BIOL 4150 Lecture Notes - Lecture 6: Song Sparrow, London Academy Of Music And Dramatic Art, Toshiyori

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10/19/17
Chapter 16&17
Predict future population size
!
Predict probability of extinction over time
!
Assess which management strategy will be most
effective at reducing extinction risk
!
Explore consequences of different assumed
mechanisms on population persistence
!
Single-species models
Consumer-resource models
Individually-based models
Can be done several ways:
!
Time series data on abundance
!
Survival rates
!
Birth rates
!
Dispersal rates
!
Interaction terms
*if unavailable for a species, use
parameters of surrogate or closely-related
species
!
Parameters drawn from real systems:
Stochastic modeling used to predict future
dynamics
Extensive replication usually employed to assess
risk of extinction or calculate confidence limits
on predictions
Common elements:
!
PVAs:
Some populations are so small or have such low growth
rates that they are unlikely to experience any major
changes in net recruitment due to increasing density
!
yearmax<-20 *set limit on number of years
!
Year<-1:yearmax *variable years: 1-20
!
N<-numeric(yearmax) *assign variable N
to open slots of data from 1-20
!
lambda<-1.649 *assign growth rate of
population (annual growth)
!
N[1]<-1.5 *assign initial population size
!
for(t in 2:yearmax){N[t]<-lamba*N[t-
l]}*calculate populations in following
years
!
plot(N~Year)
!
rm(list=ls())
!
In R:
If so, then we would expect the population to grow
according to the geometric growth model
!
Single Species Models:
'u'=mean(r)=-8.586*10^-4
*use exponential growth rate (r=ln(lambda))
!
Over the first 24 years, the mean growth rate of
Yellowstone grizzlies was below the level needed for
persistence
!
Random nature
Limitations in census
!
Small sample size (sampling variation)
Demographic stochasticity
May not look like a 'perfect bell shape' when graphing
frequency and deviations
!
Sigma = 0.08
!
Calculate the residual variation in r around the
mean (see slide for equation)
In order to predict the probability of extinction over a
particular time frame, however, we also need to know
how variable population growth can been in the past
!
We assume that 10 bears is the lower critical
threshold (roughly 20% of historic abundance
level)
yearmax<-101
!
Year<-1:yearmax
!
N<-numeric(yearmax)
!
meanr<- -0.00086
!
sdr<-0.08
!
N[1]<-41
!
r<-rnorm(yearmax,meanr,sdr) ** sets up
normal distribution
!
hist(r) *histogram of r
!
In R:
Each fresh draw will look slightly different
!
Random deviates are drawn from a bell-shaped
curve with similar 'u' and 'sigma^2' as the original
data
yearmax<-101
!
Year<-1:yearmax
!
N<-numeric(yearmax)
!
meanr<- -0.00086
!
sdr<-0.08
!
N[1]<-41
!
r<-rnorm(yearmax,meanr,sdr) ** sets up
normal distribution
!
for(t in 2:yearmax){N[t]<-exp(r[t])*N[t-1]}
!
plot(N~Year)
!
*in this simulation population hit a low
value of 10 around year 80
*some simulations recover
-small probability (but still a chance)
of going to tails on normal
distribution
Therefore, must repeat process
Using data to drawn conclusion about fictious
population:
Repetition of this process suggests roughly
a 5-10% risk of extinction within the next
century, or 1 extinction event every 1-2,000
years
!
The spread in model abundance over time
is often well approximated by models of
diffusion (---> quasi-extinction threshold)
!
Length of time of simulation (long
time --> more likely to cross
threshold)
"
Quasi-extinction threshold
"
Assumes demographics stay the same
!
In order to estimate the risk of quasi-extinction
(N<=10):
We will use these parameter estimates to set up a
simple Monte Carlo simulation of stochastic bear
population dynamics
!
Ex. Population Viability of Yellowstone grizzlies
Expectation of 80% decline over next 10 years
!
Expectation of 80% decline over next 3 generations
!
