Quantifying demographic uncertainty: Bayesian methods for integral projection models

Integral projection models (IPMs) are a powerful and popular approach to modeling population dynamics. Generalized linear models form the statistical backbone of an IPM. These models are typically fit using a frequentist approach. We suggest that hierarchical Bayesian statistical approaches offer important advantages over frequentist methods for building and interpreting IPMs, especially given the hierarchical nature of most demographic studies. Using a stochastic IPM for a desert cactus based on a 10-year study as a worked example, we highlight the application of a Bayesian approach for translating uncertainty in the vital rates (e.g., growth, survival, fertility) to uncertainty in population-level quantities derived from them (e.g., population growth rate). The best fit demographic model, which would have been difficult to fit under a frequentist framework, allowed for spatial and temporal variation in vital rates and correlated responses to temporal variation across vital rates. The corresponding posterior probability distribution for the stochastic population growth rate (lambda(S)) indicated that, if current vital rates continue, the study population will decline with nearly 100% probability. Interestingly, less supported candidate models that did not include spatial variance and vital rate correlations gave similar estimates of lambda(S). This occurred because the best fitting model did a much better job of fitting vital rates to which the population growth rate was weakly sensitive. The cactus case study highlights several advantages of Bayesian approaches to IPM modeling, including that they: (1) provide a natural fit to demographic data, which are often collected in a hierarchical fashion (e.g., with random variance corresponding to temporal and spatial heterogeneity); (2) seamlessly combine multiple data sets or experiments; (3) readily incorporate covariance between vital rates; and, (4) easily integrate prior information, which may be particularly important for species of conservation concern where data availability may be limited. However, constructing a Bayesian IPM will often require the custom development of a statistical model tailored to the peculiarities of the sampling design and species considered; there may be circumstances under which simpler methods are adequate. Overall, Bayesian approaches provide a statistically sound way to get more information out of hard-won data, the goal of most demographic research endeavors.