Estimating Extinction Risk (Population Viability Analysis)

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Estimating Extinction Risk(Population Viability
Analysis)

• Fish 458 Lecture 25

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Identifying Species at Risk of Extinction
(Dennis-type methods)

• This method estimates the probability of (quasi)
  extinction within a given time frame based on
• a time-series of counts
• a rate of change and its coefficient of variation.

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Some Examples

• Steller sea lions (pup counts)
• Marmot Island Decreasing at 13.4 p.a. (sd 1.0)
• Sugerloaf Decreasing at 5.8 p.a. (sd 0.9)
• White Sisters Increasing at 20 p.a (sd 4.3)
• Bowhead whales
• Increasing at 3.2 (SD 0.76).

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The Basic Method-I

• The basic dynamics equation is
• Note the expected population size is given by
• Therefore, if ?lt0, the population will eventually
  be rendered extinct.

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Interlude Modelling

• This model is (also) the solution of the
  diffusion equation with a constant diffusion rate
  and an absorptive boundary (at zero).
• It is not uncommon for the same mathematical
  formulation to arise from different assumptions.
• Are there any other cases we have seen when the
  same model arises from different assumptions?

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The Basic Method II

• Probability of extinction. If is the logarithm
  of the ratio of the current population size to
  the population size at quasi-extinction, then the
  probability of extinction is

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The Basic Method III

• Analytic expressions exist for
• The distribution of the time to quasi-extinction.
• The median / mean time to extinction.
• The variance of the time to extinction.
• However, we will tend to explore the method using
  numerical methods as these are more flexible.

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Computing Extinction Risk Numerically

1. Set the current log-population size, x, to the
   logarithm of the initial population size.
2. Generate a random variate, ?, from N(??2) and
   add it to x.
3. Check whether x lt the logarithm of the population
   size that defines quasi-extinction. If so,
   extinction has occurred.
4. Repeat steps 2-3 many times (say 1000 years).
5. Repeat steps 1-4 many times and count the
   frequency with which extinction occurred.

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Example Grizzly Bears-I
?-0.0075 ?0.09444
Quasi extinction level
Initial population size 47 Quasi-extinction
level 10
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Example Grizzly Bears-II
Median time to extinction 163yrs Mean time to
extinction 218 yrs
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Analytic vs Numerical Methods

• Analytic solutions are available for many
  quantities of interest / problems.
• However, numerical solutions are more flexible
  (if rather computationally intensive).
• For example (Grizzly Bears 1000 simulations)

Analytical Numerical
Mean time to extinction 207 218
Median time to extinction 152 163
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Extensions

• Allow the residuals (?) to be correlated (if
  suggested by the data).
• Use integer arithmetic (there were only 47
  Grizzlies).
• Change population size by a randomly selected
  change from the actual set of changes
  (non-parametric approach).
• Allow for multiple populations.
• Take account of measurement error when computing
  ? and ?.

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Sensitivity to ? Bowheads
Initial population size 7800 Quasi-extinction
level 10 ?0.032 This population is increasing
Ignoring measurement error shifts you to the right
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Computing time to extinction non-parametrically

1. Determine the empirical set of age changes in
   abundance.
2. Set the current log-population size, x, to the
   logarithm of the initial population size.
3. Select a change in abundance at random and add it
   to x.
4. Check whether x lt the logarithm of the population
   size that defines quasi-extinction. If so,
   extinction has occurred.
5. Repeat steps 2-3 many times (say 1000 years).
6. Repeat steps 1-4 many times and count the
   frequency with which extinction occurred.

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Example Steller sea lions at Sugarloaf-I
Rather a question of when rather than whether!
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Example Steller sea lions at Sugarloaf-II
Mean time to extinction Normal assumption 95
years Non-parametric 70 years
Does this point worry anyone?
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Multiple Populations

• If a meta-population consists of n
  sub-populations (c.f. Steller sea lions). The
  probably of extinction of the whole
  meta-population depends of how changes in
  population size are correlated over space.
• If the probability of a single sub-population
  going extinct is p then
• if all populations are independent, the
  probability of the whole meta-population going
  extinct is pn.
• if the factors impacting the populations are
  perfectly correlated, the probability of the
  whole meta-population going extinct is p.

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Key Disadvantages of the Dennis method

• The results are highly sensitive to errors in the
  estimates of ? and ?. The data series is often
  short which means that ? and ? may be very
  imprecise.
• No account is taken of changes in (past or
  future) management practices and environmental
  change.
• No allowance for density-dependence.
• The extinction risk can be very sensitive to the
  initial population age-structure (which is
  ignored).

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Explicit Modeling of Extinction Risk(if it is
that important)

• An alternative to the Dennis-type approach is to
  develop a specific model(s) of the system under
  consideration and examine the consequences of
  future management actions (etc) on extinction
  risk.
• The models can include ecological knowledge.
• This is, however, highly data intensive (but the
  consequences of (say) an ESA listing are
  substantial).

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Readings

• Dennis et al. (1991).
• Holmes (2004).
• Stobutzki et al. (200)