Functional Response Models Research Articles

Rejuvenating functional responses with renewal theory.

Functional responses are widely used to describe interactions and resource exchange between individuals in ecology. The form given to functional responses dramatically affects the dynamics and stability of populations and communities. Despite their importance, functional responses are generally considered with a phenomenological approach, without clear mechanistic justifications from individual traits and behaviours. Here, we develop a bottom-up stochastic framework grounded in renewal theory that shows how functional responses emerge from the level of the individuals through the decomposition of interactions into different activities. Our framework has many applications for conceptual, theoretical and empirical purposes. First, we show how the mean and variance of classical functional responses are obtained with explicit ecological assumptions, for instance regarding foraging behaviours. Second, we give examples in specific ecological contexts, such as in nuptial-feeding species or size-dependent handling times. Finally, we demonstrate how to analyse data with our framework, especially highlighting that observed variability in the number of interactions can be used to infer parameters and compare functional response models.

Intra-host interspecific larval parasitoid competition solved using modelling and bayesian statistics

Intraguild competition is a complex phenomenon that shapes parasitoid communities. When several species of parasitoids oviposit within the same individual host, a complex phenomenon of larval competitive interaction occurs. Within the same guild there is a specialization in competitive strategies, sometimes multiparasitism is avoided, but some species are facultative hyperparasitoids/predators of their competitors.As these interactions occur within a very small host and during a brief period of time, and that direct observation is very difficult to achieve, we used an alternative methodological approach. We analyzed intraguild host competition mechanisms via the combination of a series of competitive behavioral and functional response models, thurstonian competition model and set theory. These models were fitted via a reversible-jump bayesian model selection procedure to a series of competition experiments data using larvae of three species of Gonatocerus spp. (Hymenoptera: Mymaridae), egg parasitoids of the sharpshooter Tapajosa rubromarginata (Hemiptera: Cicadellidae) as a case-of-study. This study tests the influence of intrinsic interspecific competition between inmature stages within on an individual host, and parasitoid arrival order among the three parasitoid species.The results showed that the species differed in competitive behavior, some species were better competitors than others. Individuals arriving earlier had a competitive advantage, the weaker species were able to outcompete the stronger ones if the time advantage was longer than 18 h. All the species avoided already parasitized hosts, but in different degrees. The functional response was also different, with the best competitors having shorter estimated handling times. Using this analytical approach on a conventional experimental setup, we gained insights in the mechanism of competition, both on interference and exploitation, and in terms of host selection, all in a single analysis.

Fitting functional responses: Direct parameter estimation by simulating differential equations

AbstractThe feeding functional response is one of the most widespread mathematical frameworks in ecology, marine biology, freshwater biology, microbiology, and related scientific fields describing the resource‐dependent uptake of a consumer. Since the exact knowledge of its parameters is crucial to predict, for example, the efficiency of biocontrol agents, population dynamics, food web structure, and subsequently biodiversity, a trustworthy parameter estimation method is highly important for scientists using this framework. Classical approaches for estimating functional response parameters lack flexibility and often only provide approximations of the correct parameters.Here, we combined ordinary differential equation (ODE) models that were numerically solved using computer simulations with an iterative maximum likelihood fitting approach. We compared our method to classical approaches of fitting functional responses using data both with and without additional resource growth and mortality.We found that for classical functional response models, such as the frequently used type II and type III functional responses, the established fitting methods are reliable. However, by using more complex and flexible functional responses, our new method outperforms the traditional methods. Additionally, our method allows the incorporation of side effects such as resource growth and background mortality.Our method will enable researchers from different scientific fields who are measuring functional responses to calculate more accurate parameter estimates. These estimates will enable community ecologists to parameterize their models more precisely, thus allowing a deeper understanding of complex ecological systems, and will increase the quality of ecological prediction models.

