Fitting functional response surfaces to data: a best practice guide

Abstract

Describing 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.

Systematic bias in studies of consumer functional responses

“…how many and how large a range of species abundance levels are employed; Sarnelle &Wilson (2008); Uszko et al . (2020)), observer error in estimating species densities and the number of eaten prey (Jost and Arditi, 2000), and the behavioural variation that exists among consumer and prey individuals (Chesson, 1984; Abrams, 2010). Many such sources of variation will be present even in the well‐controlled laboratory studies that dominate the literature, and could even be magnified in such studies given the small numbers of individual animals that are typically involved at low‐abundance treatment levels (Coblentz, 2020; Novak et al ., 2017).…”

“…This will occur directly via the bias inherent in their own maximum‐likelihood estimators (Box 2), and indirectly through their functional relationships and data‐dependent covariances with the estimates of other biased parameters. For example, bias is also expected for the attack rate parameter with which the interference parameter is functionally intertwined in the Arditi–Akçakaya model (see also Gutenkunst et al ., 2007; Uszko et al ., 2020), or for any deterministic functional‐response parameter when the assumed likelihood model entails `nuisance’ parameters whose estimates are biased (e.g. the standard deviation of the Gaussian likelihood; see Supplementary Materials ).…”

“…the standard deviation of the Gaussian likelihood; see Supplementary Materials ). Even for the simple Holling Type II model, estimates of the attack rate and handling time parameters can exhibit significant covariation (Papanikolaou et al ., 2016; Uszko et al ., 2020). The effect of covariation on bias will be especially large when feeding rate variation is large and heteroskedastic, or when the range over which species abundances are varied is insufficient to constrain estimates.…”

Systematic bias in studies of consumer functional responses

“…More specifically, both model-comparison bias (Box 1) and parameter-estimation bias (Box 2) are a function of a focal model’s parametric and geometric complexity, how well these are estimated, and for consumer functional responses, how these interact with the feeding rate variation within and among the treatment levels of an experiment. Sources of variation will include experimental design (e.g., how many and how large a range of species abundance levels are employed; Sarnelle & Wilson (2008); Uszko et al (2020)), observer error in estimating species densities and the number of eaten prey (Jost & Arditi, 2000), and the behavioural variation that exists among consumer and prey individuals (Abrams, 2010; Chesson, 1984). Many such sources of variation will be present even in the well-controlled laboratory studies that dominate the literature, and could even be magnified in such studies given the small numbers of individual animals that are typically involved at low-abundance treatment levels (Coblentz & DeLong, 2020; Novak et al , 2017).…”

Estimating predator functional responses using the times between prey captures

“…This is repeated across all the prey densities multiple times typically using different predator individuals for each trial. The experimenters then fit a functional response model (or models) to the data using one of several existing statistical methods depending on whether prey were replaced as they were eaten (Royama, 1971;Rogers, 1972;Bolker, 2008;Rosenbaum & Rall, 2018;Uszko, Diehl, & Wickman, 2020).…”