stochastic population modelling qsci/ fish 454. stochastic vs. deterministic so far, all models...
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Stochastic Population Modelling
QSCI/ Fish 454
Stochastic vs. deterministic
• So far, all models we’ve 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
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 1000’s or more)
• For instance, if I flip a coin 1000 times, I’m pretty sure that I’m 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 doesn’t work so well
Why consider stochasticity?
• Stochasticity generally lowers population growth rates
• “Autocorrelated” stochasticity REALLY lowers population growth rates
• Allows for risk assessment– What’s the probability of extinction– What’s the probability of reaching a minimum
threshold size
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)
)(
)(
Nfdt
dN
Nft
N
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.
)())(()(
tvtNft
tN
Density-Independent Model
• Deterministic Model:
• We can predict population size 2 time steps into the future:
• Or any ‘n’ time steps into the future:
)()1(
)()1()1(
)()()(
tNtN
tNdbtN
tNdbt
tN
)()()1()2( 2 tNtNtNtN
)()( tNntN n
Adding Stochasicity
• Presume that varies over time according to some distributionN(t+1)=(t)N(t)
• Each model run is unique
• We’re interested in the distributionof N(t)s
Why does stochasticity lower overall growth rate
• Consider a population changing over 500 years: N(t+1)=(t)N(t)– During “good” years, = 1.16– During “bad” years, = 0.86
• The probability of a good or bad year is 50%• N(t+1)=[tt-1t-2….2 1 o]N(0)
• The “arithmetic” mean of (A)equals 1.01 (implying slight population growth)
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 doesn’t tell us much about the actual population trajectory!
Why does stochasticity lower overall growth rate
• N(t+1)=[tt-1t-2….2 1 o]N(0)• There are 250 good and 250 bad • N(500)=[1.16250 x 0.86250]N(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
t
tG t
1
)(
Calculating Geometric Mean
• Remember:ln (1 x 2 x 3 x 4)=ln(1)+ln(2)+ln(3)+ln(4)
So that geometric mean G = exp(ln(t))
It is sometimes convenient to replace ln() with r
Mean and Variance of N(t)
• If we presume that r is normally distributed with mean r and variance 2
• Then the mean and variance of the possible population sizes at time t equals
1)exp()2exp()0(
)exp()0()(222
)(
ttrN
trNtN
rtN
Probability Distributions of Future Population Sizes
r ~ N(0.08,0.15)
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
Trends in Grizzly Bear Abundance
• From the N(t),we can calculate the ln (N(t+1)/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 σ2 = 0.0123
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
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)
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:
• is the “autocorrelation” coefficient. 0 means no temporal correlation
),0(~)(
)()1()(2
Ntv
tvrrtrtr
Three time series of r
• For all, v(t) had mean 0 and variance 0.06
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
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”
)(')( NNfNft
N
Compensatory vs. depensatory
)(')( NNfNft
N
Per-
Capi
ta G
row
th R
ate,
f’(N
)
Population Size (N)
Below this point, population growth rate will be negative
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
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