count based pva: density-independent models. count data of the entire population of a subset of the...
DESCRIPTION
The theoretical results that underlie the simplest count-based methods in PVA. The model for discrete geometric population growth in a randomly varying environment N t+1 =λ t N t Assumes that population growth is density independent (i.e. is not affected by population size, N t ) Population dynamics in a random environmentTRANSCRIPT
![Page 1: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/1.jpg)
Count Based PVA:
Density-Independent Models
![Page 2: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/2.jpg)
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
![Page 3: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/3.jpg)
• 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)
Population dynamics in a random environment
![Page 4: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/4.jpg)
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
Geometric increase
Geometric decline
Stasis
λ>1
λ<1
λ=1
![Page 5: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/5.jpg)
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
![Page 6: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/6.jpg)
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
![Page 7: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/7.jpg)
t=10 t=20
t=40t=50
![Page 8: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/8.jpg)
The best predictor of whether Nt will increase or decrease over the long term is
λG
Nt+1=(λt λt-1 λt-2 …λ1 λ0) No
(λG)t =λt λt-1 λt-2 …λ1 λ0 ;or
• Since
λG is defined as
λG =(λt λt-1 λt-2 …λ1 λ0)(1/t)
![Page 9: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/9.jpg)
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 growμ<0, then λ<1 the most populations will decline
![Page 10: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/10.jpg)
t=15
1 2 3 4 5 6 7 80
1
2
3
4
5
6
-200 0 200 400 600 800 1000 1200 1400 16000
1
2
3
4
5
6
1 2 3 4 5 6 7 80
1
2
3
4
5
6
-200 0 200 400 600 800 1000 1200 1400 16000
1
2
3
4
5
6
t=30
N ln(N)
N Ln(N)
0 10 20 30 40 503
4
5
6
7
8
ln(N)
t
![Page 11: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/11.jpg)
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
![Page 12: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/12.jpg)
0 2 4 6 8 10 12 14 16 18 20
-6
-4
-2
0
2
4
6
8
![Page 13: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/13.jpg)
0 2 4 6 8 10 12 14 16 18 20
-6
-4
-2
0
2
4
6
8
![Page 14: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/14.jpg)
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
![Page 15: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/15.jpg)
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 Cumulative distribution function for the time to quasi-extinction
![Page 16: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/16.jpg)
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
The probability of ultimate extinction
![Page 17: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/17.jpg)
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
![Page 18: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/18.jpg)
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
![Page 19: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/19.jpg)
Estimating μ,σ2
• μ as the arithmetic mean• σ2 as the sample variance
• Of the log(Ni+1/Ni)
![Page 20: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/20.jpg)
Female Grizzly bears in the Greater Yellowstone
0
20
40
60
80
100
120
1950 1970 1990
Year
Adul
t Fem
ales
.
![Page 21: 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](https://reader036.vdocument.in/reader036/viewer/2022062401/5a4d1b1b7f8b9ab0599936f1/html5/thumbnails/21.jpg)
Estimating μ,σ2
• μ =0.02134; σ2 =0.01305
Model R R SquareAdjusted R
Square
Std. Error of the Estimate Durbin-Watson
1 0.186005 0.034598 0.008506 0.114241 2.570113
ANOVA
Model Sum of Squares df Mean Square F Sig.
1 Regression 0.017305 1 0.017305 1.325996 0.256906
Residual 0.482884 37 0.013051
Total 0.500189 38
Coefficients
Model Unstandardized Coefficients Standardized Coefficients t Sig.
B Std. Error Beta
1 INTERVAL 0.02134 0.018532 0.186005 1.151519 0.256906