Stochastic Population Modelling

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Stochastic Population Modelling

• QSCI/ Fish 454

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Stochastic vs. deterministic

• So far, all models weve explored have been
  deterministic
• Their behavior is perfectly determined by the
  model equations
• Alternatively, we might want to include
  stochasticity, or some randomness to our models
• Stochasticity might reflect
• Environmental stochasticity
• Demographic stochasticity

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Demographic stochasicity

• We often depict the number of surviving
  individuals from one time point to another as the
  product of Numbers at time t (N(t)) times an
  average survivorship
• This works well when N is very large (in the
  1000s or more)
• For instance, if I flip a coin 1000 times, Im
  pretty sure that Im going to get around 500
  heads (or around p N 0.5 1000)
• If N is small (say 10), I might get 3 heads, or
  even 0 heads
• The approximation N p 10 doesnt work so well

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Why consider stochasticity?

• Stochasticity generally lowers population growth
  rates
• Autocorrelated stochasticity REALLY lowers
  population growth rates
• Allows for risk assessment
• Whats the probability of extinction
• Whats the probability of reaching a minimum
  threshold size

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Mechanics Adding Environmental Stochasticity

• Recall our general form for a dynamic model
• So that N(t) can be derived by
• Creating a recursive equation (for difference
  equations)
• Integrating (for differential equations)

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Mechanics Adding Environmental Stochasticity

• In stochastic models, we presume that the dynamic
  equation is a probability distribution, so that
• Where v(t) is some random variable with a mean 0.

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Density-Independent Model

• Deterministic Model
• We can predict population size 2 time steps into
  the future
• Or any n time steps into the future

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Adding Stochasicity

• Presume that l varies over time according to some
  distribution
• N(t1)l(t)N(t)
• Each model run is unique
• Were interested in the distributionof N(t)s

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Why does stochasticity lower overall growth rate

• Consider a population changing over 500 years
  N(t1)l(t)N(t)
• During good years, l 1.16
• During bad years, l 0.86
• The probability of a good or bad year is 50
• N(t1)ltlt-1lt-2. l2 l1 loN(0)
• The arithmetic mean of l (lA)equals 1.01
  (implying slight population growth)

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Model Result
There are exactly 250 good and 250 bad
years This produces a net reduction in
population size from time 0 to t 500 The
arithmetic mean l doesnt tell us much about the
actual population trajectory!
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Why does stochasticity lower overall growth rate

• N(t1)ltlt-1lt-2. l2 l1 loN(0)
• There are 250 good l and 250 bad l
• N(500)1.16250 x 0.86250N(0)
• N(500)0.9988 N(0)
• Instead of the arithmetic mean, the population
  size at year 500 is determined by the geometric
  mean
• The geometric mean is ALWAYS less than the
  arithmetic mean

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Calculating Geometric Mean

• Remember
• ln (l1 x l2 x l3 x l4)ln(l1)ln(l2)ln(l3)ln(l4)
  
• So that geometric mean lG exp(ln(lt))
• It is sometimes convenient to replace ln(l) with r

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Mean and Variance of N(t)

• If we presume that r is normally distributed with
  mean r and variance s2
• Then the mean and variance of the possible
  population sizes at time t equals

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Probability Distributions of Future Population
Sizes
r N(0.08,0.15)
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Application

• Grizzly bears in the greater Yellowstone
  ecosystem are a federally listed species
• There are annual counts of females with cubs to
  provide an index of population trends 1957 to
  present
• We presume that the extinction risk becomes very
  high when adult female counts is less than 20

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Trends in Grizzly Bear Abundance

• From the N(t),we can calculate the ln
  (N(t1)/N(t)) to get r(t)
• From this, we can calculate the mean and
  variance of r
• For these data, mean r 0.02 and variance s2
  0.0123

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Apply stochastic population model

• This is a result of 100 stochastic simulations,
  showing the upper and lower 5th percentiles
• This says it is unlikely that adult female
  grizzly numbers will drop below 20

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But wait!

• That simulation presumed that we knew the mean of
  r perfectly

95 confidence interval for r -0.015
0.58 We need to account for uncertainty in r as
well (much harder) Including this uncertainty
leads to a much less optimistic outlook (95
confidence interval for 2050 includes 20)
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Other issues autocorrelated variance

• The examples so far assumed that the r(t) were
  independent of each other
• That is, r(t) did not depend on r(t-1) in any way
• We can add correlation in the following way
• r is the autocorrelation coefficient.
• r 0 means no temporal correlation

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Three time series of r

• For all, v(t) had mean 0 and variance 0.06

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Density Dependence

• In a density-dependent model, we need to account
  for the effect of population size on r(t)
  (per-capita growth rate)
• Typically, we presume that the mean r(t)
  increases as population sizes become small
• This is called compensation because r(t)
  compensates for low population size
• This should rescue declining populations

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Compensatory vs. depensatory

• Our general model
• f(N) is the per capita growth rate
• In a compensatory model f(N) is always a
  decreasing function of N
• In a depensatory model, f(N) may be an
  increasing function of N
• Also sometimes called an Allee effect

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Compensatory vs. depensatory
Per-Capita Growth Rate, f(N)
Population Size (N)
Below this point, population growth rate will be
negative
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Lab this week

• Create your own stochastic density-independent
  population model and evaluate extinction risk
• Evaluate the effects of autocorrelated variance
  on extinction risk
• Evaluate the interactive effect of stochastic
  variance and Allee effects on extinction risk