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

1. 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. 2. 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. 3. 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. 4. 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.