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.

The timing of many biological events, including butterfly imago emergence, has advanced under climate change, with the rate of these phenological changes often differing among taxonomic groups. Such inter-taxa variability can lead to phenological mismatches. For example, the timing of a butterfly's flight period may become misaligned with a key nectar resource, potentially increasing the extinction risk to both species. Here we fit statistical models to field data to determine how the phenology of the marbled white butterfly, Melanargia galathea, and its main nectar source, greater knapweed, Centaurea scabiosa, have changed over recent years at three sites across the UK. We also consider whether topographical diversity affects C. scabiosa's flowering period. At our focal site, on the species northern range limit, we find that over a 13year period the onset of C. scabiosa's flowering period has become later whilst there is no obvious trend over time in the onset of M. galathea's flight period. In recent years, butterflies have started to emerge before their key nectar source was available across most of the site. This raises the intriguing possibility that phenological mismatch could be an unrecognised determinant of range limits for some species. However, the presence of topographical diversity within the site decreased the chance of a mismatch occurring by increasing the length of the flowering period by up to 14 days. We suggest that topographical diversity could be an important component in minimising phenological mismatches under future climate change.

Predicting how species will be affected by future climatic change requires the underlying environmental drivers to be identified. As vital rates vary over the lifecycle, structured population models derived from statistical environment-demography relationships are often used to inform such predictions. Environmental drivers are typically identified independently for different vital rates and demographic classes. However, these rates often exhibit positive temporal covariance, suggesting that vital rates respond to common environmental drivers. Additionally, models often only incorporate average weather conditions during a single, a priori chosen time window (e.g. monthly means). Mismatches between these windows and the period when the vital rates are sensitive to variation in climate decrease the predictive performance of such approaches. We used a demographic structural equation model (SEM) to demonstrate that a single axis of environmental variation drives the majority of the (co)variation in survival, reproduction, and twinning across six age-sex classes in a Soay sheep population. This axis provides a simple target for the complex task of identifying the drivers of vital rate variation. We used functional linear models (FLMs) to determine the critical windows of three local climatic drivers, allowing the magnitude and direction of the climate effects to differ over time. Previously unidentified lagged climatic effects were detected in this well-studied population. The FLMs had a better predictive performance than selecting a critical window a priori, but not than a large-scale climate index. Positive covariance amongst vital rates and temporal variation in the effects of environmental drivers are common, suggesting our SEM-FLM approach is a widely applicable tool for exploring the joint responses of vital rates to environmental change.