count based pva incorporating density dependence, demographic stochasticity, correlated...
TRANSCRIPT
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Count based PVA
Incorporating Density Dependence, Demographic Stochasticity, Correlated Environments,
Catastrophes and Bonanzas
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Assumptions of the diffusion appoximation
• Population growth • Is unaffected by population density • Its only source of variability is environmental
stochasticity• No trends in its mean and the variance • Its values are not correlated in successive years• Moderate variability• No observation error
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But..
• Incorporating these effects into PVA models require:
• more and better data
• more mathematically complex models
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Negative density dependence
• The simplest way to incorporate negative density dependence is introduce a population ceiling to the density-independent population growth model
Nt+1=λtNt ;if Nt < K
K ;if Nt > K
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The ceiling model
Program algo2 (prepared by Matt; 10,50,.55,.45,60)
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Mean time to extinction
cdeec
T cdck 212
1 22
KcKc
T c log212
1 2
Where c=μ/σ2, d=log(Nc/Nx), and k=log(K/Nx)
If Nc=K and Nx =1 then:
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Extinction risk predicted by the Ceiling Model
μ= 0.1
μ= 0.001
μ= -0.1σ2= μ
σ2= 2μ
σ2= 4μ
σ2= 8μ
Program tbarpedro
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The theta logistic model
• A gradually changing growth rate
tt
tt K
NrNN 1exp1
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0
20
40
60
80
100
120
0 10 20 30 40 50 60
Tiempo
Ta
ma
ño
de
po
bla
cio
nK = 100
r = 0.2
Theta:
4
1
0.3
tt
tt K
NrNN 1exp1
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The theta logistic model
0
1
2
3
4
5
6
0 20 40 60 80 100 120
Population size
log(lam
bda)
.
Θ = 0.3
Θ = 1
Θ = 4
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0
20
40
60
80
100
120
140
0 10 20 30 40 50 60
Tiempo
Ta
ma
ño
de
po
bla
cio
nK = 100
r = 0.8
Theta:
4
1
0.3
tt
tt K
NrNN 1exp1
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The Bay checkerspot butterfly
Euphydryas editha bayensis
0
1000
2000
3000
4000
5000
6000
7000
8000
1955 1960 1965 1970 1975 1980 1985 1990
Year
Estim
ate
d n
um
ber
of fe
male
s .
front
Harrison et al., 1991
JRC population
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The negative association remains after removing the outlier in the right
-3
-2
-1
0
1
2
3
0 1000 2000 3000 4000 5000 6000 7000 8000
Tamaño de la poblacion
ln(la
mbd
a)
back
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Density Dependent model
• Find the best model: Fit three models to the data using nonlinear least-squares regression of log(Nt+1/Nt) against Nt
Models to be tested:Density independent model: log(Nt+1/Nt)=r
The Ricker model log(Nt+1/Nt)=r(1-Nt/K)
The theta logistic model log(Nt+1/Nt)=r[1-(Nt/K)Θ]
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Estimate the parameters of each model
Model Least-squares parameter estimates
r K Θ Residual Variance
Density independent
0.001673 1.3999
Ricker 0.3488 846.02 1.0722
Theta logistic 0.9941 551.38 0.4566 1.0165
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Model maximum likelihood of a model
assuming normally distributed deviations is
• ln(Lmax) = -(q/2)[ln(2Vr) +1)
• Vr = residual variance
• q= Sample number
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Maximum log likelihood
• The probability of obtaining the observed data given a particular set of parameter values for a particular model
• Information criterion statistics combine the maximum log likelihood for a model with the number of parameters it include to provide a measure of “support”
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“Support” is higher for:
• models with higher likelihoods, and • models with fewer parameters
More complex models are penalized because more parameters will always lead to a better fit to the data, but at the cost of less precision in the estimate of each parameter and incorporation of spurious patterns from the data into future populations
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Akaike Information Criteria
• To identify the best model:• AICc = -2 ln(Lmax ) + (2pq)/ (q-p-1)
