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?

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

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)

Empirically-based stage 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

A is population projection matrix F is reproduction death is an absorbing state

Q = A – F S = 1- death = column sum of Q

Q’s and S’s in a variable environment At each age, A(x) is one of {A 1, A 2, A 3 …A k } and Q(x) is one of {Q 1, Q 2, Q 3 …Q k } and S(x) is one of {S 1, S 2, S 3 …S K } Stage-specific one-period survival

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 Q 1, Q 2,…,Q K Cohort dynamics with stage structure, variable environment

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

one period survival of cohort at age x = stage-specific survivals weighted by stage structure l(x+1)/l(x) = l(x+1)/l(x) = Z is a random variable that takes on values S 1, S 2,…,S K Mortality rate at age x μ(x) = - log [ l(x+1)/l(x) ] Mortality from weighted average of one-period survivals

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 λ Q __________________________________ *assuming individuals are born in stage 1

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

Survivorship and N in Markovian environment

Mortality plateau in variable environments Megamatrix μ m = - log λ m Before the plateau things are a little messier, powers of the megamatrix pre-multiplied by the initial environment’s Q c 22

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.

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

Mean matrix as if it were a constant environ- ment μ mean =

Expected mortality and survivorship by birth state

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

Enough theory Let’s do it!