s to chas tic population models
TRANSCRIPT
8/12/2019 s to Chas Tic Population Models
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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 ourmodels
• Stochasticity might reflect: – Environmental stochasticity
– Demographic stochasticity
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Demographic stochasicity
• We often depict the number of surviving individualsfrom one time point to another as the product ofNumbers at time t (N(t)) times an average survivorship
• This works well when N is very large (in the 1000’s ormore)
• For instance, if I flip a coin 1000 times, I’m pretty surethat 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 0heads
– The approximation N = p * 10 doesn’t 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
– What’s the probability of extinction
– What’s 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 differenceequations)
– Integrating (for differential equations)
)(
)(
N f dt
dN
N f t
N
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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.
)())(()(
t vt N f t
t N
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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:
)()1(
)()1()1(
)()()(
t N t N
t N d bt N
t N d bt
t N
)()()1()2( 2 t N t N t N t N
)()( t N nt N n
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Adding Stochasicity
• Presume that varies over time according to
some distribution
N(t+1)=(t)N(t)
• Each model run
is unique
• We’re interested
in the distribution
of N(t)s
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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)=[t
t-1
t-2
…. 2
1
o
]N(0)
• The “arithmetic” mean of (A)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 doesn’t
tell us much
about the actual
population
trajectory!
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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 atyear 500 is determined by the geometric mean:
• The geometric mean is ALWAYS less than thearithmetic mean
t
t G
t
1
)(
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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
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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
1)exp()2exp()0(
)exp()0()(
222
)(
t t r N
t r N t N
r t N s s
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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 greaterYellowstone ecosystem are afederally listed species
• There are annual counts offemales with cubs to providean index of population trends1957 to present
• We presume that the
extinction risk becomes veryhigh when adult femalecounts is less than 20
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Trends in Grizzly Bear Abundance
• From the N(t),wecan calculate theln (N(t+1)/N(t)) toget r(t)
• From this, we cancalculate the meanand variance of r
•
For these data,mean r = 0.02 andvariance σ2 = 0.0123
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Apply stochastic population model
• This is a result of 100
stochastic
simulations, showing
the upper and lower5th percentiles
• This says it is unlikely
that adult female
grizzly numbers willdrop 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) wereindependent 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
),0(~)(
)()1()(
2s
r
N t v
t vr r t r t r
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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 anincreasing function of N
– Also sometimes called an “Allee effect”
)(')( N Nf N f t
N
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Compensatory vs. depensatory
)(')( N Nf N f t
N
P e r - C a p i t a G r o w t h R a
t e ,
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