A multi-factor empirical model for calculation of naupliar ingestion rate of the embayment copepod Acartia steueri Smirnov (Copepoda: Calanoida)

Mathematical approaches with multiple factors are essential for estimating in situ ingestion rates of copepod nauplii. The present study aimed to construct a multi-factor empirical model that could calculate naupliar carbon-specific ingestion rates using food concentrations and individual carbon weights. Laboratory feeding experiments were conducted to derive the functional response model and temperature quotient (Q10) of the naupliar stages of the embayment copepod Acartia steueri from NIII to NVI. Experiments were conducted with haptophyte Isochrysis galbana at different concentrations for all stages and at different temperatures for NV. Carbon-specific ingestion rates in relation to food concentration followed a type III functional response model at every developmental stage, with two constants of a maximum carbon-specific ingestion rate and a half-saturation food concentration. Each of the two constants varied and was significantly related to individual carbon weights, and two relational equations were obtained from the relationships. By substituting the equations into a type III functional response model, an empirical model was constructed. Carbon-specific ingestion rates of NV increased with increasing temperature and Q10 was determined to be 2.37. Estimated carbon-specific ingestion rates calculated using the empirical model and Q10 were significantly related to the measured carbon-specific ingestion rates. This trend was also confirmed in a dataset of ingestion rates of Oithona davisae nauplii quoted from a previous study. This suggests that the empirical model proposed here might be widely applicable for other species of copepod nauplii.

Optimal experimental design for predator-prey functional response experiments.

Functional response models are important in understanding predator-prey interactions. The development of functional response methodology has progressed from mechanistic models to more statistically motivated models that can account for variance and the over-dispersion commonly seen in the datasets collected from functional response experiments. However, little information seems to be available for those wishing to prepare optimal parameter estimation designs for functional response experiments. It is worth noting that optimally designed experiments may require smaller sample sizes to achieve the same statistical outcomes as non-optimally designed experiments. In this paper, we develop a model-based approach to optimal experimental design for functional response experiments in the presence of parameter uncertainty (also known as a robust optimal design approach). Further, we develop and compare new utility functions which better focus on the statistical efficiency of the designs; these utilities are generally applicable for robust optimal design in other applications (not just in functional response). The methods are illustrated using a beta-binomial functional response model for two published datasets: an experiment involving the freshwater predator Notonecta glauca (an aquatic insect) preying on Asellus aquaticus (a small crustacean), and another experiment involving a ladybird beetle (Propylea quatuordecimpunctata L.) preying on the black bean aphid (Aphis fabae Scopoli). As a by-product, we also derive necessary quantities to perform optimal design for beta-binomial regression models, which may be useful in other applications.

What drives interaction strengths in complex food webs? A test with feeding rates of a generalist stream predator.

Describing the mechanisms that drive variation in species interaction strengths is central to understanding, predicting, and managing community dynamics. Multiple factors have been linked to trophic interaction strength variation, including species densities, species traits, and abiotic factors. Yet most empirical tests of the relative roles of multiple mechanisms that drive variation have been limited to simplified experiments that may diverge from the dynamics of natural food webs. Here, we used a field-based observational approach to quantify the roles of prey density, predator density, predator-prey body-mass ratios, prey identity, and abiotic factors in driving variation in feeding rates of reticulate sculpin (Cottus perplexus). We combined data on over 6,000 predator-prey observations with prey identification time functions to estimate 289 prey-specific feeding rates at nine stream sites in Oregon. Feeding rates on 57 prey types showed an approximately log-normal distribution, with few strong and many weak interactions. Model selection indicated that prey density, followed by prey identity, were the two most important predictors of prey-specific sculpin feeding rates. Feeding rates showed a positive relationship with prey taxon densities that was inconsistent with predator saturation predicted by current functional response models. Feeding rates also exhibited four orders-of-magnitude in variation across prey taxonomic orders, with the lowest feeding rates observed on prey with significant anti-predator defenses. Body-mass ratios were the third most important predictor variable, showing a hump-shaped relationship with the highest feeding rates at intermediate ratios. Sculpin density was negatively correlated with feeding rates, consistent with the presence of intraspecific predator interference. Our results highlight how multiple co-occurring drivers shape trophic interactions in nature and underscore ways in which simplified experiments or reliance on scaling laws alone may lead to biased inferences about the structure and dynamics of species-rich food webs.