*IUCN criteria for critical endangerment
10/24/17
*deviations around regression line --> sdr
!
yearmax<-50
Year<-1:yearmax
N<-numeric(yearmax)
rmax<-0.493
K<-12609
sdr<-0.179
N[1]<-10000
noise<-rnorm(yearmax,0,sdr) *noise = vector
for(t in 2:yearmax){N[t]<-N[t-1]*exp(rmax*(1-
N[t-1]/K) + noise[t-1])}
plot(N~Year)
For(i in 1:10){
N[1]<-10000
noise<-rnorm(yearmax,0,sdr)
for(t in 2:yearmax){N[t]<-N[t-1]*exp(rmax*(1-
N[t-1]/K) + noise[t-1])}
lines(Year-1,N)
R:
!
*see stochastic changes in population size
!
Helps determine population viability for
management decisions
Embracing variability
Shows potential changes in population size based on
density-independent factors
!
Ex. Elk Population
Age-specific survivorship and fecundity are
probabilities
!
As in any game of chance, the probability of
extraordinary runs of good or bad luck for a population
depend critically on population size
!
We term such processes = demographic stochasticity,
because uncertainty in population dynamics from one
time to another depends heavily on unpredictable
demographic events
!
Demographic Stochasticity
p = probability of survival
1-p = probability of death
N = initial population size
z = number of survivors
Let…
!
Probability of survival increases with number of
survivors
Pr(z) = (N! / (z!*N -z!)) * p^z * (1-p)^(N-z)
!
Probability of extinction decreases with
increasing population size
Extinct(N) = (1-p)^N
!
Binomial Probability Distribution
Density = 0.01 /km^2
rmax = 0.28
Each female can produce 1 offspring/year, with
probabilitiy = 0.95
Adult survival = 0.9
Population characteristics:
!
100,000 km^2
Density --> 1,000 adults
Next year, expect 900 adults and 428 offspring =
1,328
Potential outcomes in a large reserve:
!
200 km^2
Density --> 2 adults (1 male, 1 female?)
Both die (r= -infinity) -0 < p <0.05
!
One dies (r=-0.61) -p ~0.2
!
Both live, no offspring (r=0) -p ~ 0.05
!
Both live and produce 1 offspring (r=
0.41) -p ~ 0.75
!
Next year has a range of possibilities
Potential outcomes in a small reserve:
!
Hypothetical Case:
This threshold will be higher in species with low
survival rates and variable fecundity than those with
more constant demographic parameters
!
Wildlife population reduced to <30 individuals have a
meaningful chance of going extinct though chance
demographic events
Population size (male and female) fluctuate over time
!
Song sparrow show an increase in non-territorial males
with increased density
!
Song sparrows also show decreased reproductive rates
with increased density
!
Juvenile survival also declines with density
!
Var(rd) = (1/ n-1)* sum of (wi -wmean)
!
Var(rd) = 0.66
!
Variation in fitness among individual song
sparrows over several years
Var(rd) = 0.66
!
Var(re) = ( (deltaN/N) -g(N) +1)^2 -
(Var(rd) / N)
"
Var(re) = 0.41
!
*where g(N) is the annual growth rate
(lamda) predicted by the logistic model
!
Sigmat = sqrt( Var(re) + Var(rd)/N)
!
Relative contributions of demographic vs
environmental stochasticity can be teased apart
using:
Perturbation experiment to test food hypothesis:
!
tmax<-31
runmax<-20
Year<-0:(tmax-1)
N<-numeric(tmax)
N<-N+1
epsilon<-numeric(tmax)
rmax<-0.99
K<-41.5
theta<-1.09
vard<-0.66
vare<-0.41
N[1]<-K
extinct<-0
for(t in 2:tmax){
sdr<-sqrt(vare + vard/N[t-1])
epsilon[t-1]<-rnorm(1,0,sdr)
N[t]<-exp(rmax*(1-(N[t-1]/K)^theta) +
epsiolon[t-1])*N[t-1]
if(N[t]<1.0){N[t]<-1.0;break}
}
extinct<-extinct + ifelse(N[tmax]<=1,1,0)
plot(Year,N,type='l',ylim=c(1,200),log='y')
Lines(Year,N,col=i)
}
runmax
extinct
extinct/runmax
R Coding:
!
