Population dynamics with Matrices. A is the population projection matrix.
Population dynamics with Matrices
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Transcript of Population dynamics with Matrices
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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
• 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
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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 =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 =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?
ANdt
dN
Four points are worth noting here about the eigenvalues, r , for transport matricesIn 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
ANdt
dN
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What is the transpose of the matrix below?
153
726
241
A
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What is the transpose of the matrix below?
172
524
361TA
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The population projection matrix and initial population are shown below. What is the population after 1 year?
____
____
____
0
0
10
N
110
115.
241
1
0
N
A
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The population projection matrix and initial population are shown below. What is the population after 1 year?
Assume N1=AN0
_0
5
10
0
0
10
N
110
115.
241
1
0
N
A
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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.