Count Based PVA:

1. Count Based PVA: Density-Independent Models

2. 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

3. 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)

4. Nt+1=λtNt λ>1 Geometric increase • 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 Stasis λ=1 Geometric decline λ<1

5. 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

6. 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

7. t=10 t=20 t=50 t=40

8. The best predictor of whether Nt will increase or decrease over the long term is λG • Since λG is defined as Nt+1=(λt λt-1 λt-2 …λ1 λ0)No (λG)t=λt λt-1 λt-2 …λ1 λ0 ;or λG=(λt λt-1 λt-2 …λ1 λ0)(1/t)

9. Converting this formula for λG to the log scale μ= lnλG=lnλt+lnλt-1+…lnλ1 +lnλ0 t The correct measure of stochastic population growth on a log scale, μ, is equal to the lnλGor 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

10. t=30 t=15 N Ln(N) N ln(N) ln(N) t

11. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. Estimating μ,σ2 • μ as the arithmetic mean • σ2 as the sample variance • Of the log(Ni+1/Ni)

20. Female Grizzly bears in the Greater Yellowstone

21. Estimating μ,σ2 • μ =0.02134; σ2 =0.01305