Age- and stage-structured population models are central to population ecology, but most applications omit biologically realistic variation in organismal development. Individual variation or stochasticity in development times, whether between distinct stages, to reproductive maturity, or to time of death is the norm across taxa in nature. Development times may be correlated across stages, or with fecundity or mortality, if they reflect differences in individual quality or resource acquisition. Stochasticity in development times can have a strong impact on population growth rate, sensitivities to demographic parameters, and stable age-stage distribution. Lack of a routine, flexible method to analyze even the density-independent demographic implications of stochastic development times is a key gap in population dynamics theory. Matrix models, the most widely used tool in applied population modeling, are very limited in their accommodation of stochastic development times. Other methods, such as the "linear chain" trick of modeling a stage as a series of virtual stages, lack flexibility and require independent stages.
Results/Conclusions
I give a general approach for formulating and analyzing stochastic development in density-independent population models using the framework of integral projection models. Integral projection models are a generalization of matrix models to allow continuous state variables that change over discrete time steps between surveys. The approach allows flexible assumptions, including correlated development times among multiple life stages, but requires "bookkeeping" involving both age and stage. I give a general implementation based on simulations of individual life schedules, rather than iteration of full population dynamics, which are relatively intuitive for many ecologists. The connection between individual simulations and solution of population growth rate, its sensitivities, stable age-stage distributions and reproductive value is formalized using Monte Carlo numerical integration theory for integration of demographic equations such as the Euler-Lotka equation. In particular I give a sequential Monte Carlo method that is very efficient.
I show that stochastic development is demographically important using two examples. For a desert cactus, many stochastic development models, with independent or correlated stage durations, can generate the same stable stage distribution as the real data, but stable age-within-stage distributions and sensitivities of growth rate to demographic rates differ greatly among stochastic development scenarios. For Mediterranean fruit flies, empirical variation in maturation time has a large impact on population growth. The systematic model formulation and analysis approach given here should make consideration of variable development models widely accessible and readily extendible. The method is available as an R package.