Role of different habitats for the functional response of Crytorrhinus lividipennis (Hemiptera: Miridae) on Nilaparvata lugens (Hemiptera: Delphacidae)

ABSTRACTThe functional response of a predator, Cyrtorrhinus lividipennis, on the brown planthopper, Nilaparvata lugens, was investigated in two different experimental habitats: a piece of tiller in a petri dish (5.5 cm in diameter) and a rice plant in a net cage (5.5 cm diameter × 43 cm height) for 24 h at room temperature. In the petri dish experiment, 2nd–5th instar nymphs, adult male, and female C. lividipennis were introduced in separate experiments to eggs of N. lugens at densities of five, 10, 20, 30, and 40 eggs per piece of rice. In the rice plant experiment, each C. lividipennis was introduced to a cage with a rice plant containing N. lugens eggs. After 24 h, the number of dead eggs and remaining eggs were counted and data fitted to three functional response models. Among the three types of functional responses, Type II best described the predator response to host densities in N. lugens in both experimental habitats, according to the logistic regression analysis value. The results showed that C. lividipennis was a more effective predator in the rice plant experiment compared to the disc experiment. Additionally, the searching efficiency and handling time parameters were different in the two different experimental habitats. This may cause errors when applying the functional response to biological control and predator–prey models. Different habitats and other environmental conditions from the experiment and natural rice field have to be considered.

Testing the validity of functional response models using molecular gut content analysis for prey choice in soil predators

Analysis of predator–prey interactions is a core concept of animal ecology, explaining structure and dynamics of animal food webs. Measuring the functional response, i.e. the intake rate of a consumer as a function of prey density, is a powerful method to predict the strength of trophic links and assess motives of prey choice, particularly in arthropod communities. However, due to their reductionist set‐up, functional responses, which are based on laboratory feeding experiments, may not display field conditions, possibly leading to skewed results. Here, we tested the validity of functional responses of centipede predators and their prey by comparing them with empirical gut content data from field‐collected predators. Our predator–prey system included lithobiid and geophilomorph centipedes, abundant and widespread predators of forest soils and their soil‐dwelling prey. First, we calculated the body size‐dependent functional responses of centipedes using a published functional response model in which we included natural prey abundances and animal body masses. This allowed us to calculate relative proportions of specific prey taxa in the centipede diet. In a second step, we screened field‐collected centipedes for DNA of eight abundant soil‐living prey taxa and estimated their body size‐dependent proportion of feeding events. We subsequently compared empirical data for each of the eight prey taxa, on proportional feeding events with functional response‐derived data on prey proportions expected in the gut, showing that both approaches significantly correlate in five out of eight predator–prey links for lithobiid centipedes but only in one case for geophilomorph centipedes. Our findings suggest that purely allometric functional response models, which are based on predator–prey body size ratios are too simple to explain predator–prey interactions in a complex system such as soil. We therefore stress that specific prey traits, such as defence mechanisms, must be considered for accurate predictions.

Size-dependent predation and intraspecific inhibition of an estuarine snail feeding on oysters

Predator outbreaks have increased in the past two decades in many ecosystems and are predicted to become more common with climate change. During these outbreaks, predator densities increase rapidly, and can cause large reductions in prey populations or shifts in prey size structure. However, unexpected interactions may occur at high predator densities, necessitating a mechanistic understanding of how increased predator density affects predator-prey dynamics. In the northern Gulf of Mexico, outbreaks of southern oyster drill, Stramonita haemastoma (Linnaeus, 1767), occur during high salinity events, and can greatly reduce eastern oyster, Crassostrea virginica (Gmelin, 1791), populations. A large outbreak of drills occurred from 2013 to 2015 in Apalachicola Bay, FL which corresponded with an oyster fishery collapse in that bay. To improve our mechanistic understanding of predation during such a high-density outbreak, laboratory experiments based on field observations were used to quantify the prey size selection by drills and the drill functional response, as a function of drill abundance. Drills fed on medium-sized oysters (50–75 mm) more often than small and larger-sized oysters, and often formed aggregations during feeding events. However, despite this aggregative response, there was a negative relationship between per capita feeding rates and drill abundance. Indeed, the Crowley-Martin functional response model had the most parsimonious fit to the data, suggesting that predator-predator inhibition reduced attack rates and increased handling times. Due to an increase in regional drought conditions and water usage in the southeastern United States, drill outbreaks will likely increase in frequency and duration. A greater understanding of how predation rates change with predator densities during outbreaks will improve predictions of oyster mortality, and strengthen the scientific framework for oyster fishery decisions.

