Age-from-stage theory

1. Age-from-stage theory What is the probability an individual will be in a certain state at time t, given initial state at time 0?

2. Age-from-stage theory • Markov chains, absorbing states • An individual passes through various stages before being absorbed, e.g. dying • What is the probability it will be in certain stage at age x (time t), given initial stage? • The answer can be found by extracting information from stage-based population projection matrices Cochran and Ellner 1992, Caswell 2001 Tuljapurkar and Horvitz 2006, Horvitz and Tuljapurkar in review

3. Age-specific demographic rates from stage based models? • Life Expectancy • Stage structure at each age • Survivorship to age x, l(x) • Mortality at age x, μ(x)

4. Empirically-basedstage structured demography • Cohorts begin life in particular stage • Ontogenetic stage/size/reproductive status are known to predict survival and growth • Survival rate does not determine order of stages

5. A is population projection matrixF is reproductiondeath is an absorbing state

6. Q =A –FS = 1- death = column sum of Q

7. Q’s and S’s in a variable environment At each age, A(x) is one of {A1, A2, A3…Ak} and Q(x) is one of {Q1, Q2, Q3…Qk} and S(x) is one of {S1, S2, S3…SK} Stage-specific one-period survival

8. Cohort dynamicswith stage structure, variable environment Individuals are born into stage 1 N(0) = [1, 0, … ,0]’ As the cohort ages, its dynamics are given by N(x+1) = X (t) N (x), X is a random variable that takes on values Q1, Q2,…,QK

9. Cohort dynamicswith stage structure As the cohort ages, it spreads out into different stages and at each age x, we track l(x) = ΣN(x) survivorship of cohort U(x) = N(x)/l(x) stage structure of cohort

10. Mortality from weighted average of one-period survivals one period survival of cohort at age x = stage-specific survivals weighted by stage structure l(x+1)/l(x) = < Z (t), U(x) > Z is a random variable that takes on values S1, S2,…,SK Mortality rate at age x μ(x) = - log [ l(x+1)/l(x) ]

11. Mortality directly from survivorship • Survivorship to age x , l(x),is given by the sum of column 1* of • Powers of Q (constant environment) • Random matrix product of Q(x)’s (variable environment) • Age-specific mortality, the risk of dying soon after reaching age x, given that you have survived to age x, is calculated as, μ(x) = - log [ l(x+1)/l(x)] asymptotically, μ(x) = - log λQ __________________________________ *assuming individuals are born in stage 1

12. N, “the Fundamental Matrix”and Life Expectancy • Constant: • N = I + Q1 + Q2 + Q3 + …+QX • which converges to (I-Q) -1 • Life expectancy: column sums of N • e.g., for stage 1, column 1 • Variable: • N = I + Q(1) + Q(2)Q(1) + Q(3)Q(2)Q(1)+ …etc • which is NOT so simple; described for several cases in Tuljapurkar and Horvitz 2006 • Life expectancy: column sums of N • e.g., for stage 1, column 1

13. Survivorship and N in Markovian environment

14. Mortality plateau in variable environments Megamatrix μm= - logλm Before the plateau things are a little messier, powers of the megamatrix pre-multiplied by the initial environment’s Q c22

15. Environmental variability: types Each diagram represents a matrix of transitions among environmental states; the dots show the relative probability of changing states or remaining (indicated with a +) in a state over one time step.

16. Variable environment: Example Understory subtropical shrub • 8 life history stages • Seeds, seedlings, juveniles, pre-reproductives, reproductives of 4 sizes • Markovian environment: hurricane driven canopy dynamics • 7 environmental states • State 1 is very open canopy, lots of light • State 7 is closed canopy, quite dark

18. Mean matrix as if it were a constant environ-ment μmean= 0.01584

19. Expected mortality and survivorship by birth state

20. oct04pmrun_ares_sim_path59

21. oct04pmrun_ares_sim_path41

22. oct04pmrun_ares_sim_path44

23. Long run dynamics: stationary distribution of stage distributions Time after 39,000

24. Enough theory • Let’s do it!