Population dynamics with Matrices

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Population dynamics with Matrices
2

• A is the population projection matrix

3

• Leslie 1945 summarized the existing theory at the
  time for populations with a certain age
  structure. Each age was one unit of time apart

4

• F is the stage specific Fecundity.
• G is the survival from stage i to stage i1

5

• Lefkovitch (1965) proposed that the population
  stages need not have the same duration and that
  some in a given stage will survive and stay in
  the same stage after one year (or time interval).

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• Lefkovitch (1965) proposed that the population
  stages need not have the same duration and that
  some in a given stage will survive and stay in
  the same stage after one year (or time interval).
• In the above P1, P2, P3, P4 is the probability
  that females in stages 1-4 will remain in the
  same stage the following year.

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Northern Spotted Owl
8
Northern Spotted Owl

• http//www.fs.fed.us/psw/rsl/projects/wild/lambers
  on1.PDF
• ROLAND H. LAMBERSON, ROBERT McKELVEY, BARRY R.
  NOON, CURTIS VOSS, 1992. A Dynamic Analysis of
  Northern Spotted Owl
• Viability in a Fragmented Forest Landscape.
  Conservation Biology
• Volume 6, No. 4, December 1992
• Or http//www.fs.fed.us/psw/publications/documents
  /gtr-133/chap8.pdf

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• For the questions to follow we will assume a
  Lefkovitch population projection matrix
  structure as shown above

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4 years of population data for the spotted owl is
shown below.

• Using the 1991 to 1992 data what is the fecundity
  F of the pairs? (F20)
• Assume that P1P20 i.e. Owls in stage 1 or 2
  automatically advance to the next stage and that
  P30.94 i.e. 94 survival rate of mating pairs.

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4 years of population data for the spotted owl is
shown below.

• Using the 1991 to 1992 data what is the fecundity
  F of the pairs? (F20)
• FF333/880.38
• Assume that P1P20 i.e. Owls in stage 1 or 2
  automatically advance to the next stage and that
  P30.94 i.e. 94 survival rate of mating pairs.

12
4 years of population data for the spotted owl is
shown below.

• Using the 1991 to 1992 data what is the value of
  G1? G1 is the fraction of stage 1 individuals
  advancing to stage 2.
• Assume that P1P20 i.e. Owls in stage 1 or 2
  automatically advance to the next stage and that
  P30.94 i.e. 94 survival rate of mating pairs.

13
4 years of population data for the spotted owl is
shown below.

• Using the 1991 to 1992 data what is the value of
  G1? G1 is the fraction of stage 1 individuals
  advancing to stage 2.
• G17/360.19
• Assume that P1P20 i.e. Owls in stage 1 or 2
  automatically advance to the next stage and that
  P30.94 i.e. 94 survival rate of mating pairs.

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4 years of population data for the spotted owl is
shown below.

• Using the 1991 to 1992 data what is the value of
  G2? G2 is the fraction of stage 2 individuals
  advancing to stage 3.
• Assume that P1P20 i.e. Owls in stage 1 or 2
  automatically advance to the next stage and that
  P30.94 i.e. 94 survival rate of mating pairs.

15
4 years of population data for the spotted owl is
shown below.

• Using the 1991 to 1992 data what is the value of
  G2? G2 is the fraction of stage 2 individuals
  advancing to stage 3.
• G2(87-88.94)/90.48
• Assume that P1P20 i.e. Owls in stage 1 or 2
  automatically advance to the next stage and that
  P30.94 i.e. 94 survival rate of mating pairs.

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• Four points are worth noting here about the
  eigenvalues, r for population projection matrices
  Nt1ANt
• When r1.0 the exponential term is a constant
  term,
• when r less than 1.0 the exponential term
  eventually goes to zero
• if r is greater than 1.0 will be exponential
  growth.
• If r is a complex number this corresponds to
  oscillations

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Question

• Using a difference equation
• Nt1Ant
• The dominant eigenvalue is l1.04.
• What is the implied population rate of increase?
• Will this population grow or get smaller?

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Question

• Using a difference equation
• Nt1Ant
• The dominant eigenvalue is l1.04.
• What is the implied population rate of increase?
• 4 increase each year

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Question

• Using a flow equation
• The dominant eigenvalue is r.02. What is the
  implied population rate of increase?

Four points are worth noting here about the
eigenvalues, r , for transport matrices In flow
equations like above When r0 the exponential
term is a constant term, when r is negative the
exponential term eventually goes to zero if r is
positive there will be exponential growth. If r
is a complex number this corresponds to
oscillations
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Question

• Using a flow equation
• The dominant eigenvalue is r.02. What is the
  implied population rate of increase?
• 2 increase each year

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What is the transpose of the matrix below?
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What is the transpose of the matrix below?
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The population projection matrix and initial
population are shown below. What is the
population after 1 year?
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The population projection matrix and initial
population are shown below. What is the
population after 1 year? Assume N1AN0
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The last four years of a long population model
simulation are shown below.

• What is the dominant eigenvalue for this
  population? And what is the percent growth rate?

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The last for years of a long population model
simulation are shown below.

• What is the dominant eigenvalue for this
  population? 1.11
• And what is the percent growth rate? 11

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• Deborah T.Crouse, L.B. Crowder, and H. Caswell.
  1987. A stage-based population Model for
  Loggerhead Sea Turtles and implications for
  conservation. Ecology, 68 (5), 1412 1423.