Demographic PVA

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Demographic PVAs

• Assessing Population Growth and Viability

2
Structured populations in a deterministic
environment

• Deterministic projection models, when we do not
  have (or use) estimates of variation

3
vector a representation of the population
structure

• It is a column of numbers that indicates the
  densities of individuals in each class in the
  population at one point in time

n(t)
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Each entry aij(t) in a projection matrix A(t)
gives the number of individuals in class i at
census (t1) produced on average by a single
individual in class j at census (t)
A(t)
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If we know the densities at census t n(t), we can
project the densities at the next census n(t1)

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If we know the densities at census t n(t), we can
project the densities at the next census n(t1)

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In a constant environment

• A(t)A
• Population convergence

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The stable distribution (w) is
The unique vector containing the ultimate
proportions of the population in each class given
the constant projection matrix A
The ultimate or long term growth rate (?) is
?10.6389
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The reproductive values (v) is
The relative contribution to future population
growth an individual currently in a particular
class is expected to make
Reproductive values take into account the number
of offspring an individual might produce in each
of the classes it passes through the future, the
likelihood of the individual reaching those
classes, the time required to do so, and the
population growth rate
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Eigenvalue sensitivities

• The ultimate rate of population growth in a
  constant environment, ?1, depends on the
  magnitudes of all the elements in A, so changing
  any of them will change ?1.
• Sij Sensitivity is a useful measure of how much
  changes in a particular matrix element will
  change ?1

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Sij Sensitivity

• It is the partial derivative of ?1 with respect
  to aij
• It measures the change in ?1 that would result
  from a small change in aij , keeping all other
  elements of the matrix A fixed at their present
  values

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Sensitivities
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Sensitivity
Slope0.787
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How to include stochastic environmental effects
on matrix models?

• Matrix selection

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If environmental conditions are aperiodic and
uncorrelated, and moreover the probability of
choosing a particular matrix does not change over
time then environmental conditions are said to
be independently and identically distributed
or iid
Year 0 Matrix 2
Yar 1 matrix 2
Modeling using matrix selection
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Mountain golden heather
Number of realizations
Population size at t 50
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Modeling samples from matrices by time since
fire. In this (simplified) example, the fire
return interval is 3 years
Year 0 fire matrix
Use this
Year 1 matrix 1.1
Year 1 matrix 1.2
Year 1 matrix 1.3
or
or
Choose 1
Other years
or
Use this
Beyond interpolation, input pooled matrices
reset
fire
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Estimating the Stochastic log Growth rate ?s

• By simulation

LogN(t1)/N(t) over all pairs of adjacent years
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Estimating the Stochastic log Growth rate ?s

• Tuljapurkars approximation (an analytical
  solution)

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Calculating Quasi-Extinction probability

• Box 7.5