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 model's 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 model's geometric complexity varies with the data's 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.

Statistical Issues in the Estimation of Functional Responses

In this chapter I cover some key issues in fitting functional response models to data and determining the values of parameters. Because some of these issues have been covered elsewhere, here I focus on the nature of foraging trial data and why noise, stochasticity, and individual variation pose particular challenges for understanding functional responses. I examine several data sets to illustrate methods of determining differences in functional response parameters and types. I also show through simulations that individual variation in functional response parameters may account for the noisiness of foraging data and also lead to underestimates of both space clearance rate and handling time in curve-fitting approaches.

The Basics and Origin of Functional Response Models

This chapter is the essential beginner’s guide to the functional response, its derivation, the various forms, its connection to other models in the literature, and what the parameters mean. It is the ground floor for the rest of the book, covering the four main types of functional response, what the parameters mean in biological terms, and how we arrived at these equations. Surprisingly, our understanding of the functional response as represented in the literature is quite muddled, with confusion ranging from the terminology used, to the various mathematical forms the functional response takes, to the biological interpretation of functional response model parameters. I provide a summary and forward-looking perspective on these issues.

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 model's 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 criterion 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 levels of phenomenological generality and varying apparent degrees and forms of non-linearity. Because a model's geometric complexity varies with the data's 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 geometric complexity 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 the relative effects of geometric complexity are strongest. We conclude by discussing the statistical and epistemological challenges that geometric complexity poses for the study of functional responses as it relates to the attainment of biological truth and predictive ability.

Influence of competition and intraguild predation between two candidate biocontrol parasitoids on their potential impact against Harrisia cactus mealybug, Hypogeococcus sp. (Hemiptera: Pseudococcidae)

When two or more parasitoid species, particularly candidates for biocontrol, share the same target in the same temporal window, a complex of behaviors can occur among them. We studied the type of interactions (competition and intraguild predation) that existed between the nymphal parasitoids Anagyrus cachamai and A. lapachosus (Hymenoptera: Encyrtidae), two candidate neoclassical biocontrol agents against the Puerto Rican cactus pest mealybug, Hypogeococcus sp. (Hemiptera: Pseudococcidae). The surrogate native congener host in Argentina, the cactus mealybug Hypogeococcus sp., was studied to predict which species should be released; in the case that both should be released, in which order, and their potential impact on host suppression. In the laboratory we conducted experiments where different densities of the host mealybug were exposed to naive females of A. cachamai and A. lapachosus sequentially in both directions. Experiments were analyzed by combining a series of competitive behavioral and functional response models. A fully Bayesian approach was used to select the best explaining models and calculate their parameters. Intraguild predation existed between A. cachamai, the species that had the greatest ability to exploit the resource, and A. lapachosus, the strongest species in the interference competition. The role that intraguild predation played in suppression of Hypogeococcus sp. indicated that a multiple release strategy for the two biocontrol agents would produce better control than a single release; as for the release order, A. lapachosus should be released first.

Influence Of Competition And Intraguild Predation Between Two Candidate Biocontrol Parasitoids On Their Potential Impact Against Harrisia Cactus Mealybug, Hypogeococcus sp. (Hemiptera: Pseudococcidae)

When two or more parasitoid species, particularly candidates for biocontrol, share the same target in the same temporal window, a complex of behaviors can occur among them. We studied the type of interactions (competition and intraguild predation) that existed between the nymphal parasitoids Anagyrus cachamai and A. lapachosus (Hymenoptera: Encyrtidae), two candidate neoclassical biocontrol agents against the Puerto Rican cactus pest mealybug, Hypogeococcus sp. (Hemiptera: Pseudococcidae). The surrogate native congener host in Argentina, the cactus mealybug Hypogeococcus sp., was studied to predict which species should be released; in the case that both should be released, in which order, and their potential impact on host suppression. In the laboratory we conducted experiments where different densities of the host mealybug were exposed to naive females of A. cachamai and A. lapachosus sequentially in both directions. Experiments were analyzed by combining a series of competitive behavioral and functional response models. A fully Bayesian approach was used to select the best explaining models and calculate their parameters. Intraguild predation existed between A. cachamai, the species that had the greatest ability to exploit the resource, and A. lapachosus, the strongest species in the interference competition. The role that intraguild predation played in suppression of Hypogeococcus sp. indicated that a multiple release strategy for the two biocontrol agents would produce better control than a single release; as for the release order, A. lapachosus should be released first.