Long-term study of song sparrows on Mandarte Island (BC):
10/26/17
Eastern wolves (78%)
Better represented in areas of high moose
abundance and fewer roads
!
Gray wolves
Coyotes
Canids of Algonquin:
!
p=0.14, r2=0.25
!
A study completed from 1989-1999 determined
weak density dependence
Human-caused mortality was leading source of
death
Wolf density: has fluctuated from 1988 to 2000
!
yearmax<-100
Year<-1:yearmax
N<-numeric(yearmax)
rmean<--0.11
sdr<-0.4243
N[1]<-5
noise<-rnorm(yearmax,0,sdr)
#######geometric model#######
for(t in 2:yearmax){N[t]<-
N[t-1]*exp(rmean+noise[t-1])}
plot(N~Year,type='l',ylim=c(0,20))
for(i in 1:20){
noise<-rnorm(yearmax,0,sdr)
for(t in 2:yearmax){N[t]<-
N[t-1]*exp(rmean+noise[t-1])}
}
R-coding: Geometric Model
!
The cumulative probability of quasi-extinction
increases (to almost 1.0 in year 20)
Geometric model: shows N drastically fluctuating but
decreasing to a very low population size by year 100
!
Outcome: ban on human hunting of wolves outside the
park for 36 months
!
Rate of chance decreases (to negative values)
with increasing population density
*density dependent (Ricker) model is the
most parismonious (w=0.796)
!
Model Evaluation:
Equivocal:
!
Rmax<-0.92
K<-2.63
For(t in 2:yearmax){N[t]<-N[t-1]*exp(rmax*(1-
N[t-1]/K) + noise[t-1])}
Plot(N~Year, type='l',ylim=c(0,20))
For(I in 1:20){
N[1]<-5
noise<-rnorm(yearmax,0,sdr)
for(t in 2:yearmax){N[t]<-N[t-1]*exp(rmax*(1-
N[t-1]/K)+noise[t-1])}
lines(N~Year)
}
R-coding: Ricker Logistic Model
!
The cumulative probability of quasi-extinction
varies (from 0.5 to 0.9) over 20 years
Ricker Logistic model shows fluctuating population
size between 1-9 over 100 years (no population crash)
!
If the logistic model predicts a much lower risk of
population collapse, which interpretation is correct?
!
Case Study: PVA of Algonquin wolves
Larger probabilities of
r being small or large
with smaller N
!
*deviations around regression
become smaller as N gets
larger (r vs. N)
Population Viability Analysis
#$%&'()*+, -./012&, 34+,5637
3389:,;<
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10/19/17
Chapter 16&17
Predict future population size
!
Predict probability of extinction over time
!
Assess which management strategy will be most
effective at reducing extinction risk
!
Explore consequences of different assumed
mechanisms on population persistence
!
Single-species models
Consumer-resource models
Spatially-structured models
Individually-based models
Can be done several ways:
!
Time series data on abundance
!
Survival rates
!
Birth rates
!
Dispersal rates
!
Interaction terms
*if unavailable for a species, use
parameters of surrogate or closely-related
species
!
Parameters drawn from real systems:
Stochastic modeling used to predict future
dynamics
Extensive replication usually employed to assess
risk of extinction or calculate confidence limits
on predictions
Common elements:
!
PVAs:
Some populations are so small or have such low growth
rates that they are unlikely to experience any major
changes in net recruitment due to increasing density
!
yearmax<-20 *set limit on number of years
!
Year<-1:yearmax *variable years: 1-20
!
N<-numeric(yearmax) *assign variable N
to open slots of data from 1-20
!
lambda<-1.649 *assign growth rate of
population (annual growth)
!
N[1]<-1.5 *assign initial population size
!
for(t in 2:yearmax){N[t]<-lamba*N[t-
l]}*calculate populations in following
years
!
plot(N~Year)
!
rm(list=ls())
!
In R:
If so, then we would expect the population to grow
according to the geometric growth model
!
Single Species Models:
'u'=mean(r)=-8.586*10^-4
*use exponential growth rate (r=ln(lambda))
!
