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**Population dynamics with Matrices**

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**A is the population projection matrix**

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

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**F is the stage specific Fecundity.**

G is the survival from stage i to stage i+1

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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).

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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

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Northern Spotted Owl ROLAND H. LAMBERSON, ROBERT McKELVEY, BARRY R. NOON, CURTIS VOSS, A Dynamic Analysis of Northern Spotted Owl Viability in a Fragmented Forest Landscape*. Conservation Biology Volume 6, No. 4, December 1992 Or

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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? (F2=0) Assume that P1=P2=0 i.e. Owls in stage 1 or 2 automatically advance to the next stage and that P3=0.94 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? (F2=0) F=F3=33/88=0.38 Assume that P1=P2=0 i.e. Owls in stage 1 or 2 automatically advance to the next stage and that P3=0.94 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 G1? G1 is the fraction of stage 1 individuals advancing to stage 2. Assume that P1=P2=0 i.e. Owls in stage 1 or 2 automatically advance to the next stage and that P3=0.94 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 G1? G1 is the fraction of stage 1 individuals advancing to stage 2. G1=7/36=0.19 Assume that P1=P2=0 i.e. Owls in stage 1 or 2 automatically advance to the next stage and that P3=0.94 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 P1=P2=0 i.e. Owls in stage 1 or 2 automatically advance to the next stage and that P3=0.94 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. G2=(87-88*.94)/9=0.48 Assume that P1=P2=0 i.e. Owls in stage 1 or 2 automatically advance to the next stage and that P3=0.94 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 Nt+1=ANt: When r=1.0 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 Nt+1=Ant**

The dominant eigenvalue is l=1.04. What is the implied population rate of increase? Will this population grow or get smaller?

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**Question Using a difference equation Nt+1=Ant**

The dominant eigenvalue is l=1.04. 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 r=0 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 N1=AN0

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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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**What is the dominant eigenvalue for this population? 1.11 **

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**

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

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