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Count based PVA

Count based PVA. Incorporating Density Dependence, Demographic Stochasticity, Correlated Environments, Catastrophes and Bonanzas. Assumptions of the diffusion appoximation. Population growth Is unaffected by population density Its only source of variability is environmental stochasticity

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Count based PVA

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  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 λtNt ;if Nt < K Nt+1= K;if Nt > K

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

  6. Mean time to extinction Where c=μ/σ2, d=log(Nc/Nx), and k=log(K/Nx) If Nc=K and Nx =1 then:

  7. Extinction risk predicted by the Ceiling Model μ= 0.1 Program tbarpedro μ= 0.001 μ= -0.1 σ2= μ σ2= 2μ σ2= 4μ σ2= 8μ

  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 butterflyEuphydryas 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(Nt+1/Nt) against Nt Models to be tested: Density independent model: log(Nt+1/Nt)=r The Ricker model log(Nt+1/Nt)=r(1-Nt/K) The theta logistic model log(Nt+1/Nt)=r[1-(Nt/K)Θ]

  15. Estimate the parameters of each model

  16. Model maximum likelihood of a model assuming normally distributed deviations is • ln(Lmax) = -(q/2)[ln(2Vr) +1) • Vr = 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: • AICc = -2 ln(Lmax) + (2pq)/ (q-p-1) p = Number of estimated parameters (including the residual variance) q= sampling number

  20. Akaike weights exp[-0.5(AICc,i-AICc,best)] Wi =  exp[-0.5(AICc,i-AICbest)]

  21. Compute the maximum log likelihood and Akaike weights for each model

  22. Simulate the model to predict population viability qVr σ2 = q-1 Program extprobpedro Program theta_logistic

  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 Nt2 еr-βNt Nt+1 = A+Nt

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

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

  27. 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=.1 r=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)]σ2

  32. [(1+ρ)/(1-ρ)]σ2 Variance without correlation

  33. Generate the correlated environmental variation • Є= ρ Єt-1 +√σ2√ (1-ρ2)zt] ρ = correlation coefficient zt= random number drawn from a normal distribution with mean 0 and variance 1 Єt-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 σ2

  34. Extinction risk and correlation r=0.8 r=1.4 Nt+1 Nt Nt

  35. r=0.8 r=1.4

  36. Catastrophes and Bonanzas

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