Geometric complexity and the information-theoretic comparison of functional-response models
The assessment of relative model performance using information criteria like AIC and BIC has become routine among functional-response studies, reflecting trends in the broader ecological literature. Such information criteria allow comparison across diverse models because they penalize each models fit by its parametric complexity -- in terms of their number of free parameters -- which allows simpler models to outperform similarly fitting models of higher parametric complexity. However, criteria like AIC and BIC do not consider an additional form of model complexity, referred to as geometric complexity, which relates specifically to the mathematical form of the model. Models of equivalent parametric complexity can differ in their geometric complexity and thereby in their ability to flexibly fit data. Here we use the Fisher Information Approximation to compare, explain, and contextualize how geometric complexity varies across a large compilation of single-prey functional-response models -- including prey-, ratio-, and predator-dependent formulations -- reflecting varying apparent degrees and forms of non-linearity. Because a models geometric complexity varies with the datas underlying experimental design, we also sought to determine which designs are best at leveling the playing field among functional-response models. Our analyses illustrate (1) the large differences in geometric complexity that exist among functional-response models, (2) there is no experimental design that can minimize these differences across all models, and (3) even the qualitative nature by which some models are more or less flexible than others is reversed by changes in experimental design. Failure to appreciate model flexibility in the empirical evaluation of functional-response models may therefore lead to biased inferences for predator-prey ecology, particularly at low experimental sample sizes where its impact is strongest. We conclude by discussing the statistical and epistemological challenges that model flexibility poses for the study of functional responses as it relates to the attainment of biological truth and predictive ability. Contribution to Field StatementThe use of criteria like AIC and BIC for selecting among functional-response models is now standard, well-accepted practice, just as it is in the ecological literature as a whole. The generic desire underlying the use of these criteria is to make the comparison of model performance an unbiased and equitable process by penalizing each models fit to data by its parametric complexity (relating to its number of free parameters). Here we introduce the Fisher Information Approximation to the ecological literature and use it to understand how the geometric complexity of models -- a form of model complexity relating to a models functional flexibility that is not considered by criteria like AIC and BIC -- varies across a large compilation of 40 different single-prey functional-response models. Our results add caution against the simplistic use and interpretation of information-theoretic model comparisons for functional-response experiments, showing just how large an effect that model flexibility can have on inferences of model performance. We therefore use our work to help clarify the challenges that ecologists studying functional responses must face in the attainment of biological truth and predictive ability.