1. Count based PVA Incorporating Density Dependence, Demographic Stochasticity, Correlated Environments, Catastrophes and Bonanzas

2. Assumptions of the diffusion appoximation Population growth Is unaffected by population density Its only source of variability is environmental stochasticity No trends in its mean and the variance Its values are not correlated in successive years Moderate variability No observation error

3. But.. Incorporating these effects into PVA models require: more and better data more mathematically complex models

4. Negative density dependence The simplest way to incorporate negative density dependence is introduce a population ceiling to the density- independent population growth model N t+1 = λ t N t ;if N t < K K ;if N t > K

5. The ceiling model Program algo2 (prepared by Matt; 10,50,.55,.45,60)

6. Mean time to extinction Where c=μ/σ 2, d=log(N c /N x ), and k=log(K/ Nx) If N c =K and Nx =1 then:

7. Extinction risk predicted by the Ceiling Model μ= 0.1 μ= 0.001 μ= -0.1 σ 2 = μ σ 2 = 2μ σ 2 = 4μ σ 2 = 8μ Program tbarpedro

8. The theta logistic model A gradually changing growth rate

9. K = 100 r = 0.2 Theta: 4 1 0.3

10. The theta logistic model

11. K = 100 r = 0.8 Theta: 4 1 0.3

12. The Bay checkerspot butterfly Euphydryas editha bayensis front Harrison et al., 1991 JRC population

13. The negative association remains after removing the outlier in the right back

14. Density Dependent model Find the best model: Fit three models to the data using nonlinear least-squares regression of log(N t+1 /N t ) against N t Models to be tested: Density independent model: log(N t+1 /N t )=r The Ricker model log(N t+1 /N t )=r(1-N t /K) The theta logistic model log(N t+1 /N t )=r[1-(N t /K) Θ ]

15. Estimate the parameters of each model ModelLeast-squares parameter estimates r KΘResidual Variance Density independent 0.0016731.3999 Ricker0.3488846.021.0722 Theta logistic0.9941551.380.45661.0165

16. Model maximum likelihood of a model assuming normally distributed deviations is ln(L max ) = -(q/2)[ln(2  V r ) +1) V r = residual variance q= Sample number

17. Maximum log likelihood The probability of obtaining the observed data given a particular set of parameter values for a particular model Information criterion statistics combine the maximum log likelihood for a model with the number of parameters it include to provide a measure of “support”

18. “Support” is higher for: models with higher likelihoods, and models with fewer parameters More complex models are penalized because more parameters will always lead to a better fit to the data, but at the cost of less precision in the estimate of each parameter and incorporation of spurious patterns from the data into future populations

19. Akaike Information Criteria To identify the best model : AIC c = -2 ln(L max ) + (2pq)/ (q-p-1) p = Number of estimated parameters (including the residual variance) q= sampling number

20. Akaike weights W i = exp[-0.5(AIC c,i -AIC c,best )]  exp[-0.5(AIC c,i -AIC best )]

21. Compute the maximum log likelihood and Akaike weights for each model Model Number of parameters Including V r Log L max AICAkaike weights Density independent 2-41.26687.054.07 Ricker3-37.79982.689.62 Theta logistic4-37.10584.115.31

22. Simulate the model to predict population viability σ 2 = qV r q-1 Program theta_logistic Program extprobpedro

23. Simulate the model to predict population viability Program extprobpedro Program theta_denindeppedro

24. Allee effects We can simply set the quasi-extinction threshold at or above the population size at which Alee effects become important Explicitly include Alee effects in the population model N t+1 = Nt2Nt2 A+N t е r-βN t

25. The parameters The potential offspring Value at A maximum Fraction of potential reproduction that is actually achieved

26. A discrete-time model with Alee effects generated by mate-finding problems

28. Combined effects of Demographic Environmental stochasticity

29. Combined effects of Demographic and Environmental stochasticity r=0.1,K=15, Θ=1, b=.1r=0.1,K=15, Θ=1, b=1.5

30. Correlation of deviations

31. Environmental correlation When the environmental effects on the population growth rate are correlated, the “effective” environmental variance in the log population growth rate is (Foley 1994): [(1+ρ)/(1-ρ)] σ 2

32. Variance without correlation

33. Generate the correlated environmental variation Є= ρ Є t-1 +√σ 2 √ (1-ρ 2 )z t ] ρ = correlation coefficient z t = random number drawn from a normal distribution with mean 0 and variance 1 Є t-1 = is the sum of a term due to correlation with the previous environment deviation and a new random term, scaled by a factor to assure that the long string of Є is σ 2

34. Extinction risk and correlation N t+1 NtNt NtNt r=0.8r=1.4

35. r=0.8 r=1.4

36. Catastrophes and Bonanzas