Over the first 24 years, the mean growth rate of
Yellowstone grizzlies was below the level needed for
persistence
!
Random nature
Limitations in census
!
Small sample size (sampling variation)
Demographic stochasticity
May not look like a 'perfect bell shape' when graphing
frequency and deviations
!
Sigma = 0.08
!
Calculate the residual variation in r around the
mean (see slide for equation)
In order to predict the probability of extinction over a
particular time frame, however, we also need to know
how variable population growth can been in the past
!
We assume that 10 bears is the lower critical
threshold (roughly 20% of historic abundance
level)
yearmax<-101
!
Year<-1:yearmax
!
N<-numeric(yearmax)
!
meanr<- -0.00086
!
sdr<-0.08
!
N[1]<-41
!
r<-rnorm(yearmax,meanr,sdr) ** sets up
normal distribution
!
hist(r) *histogram of r
!
In R:
Each fresh draw will look slightly different
!
Random deviates are drawn from a bell-shaped
curve with similar 'u' and 'sigma^2' as the original
data
yearmax<-101
!
Year<-1:yearmax
!
N<-numeric(yearmax)
!
meanr<- -0.00086
!
sdr<-0.08
!
N[1]<-41
!
r<-rnorm(yearmax,meanr,sdr) ** sets up
normal distribution
!
for(t in 2:yearmax){N[t]<-exp(r[t])*N[t-1]}
!
plot(N~Year)
!
*in this simulation population hit a low
value of 10 around year 80
*some simulations recover
-small probability (but still a chance)
of going to tails on normal
distribution
Therefore, must repeat process
Using data to drawn conclusion about fictious
population:
Repetition of this process suggests roughly
a 5-10% risk of extinction within the next
century, or 1 extinction event every 1-2,000
years
!
The spread in model abundance over time
is often well approximated by models of
diffusion (---> quasi-extinction threshold)
!
Length of time of simulation (long
time --> more likely to cross
threshold)
"
Quasi-extinction threshold
"
Assumes demographics stay the same
!
In order to estimate the risk of quasi-extinction
(N<=10):
We will use these parameter estimates to set up a
simple Monte Carlo simulation of stochastic bear
population dynamics
!
Ex. Population Viability of Yellowstone grizzlies
Expectation of 80% decline over next 10 years
!
Expectation of 80% decline over next 3 generations
!
*IUCN criteria for critical endangerment
10/24/17
*deviations around regression line --> sdr
!
yearmax<-50
Year<-1:yearmax
N<-numeric(yearmax)
rmax<-0.493
K<-12609
sdr<-0.179
N[1]<-10000
noise<-rnorm(yearmax,0,sdr) *noise = vector
for(t in 2:yearmax){N[t]<-N[t-1]*exp(rmax*(1-
N[t-1]/K) + noise[t-1])}
plot(N~Year)
For(i in 1:10){
N[1]<-10000
noise<-rnorm(yearmax,0,sdr)
for(t in 2:yearmax){N[t]<-N[t-1]*exp(rmax*(1-
N[t-1]/K) + noise[t-1])}
lines(Year-1,N)
R:
!
*see stochastic changes in population size
!
Helps determine population viability for
management decisions
Embracing variability
Shows potential changes in population size based on
density-independent factors
!
Ex. Elk Population
Age-specific survivorship and fecundity are
probabilities
!
As in any game of chance, the probability of
extraordinary runs of good or bad luck for a population
depend critically on population size
!
We term such processes = demographic stochasticity,
because uncertainty in population dynamics from one
time to another depends heavily on unpredictable
demographic events
!
Demographic Stochasticity
p = probability of survival
1-p = probability of death
N = initial population size
z = number of survivors
Let…
!
Probability of survival increases with number of
survivors
Pr(z) = (N! / (z!*N -z!)) * p^z * (1-p)^(N-z)
!
Probability of extinction decreases with
increasing population size
Extinct(N) = (1-p)^N
!
Binomial Probability Distribution
Density = 0.01 /km^2
rmax = 0.28
Each female can produce 1 offspring/year, with
probabilitiy = 0.95
Adult survival = 0.9
Population characteristics:
!
