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Understanding Stochastic Demography and Population Growth Dynamics

This study explores the mathematical frameworks and concepts of stochastic demography, focusing on population growth dynamics through stochastic sequences and cumulative growth rates. Different growth models, such as integral projection models and time averages, are analyzed to understand how populations evolve over time in varying environments. We delve into the implications of age structure and environmental factors on growth rates, using habitat transition matrices to illustrate the complexities involved. The findings highlight the significance of sequence matters and their stochastic nature in demography.

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Understanding Stochastic Demography and Population Growth Dynamics

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  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 tth root.

  4. Projection with temporal variation n(t+1) = A1 n(t) n(t+2) = A2 n(t+1) n(t+3) = A2 n(t+2) n(t+4) = A1 n(t+3) n(t+5) = A3 n(t+4)

  5. 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, ... ∑n(t+5) / ∑n(t) =λ5 The fifth root = annual

  6. Habitat transition matrix

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

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

  9. 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,...

  10. 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

  11. …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

  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)

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