Nutritional effects on chemical defense alter predator–prey dynamics

Reservoir-based chemical defense (= reducible defense) is a widespread mechanism to repel predators in many invertebrates. We investigated the influence of macronutrients on the availability and regeneration of defensive secretions and parametrize a theoretical functional response model for reducible defense to predict nutritional effects on predator–prey dynamics. Our modeling approach showed that initially high amounts of defensive secretions provided an effective short-time defense, while higher regeneration rates were favorable under permanent predation pressure. Regeneration rates were correlated to the amount of dietary fat and carbohydrates, indicating an adaptive connection of macronutrients on chemical defense and predatory success—an effect covering two trophic levels. Our results underpin the urgent need to integrate dynamical aspects of chemical defense into the modeling of predator–prey interactions in food webs.

Experimental duration and predator satiation levels systematically affect functional response parameters

Empirical feeding studies where density‐dependent consumption rates are fitted to functional response models are often used to parameterize the interaction strengths in models of population or food‐web dynamics. However, the relationship between functional response parameter estimates from short‐term feeding studies and real‐world, long‐term, trophic interaction strengths remains largely unexamined. In a critical first step to address this void, we tested for systematic effects of experimental duration and predator satiation on the estimate of functional response parameters, namely attack rate and handling time. Analyzing a large data set covering a wide range of predator taxa and body masses, we show that attack rates decrease with increasing experimental duration, and that handling times of starved predators are consistently shorter than those of satiated predators. Therefore, both the experimental duration and the predator satiation level have a strong and systematic impact on the predictions of population dynamics and food‐web stability. Our study highlights potential pitfalls at the intersection of empirical and theoretical applications of functional responses. We conclude our study with some practical suggestions for how these implications should be addressed in the future to improve predictive abilities and realism in models of predator–prey interactions.

Functional responses of cougars (Puma concolor) in a multiple prey-species system.

The study of predator-prey interactions is commonly analyzed using functional responses to gain an understanding of predation patterns and the impact they have on prey populations. Despite this, little is known about predator-prey systems with multiple prey species in sites near the equator. Here we studied the functional response of cougars (Puma concolor) in relation to their main prey, armadillo (Dasypus novemcinctus), coati (Nasua narica) and white-tailed deer (Odocoileus virginianus). Between 2004 and 2010, cougar scats were collected along 5 transects to estimate the consumption of different prey species. A relative abundance index (RAI) was calculated for each prey species and cougar using 18 camera traps. We compared Holling type I, II and III functional response models to determine patterns in prey consumption based on the relative abundance and biomass of each prey species consumed. The 3 main prey species comprised 55% (armadillo), 17% (coati) and 8% (white-tailed deer) of the diet. Type I and II functional responses described consumption of the 2 most common prey species armadillos and coati similarly well, while a type I response best characterized consumption of white-tailed deer. A negative correlation between the proportions of armadillo versus coati and white-tailed deer biomass in cougar scats suggests switching to consume alternative prey, confirming high foraging plasticity of this carnivore. This work represents one of the few studies to compare functional responses across multiple prey species, combined with evidence for prey-switching at low densities of preferred prey.

Oviposition behaviour of Tamarixia radiata: effects of host density and exposure time

1. The number of hosts attacked as a function of host density is considered to be an important characteristic of parasitoid behaviour and is used to estimate key parameters such as handling time and ‘instantaneous rate of discovery’. However, little has been done to validate functional response models by direct observation of parasitoid oviposition behaviour.2. Tamarixia radiata is the most promising parasitoid for biological control attacking Diaphorina citri. Mass rearing and augmentative release seen as a potential strategy for suppression of D. citri has been documented in abandoned citrus, residential areas, and organic groves. Nevertheless, parasitism rates in culture and in the field are only moderate, leading to questions about oviposition behaviour in response to host density.3. Behaviours of gravid T. radiata females presented with susceptible host instars were categorised and documented by direct observation for 30 min and by camera recordings made over 12 h. Frequency of searching and antennating increased with host density during the 30 min. Probing rejection rates and search duration increased significantly with host density over 12 h. These factors resulted in significantly lower fecundity than expected, possibly due to host mark‐mediated deterrence within the small searching area. Females took approximately 3.6–4.2 min to probe and parasitise a host regardless of host density and exposure duration. These results were markedly different from the 52.2 min estimated from the functional response equation.4. Further experiments are required to assess the range and persistence of the putative host‐marking pheromone, and to better understand the relationship between functional response parameters and actual behaviour.