100,000 km^2
Density --> 1,000 adults
Next year, expect 900 adults and 428 offspring =
1,328
Potential outcomes in a large reserve:
!
200 km^2
Density --> 2 adults (1 male, 1 female?)
Both die (r= -infinity) -0 < p <0.05
!
One dies (r=-0.61) -p ~0.2
!
Both live, no offspring (r=0) -p ~ 0.05
!
Both live and produce 1 offspring (r=
0.41) -p ~ 0.75
!
Next year has a range of possibilities
Potential outcomes in a small reserve:
!
Hypothetical Case:
This threshold will be higher in species with low
survival rates and variable fecundity than those with
more constant demographic parameters
!
Wildlife population reduced to <30 individuals have a
meaningful chance of going extinct though chance
demographic events
Population size (male and female) fluctuate over time
!
Song sparrow show an increase in non-territorial males
with increased density
!
Song sparrows also show decreased reproductive rates
with increased density
!
Juvenile survival also declines with density
!
Var(rd) = (1/ n-1)* sum of (wi -wmean)
!
Var(rd) = 0.66
!
Variation in fitness among individual song
sparrows over several years
Var(rd) = 0.66
!
Var(re) = ( (deltaN/N) -g(N) +1)^2 -
(Var(rd) / N)
"
Var(re) = 0.41
!
*where g(N) is the annual growth rate
(lamda) predicted by the logistic model
!
Sigmat = sqrt( Var(re) + Var(rd)/N)
!
Relative contributions of demographic vs
environmental stochasticity can be teased apart
using:
Perturbation experiment to test food hypothesis:
!
tmax<-31
runmax<-20
Year<-0:(tmax-1)
N<-numeric(tmax)
N<-N+1
epsilon<-numeric(tmax)
rmax<-0.99
K<-41.5
theta<-1.09
vard<-0.66
vare<-0.41
N[1]<-K
extinct<-0
for(t in 2:tmax){
sdr<-sqrt(vare + vard/N[t-1])
epsilon[t-1]<-rnorm(1,0,sdr)
N[t]<-exp(rmax*(1-(N[t-1]/K)^theta) +
epsiolon[t-1])*N[t-1]
if(N[t]<1.0){N[t]<-1.0;break}
}
extinct<-extinct + ifelse(N[tmax]<=1,1,0)
plot(Year,N,type='l',ylim=c(1,200),log='y')
Lines(Year,N,col=i)
}
runmax
extinct
extinct/runmax
R Coding:
!
Long-term study of song sparrows on Mandarte Island (BC):
10/26/17
Eastern wolves (78%)
Better represented in areas of high moose
abundance and fewer roads
!
Gray wolves
Coyotes
Canids of Algonquin:
!
p=0.14, r2=0.25
!
A study completed from 1989-1999 determined
weak density dependence
Human-caused mortality was leading source of
death
Wolf density: has fluctuated from 1988 to 2000
!
yearmax<-100
Year<-1:yearmax
N<-numeric(yearmax)
rmean<--0.11
sdr<-0.4243
N[1]<-5
noise<-rnorm(yearmax,0,sdr)
#######geometric model#######
for(t in 2:yearmax){N[t]<-
N[t-1]*exp(rmean+noise[t-1])}
plot(N~Year,type='l',ylim=c(0,20))
for(i in 1:20){
noise<-rnorm(yearmax,0,sdr)
for(t in 2:yearmax){N[t]<-
N[t-1]*exp(rmean+noise[t-1])}
}
R-coding: Geometric Model
!
The cumulative probability of quasi-extinction
increases (to almost 1.0 in year 20)
Geometric model: shows N drastically fluctuating but
decreasing to a very low population size by year 100
!
Outcome: ban on human hunting of wolves outside the
park for 36 months
!
Rate of chance decreases (to negative values)
with increasing population density
*density dependent (Ricker) model is the
most parismonious (w=0.796)
!
Model Evaluation:
Equivocal:
!
