s to Chas Tic Population Models

8/12/2019 s to Chas Tic Population Models

http://slidepdf.com/reader/full/s-to-chas-tic-population-models 1/24



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



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



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 differenceequations)

– Integrating (for differential equations)

)(

)(

N f dt

dN

N f t

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.

)())(()(

t vt N f t

t N



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



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



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)



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 atyear 500 is determined by the geometric mean:

• The geometric mean is ALWAYS less than thearithmetic mean

t

t G

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 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



Probability Distributions of Future

Population Sizes

r ~ N(0.08,0.15)



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



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



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



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) 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



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 anincreasing function of N

– Also sometimes called an “Allee effect”

)(')( N Nf N f t

N



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



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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s to Chas Tic Population Models