Age Structure

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Age Structure
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Lecture Goals

• Review life tables (types, data sources, etc.)
• Develop population projections from life tables
• As a consequence of age specific birth and death
• Express population projection in matrix form
• What is the matrix?
• Examine properties of matrix population models

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Age Structure A Review of Life Tables
X lx mx 0 1.00 0 1 0.10 5 2 0.05 2 . . . . . .
X age lx survival to age X mx mean
offspring at age X
px probability of surviving from age X to X1
p0 l1/l0 0.10/1.00 0.1
p1 l2/l1 0.05/0.10 0.5
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Types of life tables from data sources
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Review of Useful Life Table Calculations
R0 average lifetime reproduction S lxmx
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Reproductive Value for Red Deer (Clutton-Brock
1984)
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Life Tables Can Also Be Used To Predict Future
Population Size
X2
?
X1
X0
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Order of Birth and Survival in Following
Development
Age 0
Age 1
Age 2
Birth(m0)
Birth(m1)
Birth(m2)
Survival (p0)
Survival (p1)
Survival (p2)
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Underlying questions/calculations
X lx mx px 0 1.00 0 0.1 1 0.10 5 0.5 2 0.05 2 0.0

• If we start with
• 100 age 0
• 10 age 1
• 5 age 2

Next year
How many offspring? How many age 1? How many age
2?
1000 105 52 60 1000.1 100 50
10 1000 100.5 50 5
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The formulae
Numbers of each class (1,2,3) are N0, N1, N2
t 1
t
N0 m0N0 m1N1 m2N2 N1 p0N0 0N1
0N2 N2 0N0 p1N1 0N2
NOTE the subscript represents age, not time
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The formulae Matrix Notation
m0 m1 m2 p0 0 0 0 p1 0
N0 N1 N2

N0' N1' N2'
Leslie MatrixTransition Matrix
population vectors
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Just Enough About Matrices
m0 m1 m2 p0 0 0 0 p1 0
N0 N1 N2
N0' N1' N2'

T x N N'

• matrices and vectors are often represented by
  bold faced type
• Index order of elements is row,column
• a1,2 is in row 1, column 2
• multiplication order is important
• T x N ? N x T

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N0 N1 N2
For any N0, N1,
T x N N', T x N' N'', T x N'' N''' ,
N

Eventually, a stable age structure occurs.
N2
N0
N1
are constant as the population changes.

,
,
SNi
SNi
SNi
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Graphically, we see the age structures stabilize
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Initialoscillations
30
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Number
Total Number
20
15
10
???
5
0
0
2
4
6
8
10
Year
Year
(4 lines 4 age classes)
What does this population pattern resemble?
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Regardless of the (non-zero) starting value,
eventually the population attains a stable age
structure, and then
T x N l x N
Where l is the parameter from the geometric
population growth model. Mathematically l is the
dominant latent root or dominant eigenvalue of T.
We could also use Nt l Nt1 and the stable age
structure to get the same result.
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When would it be necessary to use a population
matrix model instead of the much simpler
geometric model of population growth?

• Unstable age structure
• after perturbation (invasion, fire, etc.)

Will a static life table be adequate to examine a
population limited by density dependent factors?
No - it follows geometric, unlimited growth
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Variable Parameters in Leslie Matrices? Building
more complex models

• Let some or all mx and px be represented as a
  function of Nx or SNi.
• represents density dependence
• mx decline with numbers
• px decline with numbers
• General Leslie matrix results no longer apply
• Age structured model
• not solvable by matrix math
• programmable (simulation)

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Example of Poa annua (Annual Bluegrass)
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(No Transcript)
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Executing the Simulation Model

• population stabilizes (intraspecific
  competition)
• age structure stabilizes(similar to cases
  without intraspecific competition)

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If you were given all of the information required
for the Poa model, how would you calculate the
net reproductive rate (R or l)?