The importance of individual developmental variation in stage‐structured population models

Abstract

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.

Asymmetric interactions and their consequences for vital rates and dynamics: the smaller tea tortrix as a model system

“…), captures the fact that longer development times typically have greater variance, and is compatible with the Erlang distribution that is commonly used in distributed‐delay models of insect development (de Valpine et al. , Bjørnstad et al. ).…”

“…The Erlang distribution is pragmatically suited for modeling variation in insect development caused by genetic, stochastic, and/or environmental heterogeneities because it greatly simplifies numerical analysis of the distributed‐delay model by solving the underlying integro‐differential equations using chains of substage ODEs (de Valpine et al. , Bjørnstad et al. ).…”

“…To study how development time to pupation was affected in the instar and density experiments, we fit generalized linear models with a gamma distributed error. We chose a gamma distribution of development times because it describes insect development times well (Wagner et al 1984), captures the fact that longer development times typically have greater variance, and is compatible with the Erlang distribution that is commonly used in distributed-delay models of insect development (de Valpine et al 2014, Bjørnstad et al 2016. For each experiment, the effect of treatment on development time distributions was evaluated by comparing models with and without the treatment covariate using AIC.…”

A pathway for multivariate analysis of ecological communities using copulas

“…Although implemented fairly widely in other fields (Nikoloulopoulos & Karlis, 2009;Shi & Valdez, 2014), copulas have not yet been widely used in ecology (but see de Valpine, Scranton, Knape, Ram, & Mills, 2014;Popovic et al, 2018).…”

Stage duration distributions in matrix population models

“…OKUYAMA an egg stage, some eggs hatch (e.g., become larvae in some insects) before other eggs, even when they were laid at the same time (e.g., Fang, Okuyama, Wu, Feng, & Hsu, 2011;King, Brewer, & Martin, 1975). The distribution of stage duration is another important detail that affects population growth (de Valpine, Scranton, Knape, Ram, & Mills, 2014). However, in matrix models, it is uncommon to explicitly think of a probability distribution and instead use a method that captures some components (e.g., mean and variance) of a distribution.…”