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 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 environment

N t+1 =λ t N t 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<1λ<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=40 t=50

The best predictor of whether N t will increase or decrease over the long term is λ G N t+1 =(λ t λ t-1 λ t-2 …λ 1 λ 0 ) N o (λ G ) t =λ t λ t-1 λ t-2 …λ 1 λ 0 ; or Since λ G is defined as λ G =(λ t λ t-1 λ t-2 …λ 1 λ 0 ) (1/t)

Converting this formula for λ G to the log scale μ= lnλ G = lnλ t +ln λ t-1 + …lnλ 1 +ln λ 0 t The correct measure of stochastic population growth on a log scale, μ, is equal to the lnλ G or equivalently, to the arithmetic mean of the ln λ t values. μ>0, then λ>1 the most populations will grow μ<0, then λ<1 the most populations will decline

t=15 t=30 Nln(N) NLn(N) 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 σ 2 : the variance in the log population growth rate

The inverse Gaussian distribution g(t μ,σ 2,d)= (d/√2π σ 2 t 3 )exp[-(d+ μt) 2 /2σ 2 t] Where d= logN c -N x N c = current population size N x =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,σ 2 )= Φ(-d-μT/√σ 2 T)+ exp[-2μd/ σ 2 ) Φ(-d-μT/√σ 2 T) 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,σ 2 )=1 when μ< 0 exp(-2μd/ σ 2 ) 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 σ 2 do not change over time

Estimating μ,σ 2 Lets assume that we have conducted a total of q+1 annual censuses of a population at times t 0, t 1, …t q, having obtained the census counts N 0, N 1, …N q+1 Over the time interval of length (t i+1 – t i ) Years between censuses i and i+1 the logs of the counts change by the amount log(N i+1 – N i )= log(N i+1 /N i )=logλ i where λ i =N i+1 /N i

Estimating μ,σ 2 μ as the arithmetic mean σ 2 as the sample variance Of the log(N i+1 /N i )

Female Grizzly bears in the Greater Yellowstone

Estimating μ,σ 2 μ = ; σ 2 = ModelRR Square Adjusted R Square Std. Error of the EstimateDurbin-Watson ANOVA Model Sum of SquaresdfMean SquareFSig. 1Regression Residual Total Coefficients Model Unstandardized CoefficientsStandardized CoefficientstSig. BStd. ErrorBeta 1INTERVAL