Population dynamics with Matrices

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Presentation transcript:

Population dynamics with Matrices

A is the population projection matrix

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

F is the stage specific Fecundity. G is the survival from stage i to stage i+1

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).

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.

Northern Spotted Owl

Northern Spotted Owl http://www.fs.fed.us/psw/rsl/projects/wild/lamberson1.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

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

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.

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.

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.

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.

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.

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.

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

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?

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

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

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

What is the transpose of the matrix below?

What is the transpose of the matrix below?

The population projection matrix and initial population are shown below. What is the population after 1 year?

The population projection matrix and initial population are shown below. What is the population after 1 year? Assume N1=AN0

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?

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 %

Deborah T. Crouse, L. B. Crowder, and H. Caswell. 1987 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.