What’s next?

Time averages Cumulative pop growth Stochastic sequences Spatial population dynamics Age from stage Integral projection models

What’s next?

Time averages Cumulative pop growth Stochastic sequences

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 tth root.

Projection with temporal variation

n(t+1) = A1 n(t)

n(t+2) = A2 n(t+1)

n(t+5) = A3 n(t+4)

n(t+3) = A2 n(t+2)

n(t+4) = A1 n(t+3)

n(t+5) / n(t) = 5

5 n(t+5) / n(t) =

Fixed sequence: growth rate per cycle, averaged per time unit

A1 , A2, A2, A1, A3, A1 , A2, A2, A1, A3, A1 , A2, A2, A1, A3, ...

Habitat transition matrixPatchtypeattimet+1

Patch type attime t

Green Red Blue

Green 0 0.5 1.0

Red 0.5 0.5 0

Blue 0.5 0 0

Temporal mosaic of habitats

time

Fixed sequence Stochastic sequence first set

Temporal mosaic of habitats

Stochastic sequence 3rd setStochastic sequence 2nd set

time

a = mean(log 1, log 2, log 3,..log t )

e a similar to

n(t+1) / n(t) = t+1

Stochastic sequence: series of growth rates, time averaged

A1 , A3, A1, A3, A1 , A3, A1, A3, A1, A2, A1 , A3, A1 , A3, A2, A2,...

Population growth in a random environment = the stochastic growth rate

A1 , A3, A1, A3, A1 , A3, A1, A3, A1, A2, A1 , A3, A1 , A3, A2, A2,...

sae

mean

a

].)log(),log(),log(),[log( 4321

Concepts of stochastic sequence demography

Cumulative population growth rate (Nt/N0) lognormally distribution

(mean mode)

Time average (the tth root)

Sequence matters (AB BA)

Probability distribution of sequences

0 30,000 60,000 90,000 120,0000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

-5

mean = 40000

Lognormal Distribution, =40000, = 1.25

outcome

prob

abili

ty d

ensi

ty

…stochastic demography (cont’d)

Each sequence (s time steps) cumulative population growth of each (Ns/N0) time average of each (over many time steps) (sth

root)

Average cumulative population growth mean (Ns/N0) over many sequences of particular

length time average of this mean no sequence may actually experience the mean

…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

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

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

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)