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

• 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

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

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=40t=50

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)

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

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

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

0 2 4 6 8 10 12 14 16 18 20

-6

-4

-2

0

2

4

6

8

0 2 4 6 8 10 12 14 16 18 20

-6

-4

-2

0

2

4

6

8

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

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

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

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

0

20

40

60

80

100

120

1950 1970 1990

Year

Adul

t Fem

ales

.

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