SPM Journal Club: Developmental variation and stage duration in matrix models
Stage structure is fundamental in quantitative population models, but there are different approaches to deal with stage duration and individual-/cohort variation therein.
We discuss a paper that reviews three different approaches to dealing with stage duration, combine them into a new framework and discuss how individual variation and correlations in development can affect population growth rate and matrix sensitivities and elasticities:
"The importance of individual developmental variation in stage-structured population models"
(de Valpine et al. 2014, Ecology Letters)
Population stage structure is fundamental to ecology, and models of this structure have proven useful in many different systems. Many ecological variables other than stage, such as habitat type, site occupancy and metapopulation status are also modelled using transitions among discrete states. Transitions among life stages can be characterised by the distribution of time spent in each stage, including the mean and variance of each stage duration and within-individual correlations among multiple stage durations. Three modelling traditions represent stage durations differently. Matrix models can be derived as a long-run approximation from any distribution of stage durations, but they are often interpreted directly as a Markov model for stage transitions. Statistical stage-duration distribution models accommodate the variation typical of cohort development data, but such realism has rarely been incorporated in population theory or statistical population models. Delay-differential equation models include lags but no variation, except in limited cases. We synthesise these models in one framework and illustrate how individual variation and correlations in development can impact population growth. Furthermore, different development models can yield the same long-term matrix transition rates but different sensitivities and elasticities. Finally, we discuss future directions for estimating realistic stage duration models from data.