stuff in class
computing the eigenvalues and eigenvectors can be
a bugger!)
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Projecting vs. Forecasting or Prediction
So far, Ive used the term projecting what does
this mean in technical terms, and how does it
differ from a forecast or prediction. Basically,
forecasting or prediction focuses on short-term
dynamics of the population, and thus on the
transient dynamics. Projection refers to
determining the long-term dynamics if things
remained constant. Thus projection gives us a
basis for comparing different matrices without
worrying about transient dynamics.
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Projecting vs. Forecasting or Prediction
Simple (?) Analogy The speedometer of a car
gives you an instantaneous measure of a cars
velocity. You can use to compare the velocity of
two cars and indicate which one is going faster,
at the moment. To predict where a car will be in
one hour, we need more information, such as
initial conditions Where am I starting from?
What is the road ahead like? etc. Thus,
projections provide a basis for comparison,
whereas forecasts are focusing on providing
accurate predictions of the systems dynamics.
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Stage-structured ModelsLefkovitch Matrix
Instead of using an age-structured approach, it
may be more appropriate to use a stage or
size-structured approach. Some organisms (e.g.,
many insects or plants) go through stages that
are discrete. In other organisms, such as fish
or trees, the size of the individual is more
important than its age.
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Lefkovitch Matrix Example

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Lefkovitch Matrix
Note now that each of the matrix elements do not
correspond simply to survival and fecundity, but
rather to transition rates (probabilities)
between stages. These transition rates depend in
part on survival rate, but also on growth rates.
Note also that there is the possibility for an
organism to regress in stages (i.e., go to an
earlier stage), whereas in the Leslie matrix,
everyone gets older if they survive, and they
only advance one age
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Lefkovitch Matrix Example

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Example ApplicationSustainable Fishing Mortality

• An important question in fisheries management is
  How much fishing pressure or mortality can a
  population support?

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Example ApplicationSustainable Fishing Mortality

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Example ApplicationSustainable Fishing Mortality
S e-(MF) Knife-edge recruitment, meaning that
fish at a given age are either not exposed to
fishing mortality or are fully vulnerable
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Example ApplicationSustainable Fishing Mortality
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Example ApplicationSustainable Fishing Mortality
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Example ApplicationSustainable Fishing Mortality
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Example ApplicationSustainable Fishing Mortality
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Example ApplicationSustainable Fishing Mortality
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Example ApplicationReproductive Strategy of
Yellow Perch

• One of main questions was whether stunting,
  meaning very slow growth, of yellow perch was
  caused by reproductive strategy or if
  reproductive strategy resulted from adaptation to
  low prey abundance
• Our goal was to understand what stunted fish
  should do in terms of age at maturity

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Reproductive Strategy of Yellow Perch

• Basic model had a number of assumptions
• Energy intake is limited and depends on size of
  fish
• Yellow perch show an ontogenetic shift in diet
  where indivduals less than 10 grams eat
  zooplankton, individuals 10 to 30 grams eat
  benthic invertebrates, and individuals larger
  than 30 grams eat fish
• Net energy intake can only be partitioned to
  growth or reproduction
• Reproduction is all or nothing
• Reproduction may have survival costs (theta)

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Reproductive Strategy of Yellow Perch
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Reproductive Strategy of Yellow Perch
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Reproductive Strategy of Yellow Perch
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Reproductive Strategy of Yellow Perch
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Reproductive Strategy of Yellow Perch
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Reproductive Strategy of Yellow Perch
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Reproductive Strategy of Yellow Perch
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Reproductive Strategy of Yellow Perch
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Reproductive Strategy of Yellow Perch