Rmax<-0.92
K<-2.63
For(t in 2:yearmax){N[t]<-N[t-1]*exp(rmax*(1-
N[t-1]/K) + noise[t-1])}
Plot(N~Year, type='l',ylim=c(0,20))
For(I in 1:20){
N[1]<-5
noise<-rnorm(yearmax,0,sdr)
for(t in 2:yearmax){N[t]<-N[t-1]*exp(rmax*(1-
N[t-1]/K)+noise[t-1])}
lines(N~Year)
}
R-coding: Ricker Logistic Model
!
The cumulative probability of quasi-extinction
varies (from 0.5 to 0.9) over 20 years
Ricker Logistic model shows fluctuating population
size between 1-9 over 100 years (no population crash)
!
If the logistic model predicts a much lower risk of
population collapse, which interpretation is correct?
!
Case Study: PVA of Algonquin wolves
Larger probabilities of
r being small or large
with smaller N
!
*deviations around regression
become smaller as N gets
larger (r vs. N)
Population Viability Analysis
#$%&'()*+, -./012&, 34+,5637 3389:,;<
Unlock document

This preview shows pages 1-3 of the document.
Unlock all 10 pages and 3 million more documents.

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10/19/17
Chapter 16&17
Predict future population size
!
Predict probability of extinction over time
!
Assess which management strategy will be most
effective at reducing extinction risk
!
Explore consequences of different assumed
mechanisms on population persistence
!
Single-species models
Consumer-resource models
Spatially-structured models
Individually-based models
Can be done several ways:
!
Time series data on abundance
!
Survival rates
!
Birth rates
!
Dispersal rates
!
Interaction terms
*if unavailable for a species, use
parameters of surrogate or closely-related
species
!
Parameters drawn from real systems:
Stochastic modeling used to predict future
dynamics
Extensive replication usually employed to assess
risk of extinction or calculate confidence limits
on predictions
Common elements:
!
PVAs:
Some populations are so small or have such low growth
rates that they are unlikely to experience any major
changes in net recruitment due to increasing density
!
yearmax<-20 *set limit on number of years
!
Year<-1:yearmax *variable years: 1-20
!
N<-numeric(yearmax) *assign variable N
to open slots of data from 1-20
!
lambda<-1.649 *assign growth rate of
population (annual growth)
!
N[1]<-1.5 *assign initial population size
!
for(t in 2:yearmax){N[t]<-lamba*N[t-
l]}*calculate populations in following
years
!
plot(N~Year)
!
rm(list=ls())
!
In R:
If so, then we would expect the population to grow
according to the geometric growth model
!
Single Species Models:
'u'=mean(r)=-8.586*10^-4
*use exponential growth rate (r=ln(lambda))
!
Over the first 24 years, the mean growth rate of
Yellowstone grizzlies was below the level needed for
persistence
!
Random nature
Limitations in census
!
Small sample size (sampling variation)
Demographic stochasticity
May not look like a 'perfect bell shape' when graphing
frequency and deviations
!
Sigma = 0.08
!
Calculate the residual variation in r around the
mean (see slide for equation)
In order to predict the probability of extinction over a
particular time frame, however, we also need to know
how variable population growth can been in the past
!
We assume that 10 bears is the lower critical
threshold (roughly 20% of historic abundance
level)
yearmax<-101
!
Year<-1:yearmax
!
N<-numeric(yearmax)
!
meanr<- -0.00086
!
sdr<-0.08
!
N[1]<-41
!
r<-rnorm(yearmax,meanr,sdr) ** sets up
normal distribution
!
hist(r) *histogram of r
!
In R:
Each fresh draw will look slightly different
!
Random deviates are drawn from a bell-shaped
curve with similar 'u' and 'sigma^2' as the original
data
yearmax<-101
!
Year<-1:yearmax
!
N<-numeric(yearmax)
!
meanr<- -0.00086
!
sdr<-0.08
!
N[1]<-41
!
r<-rnorm(yearmax,meanr,sdr) ** sets up
normal distribution
!
for(t in 2:yearmax){N[t]<-exp(r[t])*N[t-1]}
!
plot(N~Year)
!
