Perturbation approaches for integral projection models

Perturbation analysis of population models is fundamental to elucidating mechanisms of population dynamics and examining scenarios of change. The use of integral projection models (IPMs) has increased in the last decade, and while many of the tools and approaches developed for matrix models remain relevant, the nature of IPMs expands the framework of perturbation analysis, with different approaches often requiring important considerations. This article provides a review of - and practical guide to - different perturbation approaches for IPMs, formalizes methodologies for perturbing IPM size transition probabilities, and highlights areas where researchers should be particularly careful and critical when conducting and interpreting perturbation analysis. I use a simulated dataset to compare five hierarchical perturbation approaches for IPMs found within 63 published studies, and apply a combination of approaches to the example of an invasive perennial plant. Other perturbation approaches for IPMs are also highlighted. Most perturbation analyses for IPMs to date have focused on the response of the asymptotic population growth rate () to changes in elements of the discretized projection kernel and/or the growth- survival and reproduction- recruitment sub-kernels. Perturbations to vital rate functions and regression predictions underlying these kernels provide mechanistic insight, but are less common and can require important considerations regarding the perturbation of size transitions separate from survival and the nature of the state variable (used to represent size). The second most common approach is more specific to IPMs and examines the influence of vital rate regression parameters, each of which can have broad influence on the population growth rate. Researchers using IPMs have many perturbation options available and should carefully consider which approach or combination of approaches is most applicable and interpretable for their specific questions.