Frair: an R package for fitting and comparing consumer functional responses

SummaryConsumer‐resource interactions (i.e. the functional response) underpin decades of ecological advancements. However, selecting, fitting and comparing functional response models using appropriate methods remains a non‐trivial endeavour.The R packagefrairprovides tools for selecting and differentiating various forms of consumer functional response models, a consistent interface for fitting and visualising response curves, and a selection of statistically robust methods for comparing fitted parameters.Using real data from crustacean predator‐prey systems, we demonstrate the utility offrair, highlighting best practice and common analytical mistakes.

Evaluation of alternative prey-, predator-, and ratio-dependent functional response models in a zooplankton microcosm

There is strenuous debate among ecologists regarding the inclusion of predator density into the originally prey-dependent functional response. We provided comprehensive empirical comparisons of alternative functional response models for the predatory ostracod Heterocypris incongruens (Ramdohr, 1808) and the rotifer Brachionus calyciflorus (Pallas, 1766) as its prey in small freshwater microcosms. Prey killed was measured at factorial combinations of four predator densities and five prey densities, and was recorded at 3 min intervals over 60 min experiments. To support the potential effect of predator interference on per capita kill rate, we recorded ostracod activity and aggression. Kill rate increased following a saturating function with increasing prey density and decreased with increasing predator density. Model evaluation using an information–theoretic approach indicated that the Arditi–Ginzburg type II ratio-dependent model performed best, followed by the Arditi–Akcakaya and Beddington–DeAngelis type II predator-dependent models, suggesting that predator interference was important in predicting kill rates. Interference among predators increased and their activity decreased with increasing predator density, providing confirmation that interference was responsible for the predator-dependent effect. By combining a microcosm experiment and behavioral observations, our results suggest that predator interference at realistic population densities influences ostracod kill rates and this form of interference was best accommodated by predator-dependent models.

Influence of temperature on the functional response of the predatory bug, Anthocoris minki pistaciae (Hemiptera: Anthocoridae), a predator of Agonoscena pistaciae (Hemiptera: Psyllidae)

AbstractThe common pistachio psylla, Agonoscena pistaciae Burckhardt and Lauterer (Hemiptera: Psyllidae), is one of the most destructive pests of cultivated pistachio trees in Iran. A locally available anthocorid predator, Anthocoris minki pistaciae Wagner (Hemiptera: Anthocoridae) was evaluated as a biological agent candidate for the control of this pest. The functional response of this predatory bug was investigated at temperatures of 20, 25, 27.5, and 30 °C, relative humidity of 50–60%, photoperiod L:D of 16:8 h and at densities of 2, 4, 8, 20, 40, 60, 80, and 100 nymphs of A. pistaciae during a 24-h period. A type II functional response model was fitted at all tested temperatures. The results indicated that the instantaneous attack rate for the predatory bug ranged from 0.15 to 0.36 nymphs per hour and estimated handling times were 0.41, 0.35, 0.42, and 0.27 h at 20, 25, 27.5, and 30 °C, respectively. Based on asymptotic 95% confidence intervals, searching efficiency didn’t differ significantly from 2...

Diverse foraging opportunities drive the functional response of local and landscape-scale bear predation on Pacific salmon.