*in this simulation population hit a low
value of 10 around year 80
*some simulations recover
-small probability (but still a chance)
of going to tails on normal
distribution
Therefore, must repeat process
Using data to drawn conclusion about fictious
population:
Repetition of this process suggests roughly
a 5-10% risk of extinction within the next
century, or 1 extinction event every 1-2,000
years
!
The spread in model abundance over time
is often well approximated by models of
diffusion (---> quasi-extinction threshold)
!
Length of time of simulation (long
time --> more likely to cross
threshold)
"
Quasi-extinction threshold
"
Assumes demographics stay the same
!
In order to estimate the risk of quasi-extinction
(N<=10):
We will use these parameter estimates to set up a
simple Monte Carlo simulation of stochastic bear
population dynamics
!
Ex. Population Viability of Yellowstone grizzlies
Expectation of 80% decline over next 10 years
!
Expectation of 80% decline over next 3 generations
!
*IUCN criteria for critical endangerment
10/24/17
*deviations around regression line --> sdr
!
yearmax<-50
Year<-1:yearmax
N<-numeric(yearmax)
rmax<-0.493
K<-12609
sdr<-0.179
N[1]<-10000
noise<-rnorm(yearmax,0,sdr) *noise = vector
for(t in 2:yearmax){N[t]<-N[t-1]*exp(rmax*(1-
N[t-1]/K) + noise[t-1])}
plot(N~Year)
For(i in 1:10){
N[1]<-10000
noise<-rnorm(yearmax,0,sdr)
for(t in 2:yearmax){N[t]<-N[t-1]*exp(rmax*(1-
N[t-1]/K) + noise[t-1])}
lines(Year-1,N)
R:
!
*see stochastic changes in population size
!
Helps determine population viability for
management decisions
Embracing variability
Shows potential changes in population size based on
density-independent factors
!
Ex. Elk Population
Age-specific survivorship and fecundity are
probabilities
!
As in any game of chance, the probability of
extraordinary runs of good or bad luck for a population
depend critically on population size
!
We term such processes = demographic stochasticity,
because uncertainty in population dynamics from one
time to another depends heavily on unpredictable
demographic events
!
Demographic Stochasticity
p = probability of survival
1-p = probability of death
N = initial population size
z = number of survivors
Let…
!
Probability of survival increases with number of
survivors
Pr(z) = (N! / (z!*N -z!)) * p^z * (1-p)^(N-z)
!
Probability of extinction decreases with
increasing population size
Extinct(N) = (1-p)^N
!
Binomial Probability Distribution
Density = 0.01 /km^2
rmax = 0.28
Each female can produce 1 offspring/year, with
probabilitiy = 0.95
Adult survival = 0.9
Population characteristics:
!
100,000 km^2
Density --> 1,000 adults
Next year, expect 900 adults and 428 offspring =
1,328
Potential outcomes in a large reserve:
!
200 km^2
Density --> 2 adults (1 male, 1 female?)
Both die (r= -infinity) -0 < p <0.05
!
One dies (r=-0.61) -p ~0.2
!
Both live, no offspring (r=0) -p ~ 0.05
!
Both live and produce 1 offspring (r=
0.41) -p ~ 0.75
!
Next year has a range of possibilities
Potential outcomes in a small reserve:
!
Hypothetical Case:
This threshold will be higher in species with low
survival rates and variable fecundity than those with
more constant demographic parameters
!
Wildlife population reduced to <30 individuals have a
meaningful chance of going extinct though chance
demographic events
Population size (male and female) fluctuate over time
!
Song sparrow show an increase in non-territorial males
with increased density
!
Song sparrows also show decreased reproductive rates
with increased density
!
Juvenile survival also declines with density
!
Var(rd) = (1/ n-1)* sum of (wi -wmean)
!
Var(rd) = 0.66
!
Variation in fitness among individual song
sparrows over several years
Var(rd) = 0.66
!
Var(re) = ( (deltaN/N) -g(N) +1)^2 -
(Var(rd) / N)
"
Var(re) = 0.41
!
*where g(N) is the annual growth rate
(lamda) predicted by the logistic model
!
