Count Based PVA:
Density-Independent Models
Count Data
• Of the entire population
• Of a subset of the population (“as long as
the segment of the population that is
observed is a relatively constant fraction of
the whole”)
• Censused over multiple (not necessarily
consecutive years
Population dynamics in a
random environment
• The theoretical results that underlie the simplest
count-based methods in PVA. The model for
discrete geometric population growth in a
randomly varying environment
Nt+1=λtNt
Assumes that population growth is density
independent (i.e. is not affected by population
size, Nt)
Nt+1=λtNt
• If there is no
variation in the
environment from
year to year, then
the population
growth rate λ will
be constant, and
only three
qualitative types of
population growth
are possible
λ>1
Geometric increase
Stasis
λ=1
Geometric decline
λ<1
By causing survival and reproduction to vary
from year to year, environmental variability
will cause the population growth rate, to vary
as well
• A stochastic process
Three fundamental features of
stochastic population growth
• The realizations diverge over time
• The realizations do not follow very well the
predicted trajectory based upon the
arithmetic mean population growth rate
• The end points of the realizations are
highly skewed
t=10
t=20
t=40
t=50
The best predictor of whether Nt will
increase or decrease over the long term is
λG
• Since
Nt+1=(λt λt-1 λt-2 …λ1 λ0) No
λG is defined as
(λG)t =λt λt-1 λt-2 …λ1 λ0 ;or
λG =(λt λt-1 λt-2 …λ1 λ0)(1/t)
Converting this formula for λG to the
log scale
μ= lnλG =lnλt+ln λt-1+ …lnλ1 +ln λ0
t
The correct measure of stochastic
population growth on a log scale, μ, is
equal to the lnλG or equivalently, to the
arithmetic mean of the ln λt values.
μ>0, then λ>1 the most populations will grow
μ<0, then λ<1 the most populations will decline
t=30
6
6
5
5
4
4
3
3
2
2
1
1
0
-200
t=15
6
5
5
4
4
3
3
2
2
1
1
0
200
400
600
N
800
1000
200
400
600
800
1000
1200
1400
1600 0
1
N
6
0
-200
0
1200
1400
0
1
1600
2
3
4
5
6
7
8
Ln(N)
8
7
2
3
4
5
ln(N)
6
7
8
6
ln(N)
5
4
3
0
10
20
30
t
40
50
To fully characterize the changing normal
distribution of log population size we need
two parameters:
• μ: the mean of the log population growth
rate
• σ2 : the variance in the log population
growth rate
8
6
4
2
0
-2
-4
-6
0
2
4
6
8
10
12
14
16
18
20
8
6
4
2
0
-2
-4
-6
0
2
4
6
8
10
12
14
16
18
20
The inverse Gaussian distribution
• g(t μ,σ2,d)= (d/√2π σ2t3)exp[-(d+ μt)2/2σ2t]
• Where d= logNc-Nx
• Nc = current population size
• Nx =extinction threshold
The Cumulative distribution function
for the time to quasi-extinction
To calculate the probability that the threshold is
reached at any time between the present (t=0) and
a future time of interest (t=T), we integrate
• G(T d,μ,σ2)= Φ(-d-μT/√σ2T)+
•
exp[-2μd/ σ2) Φ(-d-μT/√σ2T)
• Where Φ(z) (phi) is the standard normal cumulative
distribution function
The probability of ultimate extinction
Calculated by taking the integral of the inverse
Gaussian distribution from t=0 to t =inf
• G(T d,μ,σ2)=1 when μ< 0
•
exp(-2μd/ σ2) when μ>0
Three key assumptions
• Environmental perturbations affecting the
population growth rate are small to
moderate (catastrophes and bonanzas do
not occur)
• Changes in population size are
independent between one time interval
and the next
• Values of μ and σ2 do not change over
time
Estimating μ,σ2
• Lets assume that we have conducted a total of
q+1 annual censuses of a population at times t0,
t1, …tq, having obtained the census counts N0,
N1, …Nq+1
• Over the time interval of length (ti+1 – ti)
Years between censuses i and i+1 the logs of the
counts change by the amount
log(Ni+1 – Ni)= log(Ni+1/Ni)=logλi
where λi=Ni+1/Ni
Estimating μ,σ2
• μ as the arithmetic mean
• σ2 as the sample variance
• Of the log(Ni+1/Ni)
Female Grizzly bears in the
Greater Yellowstone
Adult Females
.
120
100
80
60
40
20
0
1950
1970
Year
1990
Estimating μ,σ2
• μ =0.02134; σ2 =0.01305
Model
R
Adjusted R
Square
R Square
1
0.186005
Std. Error of
the
Estimate
0.034598
0.008506
Durbin-Watson
0.114241
2.570113
ANOVA
Model
Sum of Squares
1
df
Mean Square
Regression
0.017305
1
0.017305
Residual
0.482884
37
0.013051
Total
0.500189
38
F
Sig.
1.325996
0.256906
Coefficients
Model
1
INTERVAL
Unstandardized Coefficients
Standardized Coefficients
B
Beta
Std. Error
0.02134
0.018532
0.186005
t
Sig.
1.151519
0.256906