1. What’s next? Time averages Cumulative pop growth Stochastic sequences Spatial population dynamics Age from stage Integral projection models

2. What’s next? Time averages Cumulative pop growth Stochastic sequences

3. Geometric mean By what factor does the population multiply each year on average? Population growth is a multiplicative process! If there are t time steps, we want the t th root.

4. Projection with temporal variation n(t+1) = A 1 n(t) n(t+2) = A 2 n(t+1) n(t+5) = A 3 n(t+4) n(t+3) = A 2 n(t+2) n(t+4) = A 1 n(t+3)

5. ∑ n(t+5) / ∑ n(t) =λ 5 The fifth root = annual Fixed sequence: growth rate per cycle, averaged per time unit A 1, A 2, A 2, A 1, A 3, A 1, A 2, A 2, A 1, A 3,...

6. Habitat transition matrix

7. Temporal mosaic of habitats time Fixed sequence Stochastic sequence first set

8. Temporal mosaic of habitats Stochastic sequence 3rd set Stochastic sequence 2nd set time

9. Population growth in a random environment = the stochastic growth rate A 1, A 3, A 1, A 3, A 1, A 3, A 1, A 3, A 1, A 2, A 1, A 3, A 1, A 3, A 2, A 2,...

10. Concepts of stochastic sequence demography Cumulative population growth rate (N t /N 0 ) lognormally distribution (mean  mode) Time average (the t th root) Sequence matters (AB  BA) Probability distribution of sequences

11. …stochastic demography (cont’d) Each sequence (s time steps) cumulative population growth of each (N s /N 0 ) time average of each (over many time steps) (s th root) Average cumulative population growth mean (N s /N 0 ) over many sequences of particular length time average of this mean no sequence may actually experience the mean

12. …stochastic demography (cont’d) Time average of one sequence  time average of the mean cumulative population growth Time average of a very long sequence is equivalent to the average of time averages and gives the long run population growth rate

13. Connecting ’s to averages Megamatrix = time average of mean cumulative population growth of a very large number of patches Stochastic sequence = mean of time averages = long run population growth rate

14. Stochastic demography Growth rates megamatrix average population mean matrix expected stochastic sequence Sensitivity of the stochastic growth rate Mean of transitions Variance of transitions Transitions in specific environments

15. Stochastic demography…more Small noise approximation Don’t forget the covariance! Temporal autocorrelation (Haridas and Tuljapurkar 2006) Elasticity to variability within and between disturbance phases (Morris et al.) Other cool stuff “real time” elasticities: separate perturbation of structure from perturbation of matrix elements (Haridas and Tuljapurkar) Age-specific survivorship and stage-conditional life expectancy (Tuljapurkar and Horvitz 2006) Integrated stochastic selection (Horvitz, Tuljapurkar, Coulson and Schemske)