Sigmat = sqrt( Var(re) + Var(rd)/N)
!
Relative contributions of demographic vs
environmental stochasticity can be teased apart
using:
Perturbation experiment to test food hypothesis:
!
tmax<-31
runmax<-20
Year<-0:(tmax-1)
N<-numeric(tmax)
N<-N+1
epsilon<-numeric(tmax)
rmax<-0.99
K<-41.5
theta<-1.09
vard<-0.66
vare<-0.41
N[1]<-K
extinct<-0
for(t in 2:tmax){
sdr<-sqrt(vare + vard/N[t-1])
epsilon[t-1]<-rnorm(1,0,sdr)
N[t]<-exp(rmax*(1-(N[t-1]/K)^theta) +
epsiolon[t-1])*N[t-1]
if(N[t]<1.0){N[t]<-1.0;break}
}
extinct<-extinct + ifelse(N[tmax]<=1,1,0)
plot(Year,N,type='l',ylim=c(1,200),log='y')
Lines(Year,N,col=i)
}
runmax
extinct
extinct/runmax
R Coding:
!
Long-term study of song sparrows on Mandarte Island (BC):
10/26/17
Eastern wolves (78%)
Better represented in areas of high moose
abundance and fewer roads
!
Gray wolves
Coyotes
Canids of Algonquin:
!
p=0.14, r2=0.25
!
A study completed from 1989-1999 determined
weak density dependence
Human-caused mortality was leading source of
death
Wolf density: has fluctuated from 1988 to 2000
!
yearmax<-100
Year<-1:yearmax
N<-numeric(yearmax)
rmean<--0.11
sdr<-0.4243
N[1]<-5
noise<-rnorm(yearmax,0,sdr)
#######geometric model#######
for(t in 2:yearmax){N[t]<-
N[t-1]*exp(rmean+noise[t-1])}
plot(N~Year,type='l',ylim=c(0,20))
for(i in 1:20){
noise<-rnorm(yearmax,0,sdr)
for(t in 2:yearmax){N[t]<-
N[t-1]*exp(rmean+noise[t-1])}
}
R-coding: Geometric Model
!
The cumulative probability of quasi-extinction
increases (to almost 1.0 in year 20)
Geometric model: shows N drastically fluctuating but
decreasing to a very low population size by year 100
!
Outcome: ban on human hunting of wolves outside the
park for 36 months
!
Rate of chance decreases (to negative values)
with increasing population density
*density dependent (Ricker) model is the
most parismonious (w=0.796)
!
Model Evaluation:
Equivocal:
!
Rmax<-0.92
K<-2.63
For(t in 2:yearmax){N[t]<-N[t-1]*exp(rmax*(1-
N[t-1]/K) + noise[t-1])}
Plot(N~Year, type='l',ylim=c(0,20))
For(I in 1:20){
N[1]<-5
noise<-rnorm(yearmax,0,sdr)
for(t in 2:yearmax){N[t]<-N[t-1]*exp(rmax*(1-
N[t-1]/K)+noise[t-1])}
lines(N~Year)
}
R-coding: Ricker Logistic Model
!
The cumulative probability of quasi-extinction
varies (from 0.5 to 0.9) over 20 years
Ricker Logistic model shows fluctuating population
size between 1-9 over 100 years (no population crash)
!
If the logistic model predicts a much lower risk of
population collapse, which interpretation is correct?
!
Case Study: PVA of Algonquin wolves
Larger probabilities of
r being small or large
with smaller N
!
*deviations around regression
become smaller as N gets
larger (r vs. N)
Population Viability Analysis
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Document Summary

Assess which management strategy will be most effective at reducing extinction risk. Explore consequences of different assumed mechanisms on population persistence. *if unavailable for a species, use parameters of surrogate or closely-related species. Extensive replication usually employed to assess risk of extinction or calculate confidence limits on predictions. Some populations are so small or have such low growth rates that they are unlikely to experience any major changes in net recruitment due to increasing density. If so, then we would expect the population to grow changes in net recruitment due to increasing density. If so, then we would expect the population to grow according to the geometric growth model. In r: yearmax

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