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

Bethan J Hindle,Andy W Sheppard,Eric S Menges,Pedro F Quintana‐Ascencio,Dylan Z Childs,Mark Rees,Satu Ramula

doi:10.1111/2041-210x.13085

Abstract

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.

Highlights

A e se equa fai ing to account for this covariation wi bias mode outputs Fieberg E ner
We compared the accuracy of popu ation growth estimates from the FA approach to those derived using an unstructured covariance matrix We considered two scenarios a re ative y simp e ife his tory with four tempora y variab e vita rates the simp e mode typica of many pub ished IPMs and a two stage juveni e and adu t ife history with a tota of seven tempora y variab e vita rates the comp ex mode Demographic rate functions in both settings were parameterised using data from a ong term study of the St Ki da Soay sheep C utton Brock Pemberton
Identifying the environmenta drivers of variation in demographic performance is cha enging A variety of approaches have been proposed e g Te er Ad er Edwards Hooker E ner

Summary

Introduction

BoyceHaridas Lee NCEAS stochastic demography working group Stenseth et aAs experimenta approaches to determining how natura popu ations are affected by environmenta variation are frequent y impractica structured demographic mode s are often used to understand the popu ation eve effects of environmen ta change Cou sonEhr en Morris von Eu er Dah gren Parmesan et aThese cha enges and the re ative y short ength of many demographic datasets Sa guero Gomez et a mean it is often difficu t to identify exp icit environmenta drivers of vita rates This restricts the abi ity of mode s to predict how popu ations wi respond to environ menta change Crone et a Environmenta variation can drive covariation amongst vita ratesDoak Morris Pfister Kenda Bruna Tomimatsu OharaA e se equa fai ing to account for this covariation wi bias mode outputs Fieberg E ner Metca f et a. Ehr en Morris von Eu er Dah gren Parmesan et a These cha enges and the re ative y short ength of many demographic datasets Sa guero Gomez et a mean it is often difficu t to identify exp icit environmenta drivers of vita rates This restricts the abi ity of mode s to predict how popu ations wi respond to environ menta change Crone et a Environmenta variation can drive covariation amongst vita rates. Using our framework we predict how changes to the average or variabi ity of the environment affect the ESS germi nation and f owering strategy We reparameterised the IPM of Rees et a Appendix A The mode is structured by the natura ogarithm of rosette area z a measure of p ant size that predicts individua performance Four stochastic vita rate functions with tempora y variab e intercepts were estimated surviva growth re cruitment and recruit size Appendix A. The prior distributions were weak y informative i e within bio ogica y reasonab e ranges to improve mixing Appendix A see Appendix A for comparison with more informative priors The credib e interva s of many parameters were re ative y wide Appendix A as a resu t of the short tempora extent years of this dataset Here to keep things simp e as this is just an examp e case study for the factor ana ytic approach we parameterise the IPM using the posterior means

Methods

Conclusion