Exploring population responses to environmental change when there is never enough data: a factor analytic approach

Abstract

Temporal variability in the environment drives variation in vital rates, with consequences for population dynamics and life‐history evolution. Integral projection models (IPMs) are data‐driven structured population models widely used to study population dynamics and life‐history evolution in temporally variable environments. However, many datasets have insufficient temporal replication for the environmental drivers of vital rates to be identified with confidence, limiting their use for evaluating population level responses to environmental change. Parameter selection, where the kernel is constructed at each time step by randomly selecting the time‐varying parameters from their joint probability distribution, is one approach to including stochasticity in IPMs. We consider a factor analytic (FA) approach for modelling the covariance matrix of time‐varying parameters, whereby latent variable(s) describe the covariance among vital rate parameters. This decreases the number of parameters to estimate and, where the covariance is positive, the latent variable can be interpreted as a measure of environmental quality. We demonstrate this using simulation studies and two case studies. The simulation studies suggest the FA approach provides similarly accurate estimates of stochastic population growth rate to estimating an unstructured covariance matrix. We demonstrate how the latent parameter can be perturbed to show how selection on reproductive delays in the monocarp Carduus nutans changes under different environmental conditions. We develop a demographic model of the fire dependent herb Eryngium cuneifolium to show how a putative driver of the variation in environmental quality can be incorporated with the addition of a single parameter. Using perturbation analyses we determine optimal management strategies for this species. This approach estimates fewer parameters than previous approaches and allows novel eco‐evolutionary insights. Predictions on population dynamics and life‐history evolution under different environmental conditions can be made without necessarily identifying causal factors. Putative environmental drivers can be incorporated with relatively few parameters, allowing for predictions on how populations will respond to changes in the environment.

Assessing seasonal demographic covariation to understand environmental‐change impacts on a hibernating mammal

“…To model temporal covariation in seasonal demography in the absence of explicit knowledge on key biotic or abiotic drivers of this covariation, we used a factor‐analytic approach. This approach has recently been proposed by Hindle & coauthors () as a structured alternative to fit and project unstructured covariances among demographic processes when factors explaining these covariances are not modelled. We implemented this novel approach parameterizing a model‐wide latent variable ( Q y ) which affected all demographic processes in a given year ( y ) (for details, see Supporting Material S1 and Hindle et al, ).…”

“…This approach has recently been proposed by Hindle & coauthors () as a structured alternative to fit and project unstructured covariances among demographic processes when factors explaining these covariances are not modelled. We implemented this novel approach parameterizing a model‐wide latent variable ( Q y ) which affected all demographic processes in a given year ( y ) (for details, see Supporting Material S1 and Hindle et al, ). Q y was incorporated as a covariate in all seven demographic‐process submodels (Table ).…”

“…The positive β q indicate that Q y effectively estimates the overall annual environmental quality or suitability, capturing both biotic and abiotic processes. A positive value of Q y then depicts an environmental condition at a given time point that increases winter survival and probability of reproducing and decreases mass loss (Hindle et al, ). The variation in Q y was in part explained by environmental variables measured at the study site, but was unrelated to population density (Supporting Material S1).…”

Assessing seasonal demographic covariation to understand environmental-change impacts on a hibernating mammal

“…This approach has recently been proposed by Hindle and coauthors (2018) as a structured alternative to fit and project unstructured covariances among demographic processes when factors explaining these covariances are not modeled. We implemented this novel approach parameterizing a model-wide latent variable ( Q y ) which affected all demographic processes in a given year ( y ) (for details see Supporting Material S1 and Hindle et al ., 2018). Q y was incorporated as a covariate in all seven demographic-process submodels (Table 1).…”

“…As a result, measures of environmental covariates (e.g., temperature or resource availability) have previously shown little effect on the covariation of marmot demographic processes (Maldonado-Chaparro et al, 2018). To address this challenge, we used a novel method, a hierarchical factor analysis (Hindle et al, 2018), to model the covariation of demographic processes as a function of a shared latent variable, quantified in a Bayesian modeling framework. We then built seasonal stage-, mass-, and environment-specific integral projection models (IPMs; Ellner et al, 2016) for the marmot population, which allowed us to simultaneously project trait distributions and population dynamics across seasons.…”