p = Number of estimated parameters (including the residual variance)
q= sampling number
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Akaike weights
Wi =exp[-0.5(AICc,i-AICc,best)]
exp[-0.5(AICc,i-AICbest)]
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Compute the maximum log likelihood and Akaike weights for each model
Model Number of parameters
Including Vr
Log Lmax AIC Akaike weights
Density independent
2 -41.266 87.054 .07
Ricker 3 -37.799 82.689 .62
Theta logistic 4 -37.105 84.115 .31
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Simulate the model to predict population viability
σ2 =qVr
q-1
Program theta_logistic
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Years into the future
Cum
ula
tive p
robabili
ty o
f quasi
-ext
inct
ion
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Years into the future
Cum
ula
tive p
robabili
ty o
f quasi
-ext
inct
ion
Program extprobpedro
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Simulate the model to predict population viability
Program extprobpedro Program theta_denindeppedro
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Years into the future
Cum
ula
tive p
robabili
ty o
f quasi
-ext
inct
ion
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Years into the future
Cum
ula
tive p
robabili
ty o
f quasi
-ext
inct
ion
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Allee effects
• We can simply set the quasi-extinction threshold at or above the population size at which Alee effects become important
• Explicitly include Alee effects in the population model
Nt+1 =Nt
2
A+Nt
е r-βNt
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The parameters
0
2
4
6
8
1 10 100 1000
N
O
The potential offspring
Value at A
maximum
0
0.2
0.4
0.6
0.8
1
1.2
1 10 100 1000
N
'-b
etaN
.
Fraction of potential reproduction that is actually achieved
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A discrete-time model with Alee effects generated by mate-finding
problems
00.5
11.5
22.5
33.5
1 10 100 1000
N
"r"
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A discrete-time model with Alee effects generated by mate-
finding problems
0
10
20
30
40
50
60
0 5 10 15 20
Time, t
Po
pu
lati
on
siz
e, N
t
.
10
15
25
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Combined effects of Demographic
Environmental stochasticity
t
m
iitt mCC
t
/1
tm
itit
td CC
mtV
1
2
1
1)(
t
dt
t
t
N
tVNf
N
Nt
)()()(
2
12
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Combined effects of Demographic and
Environmental stochasticity
5 10 15 2010
-2
10-1
100
101
102
Time (years)
Pop
ulat
ion
dens
ity
5 10 15 2010
-2
10-1
100
101
102
Time (years)P
opul
atio
n de
nsity
r=0.1,K=15, Θ=1, b=.1 r=0.1,K=15, Θ=1, b=1.5
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Correlation of deviations
y = -0.1877x - 0.0135
R2 = 0.0354
-3.000
-2.000
-1.000
0.000
1.000
2.000
3.000
-3.000
-2.000
-1.000
0.000 1.000 2.000 3.000
Deviation in year t
Devia
tio
n in
year
t+1 .
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Environmental correlation
• When the environmental effects on the population growth rate are correlated, the “effective” environmental variance in the log population growth rate is (Foley 1994):
[(1+ρ)/(1-ρ)]σ2
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[(1+ρ)/(1-ρ)]σ2
0
0.5
1
1.5
2
-1.5 -1 -0.5 0 0.5 1
rho
vari
an
ce
Variance without correlation
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Generate the correlated environmental variation
• Є= ρ Єt-1 +√σ2√ (1-ρ2)zt]
ρ = correlation coefficient
zt= random number drawn from a normal distribution with mean 0 and variance 1
Єt-1= is the sum of a term due to correlation with the previous environment deviation and a new random term, scaled by a factor to assure that the long string of Є is σ2
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Extinction risk and correlation
0
5
10
15
0 5 10 15
0
5
10
15
0 5 10 15
Nt+
1
NtNt
r=0.8 r=1.4
)/1(exp1 KNrNN ttT
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Autocorrelation in environmental effects
Pro
babi
lity
of h
ittin
g qu
asi-e
xtin
ctio
n th
resh
old
Nx
at o
r be
fore
tm
ax
r=0.8
r=1.4
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Catastrophes and Bonanzas
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Probabilty of extreme value
Pro
babi
lity
of h
ittin
g qu
asi-e
xtin
ctio
n th
resh
old
Nx
at o
r be
fore
tm
ax