The relationship between prey abundance and predation is often examined in single habitat units or populations, but predators may occupy landscapes with diverse habitats and foraging opportunities. The vulnerability of prey within populations may depend on habitat features that hinder predation, and increased density of conspecifics in both the immediate vicinity and the broader landscape. We evaluated the relative effects of physical habitat, local, and neighborhood prey density on predation by brown bears on sockeye salmon in a suite of 27 streams using hierarchical Bayesian functional response models. Stream depth and width were inversely related to the maximum proportion of salmon killed, but not the asymptotic limit on total number killed. Interannual variation in predation was density dependent; the number of salmon killed increased with fish density in each stream towards an asymptote. Seven streams in two geographical groups with ≥23years of data in common were then analyzed for neighborhood density effects. In most (12 of 18) cases predation in a stream was reduced by increasing salmon abundance in neighboring streams. The uncertainty in the estimates for these neighborhood effects may have resulted from interactions between salmon abundance and habitat that influenced foraging by bears, and from bear behavior (e.g., competitive exclusion) and abundance. Taken together, the results indicated that predator-prey interactions depend on density at multiple spatial scales, and on habitat features of the surrounding landscape. Explicit consideration of this context dependency should lead to improved understanding of the ecological impacts of predation across ecosystems and taxa.

Analysis of a predator-prey model with Lévy jumps

This paper deals with a predator-prey model of Beddington-DeAngelis type functional response with Lévy jumps. The proposed mathematical model consists of a system of two stochastic differential equations to stimulate the interactions between predator population and prey population. The dynamics of the system is discussed mainly from the point of view of persistence and extinction. To begin with, the global positivity, stochastically boundedness and other asymptotic properties have been derived. In addition, sufficient conditions for extinction, nonpersistence in the mean and weak persistence are obtained. It is proved that the variation of Lévy jumps can affect the asymptotic property of the system.

Ontogenetic shift in Daphnia-algae interaction strength altered by stressors: revisiting Jensen’s inequality

Interaction strength among species plays a crucial role in shaping the functioning of ecological communities, but it is often assumed to be insensitive to inter-individual variation in underlying parameters such as attack rates or handling time. Ecological factors including stressors exert age/size-dependent effects on such behavioral parameters, promoting shifts in the distribution of parameter values over ages. Here we analyze the effects of the pesticide methamidophos on the Daphnia-microalga interaction strength. We first analyze age-dependent effects of the pesticide on the Daphnia functional response, and then decompose the population-level effects of the stressor into contributions of shifts in elevation (i.e., vertical effect) versus shifts in nonlinearity (i.e., nonlinear effect) of the response of interaction strength over consumer age. Our results show that (1) Rogers and Holling type II functional response models best fitted the empirical functional responses of Daphnia of different ages, (2) attack rate and handling time were affected by the pesticide, (3) these effects were age-specific, shifting the average attack rate and both the mean and coefficient of variation of handling time of different age classes, (4) population level interaction strength was affected by pesticide exposure by variation in both elevation and nonlinearity of its response over consumer age. We show that both vertical and nonlinear effects were important in magnitude but opposite in sign. The consequences of factors that exert age/size dependent effects can only be evaluated through properly considering inter-individual variation.

Parasitism Rate of Habrobracon hebetor to Ephestia kuehniella Larvae: Residual Effect of Deltamethrin, Fenvalerate and Azadirachtin

Sublethal concentrations of chemical insecticides may cause changes in some behavioral characteristics of natural enemies such as functional responses. The residual effect of three synthetic insecticides including deltamethrin, fenvalerate and azadirachtin were studied on functional response of Habrobracon hebetor Say to Ephestia kuehniella Zeller larvae. Seven host densities (2, 4, 8, 16, 32, 64 and 96) were used during a 24 h period. The resulting data were appropriately fit to Type II functional response models in all treatments: (1) control (0.0916 h−1; and T h = 0.2011 h); (2) deltamethrin (a = 0.0839 h−1; and T h = 0.3560 h); (3) fenvalerate (a = 0.0808 h−1 and T h = 0.3623 h); and (4) azadirachtin (a = 0.0900 h−1 and T h = 0.2042 h). Maximum theoretical parasitism rate (T/T h ) was 119.34 estimated for control wasps. There was no significant difference between the values of attack rates (a and a + D a ) in all treatments while the handling time was statistically affected in female wasps treated with fenvalerate. Our findings will be useful in safe application of these insecticides in pest management programmes.