Fitting functional response surfaces to data: a best practice guide

Wojciech Uszko,Jonas Wickman,Sebastian Diehl

doi:10.1002/ecs2.3051

Abstract

AbstractDescribing how resource consumption rates depend on resource density, conventionally termed “functional responses,” is crucial to understanding the population dynamics of trophically interacting organisms. Yet, accurately determining the functional response for any given pair of predator and prey remains a challenge. Moreover, functional responses are potentially complex surfaces in multidimensional space, where resource density is only one of several factors determining consumption rates. We explored how three sources of error can be addressed in the design and statistical analysis of functional response experiments: ill‐chosen spacing of prey densities, heteroscedastic variance in consumption rates, and non‐independence of parameters of the function describing prey consumption in relation to prey density and additional environmental factors. We generated extensive, virtual data sets that simulated feeding experiments in which both prey density and environmental temperature were varied, and for which the true, deterministic functional response surface was known and realistic variance had been added. We compared eight different methods of functional response fitting, one of which stood out as best performing. We subsequently tested several conclusions from the simulation study against experimental data of zooplankton feeding on algae across a broad range of temperatures. We summarize our main findings in three best practice guidelines for the experimental estimation of functional response surfaces, of which the second is the most important: (1) space prey densities logarithmically, starting from very low densities; (2) log‐transform prey consumption data prior to fitting; and (3) fit a multivariate functional response surface to all data (including all prey densities and other factors, in our case temperature) in a single step. We also observed that functional response surfaces were fitted more accurately and precisely than their component parameters. The latter occurred because parameter estimates were non‐independent, which is an inevitable feature of fitting complex nonlinear functions to data: A given response surface can often be described with near‐equal accuracy by multiple parameter combinations. We therefore conclude that fitted functional response models perform better at optimizing the fit of the overall response surface than at determining how component parameters, such as the attack rate or handling time, depend on environmental factors such as temperature.

Highlights

All organisms are embedded in trophic networks in which they act as resources, consumers, or both
It has been reported that attack and maximum ingestion rates are unimodally related to temperature (Englund et al 2011) and to predator–prey body size ratio (Rall et al 2012), while other studies have reported shifts of the Hill exponent with warming (Wang and Ferro 1998, Mohaghegh et al 2001, South et al 2018, Uszko et al 2017a,b) and increasing predator–prey body size ratio (Kalinkat et al 2013, Barrios-O’Neill et al 2016)
We have shown that the accuracy and precision of functional response measurements based on feeding experiments can be greatly improved by following three rules: (1) include very low prey densities and space out prey densities logarithmically; (2) log-transform data prior to functional response fitting; (3) when other environmental variables beyond prey density are studied, fit a multivariate functional response surface in a single step

Summary

Introduction

All organisms are embedded in trophic networks in which they act as resources, consumers, or both. In addition to resource density, resource uptake rates may depend on other factors such as consumer density (Skalski and Gilliam 2001), consumer body size (Vucic-Pestic et al 2010), and environmental temperature (Englund et al 2011). When integrating these dependencies into the Holling type framework, it is customary to make one or more of the parameters in Eq 1 dependent on these other factors. Functional responses are increasingly viewed as complex response surfaces in multidimensional space (Fig. 2A), where resource density is only one of several factors determining resource uptake rates (Vucic-Pestic et al 2010, 2011)

Methods

Discussion

Conclusion