View
232
Download
3
Category
Tags:
Preview:
Citation preview
Count based PVA
Incorporating Density Dependence, Demographic Stochasticity, Correlated Environments,
Catastrophes and Bonanzas
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
But..
• Incorporating these effects into PVA models require:
• more and better data
• more mathematically complex models
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
The ceiling model
Program algo2 (prepared by Matt; 10,50,.55,.45,60)
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:
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
The theta logistic model
• A gradually changing growth rate
tt
tt K
NrNN 1exp1
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
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
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
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
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
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]
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
Model maximum likelihood of a model
assuming normally distributed deviations is
• ln(Lmax) = -(q/2)[ln(2Vr) +1)
• Vr = residual variance
• q= Sample number
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”
“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
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
Akaike weights
Wi =exp[-0.5(AICc,i-AICc,best)]
exp[-0.5(AICc,i-AICbest)]
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
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
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
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
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
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"
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
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
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
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 .
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
[(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
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
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
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
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
Recommended