Temperature-Dependent Functional Response of Harmonia Axyridis Pallas (Coleoptera: Coccinellidae) Preying on Acyrthosiphon Pisum (Harris) (Hemiptera: Aphididae) Nymphs

Functional response models are often used to understand the foraging interactions and determine the suitable biocontrol agents. We determined the functional response of Harmonia axyridis to nymph Acyrthosiphon pisum at different but constant temperatures (between 15 and 35 °C) and prey densities. Logistic regression and Roger’s random predator models were employed to determine the type and parameters of functional response. Harmonia axyridis larvae and adults exhibited Type II functional responses to different densities of A. pisum. Warming increased both the predation activity and host aphid control mortality. The 4th instar and female H. axyridis consumed the most aphids. Warming contributed markedly in accelerating the predator action. For fourth instar larvae and female H. axyridis adult, the successful attack rates were 0.234 ± 0.014 h−1 and 0.247 ± 0.015 h−1; the handling times were 0.132 ± 0.005 h and 0.156 ± 0.004 h; and the estimated maximum predation rates were 181.28 ± 14.54 and 153.85 ± 4.06, respectively. These findings accentuate the high performance of 4th instar and female H. axyridis and the role of temperature in their efficiency. Further studies exploring intraguild predation and mutual interference will be required to conclude the biocontrol potential of H. axyridis to A. pisum.

Hidden layers of density dependence in consumer feeding rates

Functional responses relate a consumer’s feeding rates to variation in its abiotic and biotic environment, providing insight into consumer behavior and fitness, and underpinning population and food-web dynamics. Despite their broad relevance and long-standing history, we show here that the types of density dependence found in classic resource- and consumer-dependent functional-response models equate to strong and often untenable assumptions about the independence of processes underlying feeding rates. We first demonstrate mathematically how to quantify non-independence between feeding and consumer interference and between feeding on multiple resources. We then analyze two large collections of functional-response datasets to show that non-independence is pervasive and borne out in previously-hidden forms of density dependence. Our results provide a new lens through which to view variation in consumer feeding rates and disentangle the biological underpinnings of species interactions in multi-species contexts.

Systematic bias in studies of consumer functional responses

Functional responses are a cornerstone to our understanding of consumer-resource interactions, so how to best describe them using models has been actively debated. Here we focus on the consumer dependence of functional responses to evidence systematic bias in the statistical comparison of functional-response models and the estimation of their parameters. Both forms of bias are universal to nonlinear models (irrespective of consumer dependence) and are rooted in a lack of sufficient replication. Using a large compilation of published datasets, we show that – due to the prevalence of low sample size studies – neither the overall frequency by which alternative models achieve top rank nor the frequency distribution of parameter point estimates should be treated as providing insight into the general form or central tendency of consumer interference. We call for renewed clarity in the varied purposes that motivate the study of functional responses, purposes that can compete with each other in dictating the design, analysis, and interpretation of functional-response experiments.

Boosting functional response models for location, scale and shape with an application to bacterial competition

We extend generalized additive models for location, scale and shape (GAMLSS) to regression with functional response. This allows us to simultaneously model point-wise mean curves, variances and other distributional parameters of the response in dependence of various scalar and functional covariate effects. In addition, the scope of distributions is extended beyond exponential families. The model is fitted via gradient boosting, which offers inherent model selection and is shown to be suitable for both complex model structures and highly auto-correlated response curves. This enables us to analyse bacterial growth in Escherichia coli in a complex interaction scenario, fruitfully extending usual growth models.