Depensation, low density dynamics and extinction risk

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Depensation, low density dynamics and extinction
risk
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Start with bluey
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References

• Sinclair, ARE, Pech, RP Dickman, CR, Hik-D
  Mahon-P Newsome-A-E . 1998. Predicting effects
  of predation on conservation of endangered prey/
  Conservation-Biology. 12 (3) 564-575..
• Fowler, C. W., and J. D. Baker. 1991. A review of
  animal population dynamics at extremely reduced
  population levels. Rep. Int. Whal. Commn. 41
  545-554.
• Myers, R. A., N. J. Barrowman, J. A. Hutchings,
  and A. A. Rosenberg. 1995. Population dynamics of
  exploited fish stocks at low population levels.
  Science. 269 1106-1108.

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More references

• Dennis, B., P.L. Munholland and J.M. Scott. 1991.
  Estimation of growth and extinction parameters
  for endangered species. Ecological Monographs,
  6 115-143.
• Morris, W. et al. 1999. A practical handbook for
  population viability analysis. The Nature
  Conservancy. ISBN 0-9624590-4-6
• Liermann, M. and R. Hilborn. 2001. Depensation,
  evidence, models and implications. Fish and
  Fisheries 2 33-58.

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Definitions

• Compensatory - the rate of increase declines at
  higher densities
• Depensatory - the rate of increase declines at
  lower densities

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Why low density

• This is where the conservation concern is!

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What happens at low densities

• Competition for resources should be at a minimum
  BUT
• Possibly reduced pregnancy/fertilization due to
  difficulty in finding mates
• Possibly reduced survival due to predation
• Possible reductions in either of the above due to
  lack of social facilitation
• Importance of random events with small numbers

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The probability of finding a mate

• Assume that the probability a female will
  encounter any individual male is p and there are
  N males in the population.
• The probability that no males will be encountered
  will be the 0 class of the Poisson distribution.

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0 Class of the Poisson
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Reparameterizing to N50
N50 the population size at which 50 are mated
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Adding depensation to recruitment function

• Excel in depensation example for lecture

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Impact on rate of increase
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Predation

• If predator population is unaffected by the
  abundance of the prey (i.e. this prey species is
  not the major part of the diet)
• then the predator functional response can lead to
  depensatory mortality

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Review results from functional responses
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Demographic stochasticity

• Assume each individual has a probability p of
  giving birth
• And a probability d of dying
• and these are random events for each individual
• EXCEL example sheet demographic stochasticity

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Bayesian fitting with demographic stochasticity

• For the population, number surviving, or number
  of births is binomially distributed with some
  probability p
• XBinom(p,n) where p is the probability n is the
  number of animals and x is the binomially
  distributed random variable
• We can treat x as a process error, as the w in
  previous process error models
• Or we can draw random numbers (the Gavin SIR
  approach)

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Myers analysis

• Show myers model Icelandic spring spawning
  herring
• Use my S/R. model, see if it is different,
• do likelihood profile on N_50

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Myers Analysis
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Myers results

• Explored over 100 data sets
• Found few significant cases of depensation
• Fewer than expected by chance
• Of these data sets about 27 had high power

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Problems with Myers method

• Parameterization has no biological interpretation
  except dgt1 implies depensation
• Used p values to test for significant
  depensation, ignores biological significance

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Using Allee effect model

• Show fits
• Show likelihood profile
• List advantages

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Simple analytic modelsDennis et al 1991
Does this look familiar?What model have we built
recently that has exactly the same property?
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Analytic results
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Estimating ? and ?2 from counts

• Choose pairs of N(i) and N(j) performed in years
  t(i) and t(j)
• calculate transformed variables

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steps

• Do a regression forcing the line to go through
  the mean
• slope is ?
• mean squared residual is ?2
• now we can plot the probability distribution
  after t periods
• Remembering that log(N) is normally distributed

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Use Dennis model excel
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Steps in calculating extinction risk

• Define model and parameters
• Exponential, logistic, with or without
  depensation, Dennis model
• Simulate population size into future
• Generate cumulative probability for population
  size at specified times
• Define threshold population size
• Pseudo extinction or critical population sizes
• Calculate proportion of simulations that fall
  below critical number

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