A functional response model of a predator population foraging in a patchy habitat

1. Functional response models (e.g. Holling's disc equation) that do not take the spatial distributions of prey and predators into account are likely to produce biased estimates of predation rates. 2. To investigate the consequences of ignoring prey distribution and predator aggregation, a general analytical model of a predator population occupying a patchy environment with a single species of prey is developed. 3. The model includes the density and the spatial distribution of the prey population, the aggregative response of the predators and their mutual interference. 4. The model provides explicit solutions to a number of scenarios that can be independently combined: the prey has an even, random or clumped distribution, and the predators show a convex, sigmoid, linear or no aggregative response. 5. The model is parameterized with data from an acarine predator-prey system consisting of Phytoseiulus persimis and Tetranychus urticae inhabiting greenhouse cucumbers. 6. The model fits empirical data quite well and much better than if prey and predators were assumed to be evenly distributed among patches, or if the predators were distributed independently of the prey. 7. The analyses show that if the predators do not show an aggregative response it will always be an advantage to the prey to adopt a patchy distribution. On the other hand, if the predators are capable of responding to the distribution of prey, then it will be an advantage to the prey to be evenly distributed when its density is low and switch to a more patchy distribution when its density increases. The effect of mutual interference is negligible unless predator density is very high. 8. The model shows that prey patchiness and predator aggregation in combination can change the functional response at the population level from type II to type III, indicating that these factors may contribute to stabilization of predator-prey dynamics.     

Mixed encounters, limited perception and optimal foraging

This article demonstrates how perceptual constraints of predators and the possibility that predators encounter prey both sequentially (one prey type at a time) and simultaneously (two or more prey types at a time) may influence the predator attack decisions, diet composition and functional response of a behavioural predator-prey system. Individuals of a predator species are assumed to forage optimally on two prey types and to have exact knowledge of prey population numbers (or densities) only in a neighbourhood of their actual spatial location. The system characteristics are inspected by means of a discrete-time, discrete-space, individual-based model of the one-predator-two-prey interaction. Model predictions are compared with ones that have been obtained by assuming only sequential encounters of predators with prey and/or omniscient predators aware of prey population densities in the whole environment. It is shown that the zero-one prey choice rule, optimal for sequential encounters and omniscient predators, shifts to abruptly changing partial preferences for both prey types in the case of omniscient predators faced with both types of prey encounters. The latter, in turn, become gradually changing partial preferences when predator omniscience is considered only local.     

Modelling Holling type II functional response in deterministic and stochastic food chain models with mass conservation

The Rosenzweig-MacArthur predator-prey model is the building block in modeling food chain, food webs and ecosystems. There are a number of hidden assumptions involved in the derivation. For instance the prey population growth is logistic without predation but also with predation. In order to reveal these we will start with modelling a resource-predator-prey system in a closed spatially homogeneous environment. This allows us to keep track of the nutrient flow. With an instantaneous remineralisation of the products excreted in the environment by the populations and dead body mass there is conservation of mass. This allows for a model dimension reduction and yields the mass balance predator-prey model. When furthermore the searching and handling processes are much faster that the population changing rates, the trophic interaction is described by a Holling type II functional response, also assumed in the Rosenzweig-MacArthur model. The derivation uses an extended deterministic model with number of searching and handling predators as model variables where the ratio of the predator/prey body masses is used as a mechanistic time-scale parameter. This extended model is also used as a starting point for the derivation of a stochastic model. We will investigate the stochastic effects of random switching between searching and handling of the predators and predator dying. Prey growth by consumption of ambient resources is still deterministic and therefore the stochastic model is hybrid. The transient dynamics is studied by numerical Monte Carlo simulations and also the quasi-equilibrium distribution for the population quantities is calculated. The body mass of the prey individual is the scaling parameter in the stochastic model formulation. This allows for a quantification of the mean-field approximation criterion for the justification of replacement of the stochastic by a deterministic model.     

Stabilizing effects of spatial heterogeneity in predator-prey systems

A general predator is assumed to divide its hunting time between two sub-habitats with different prey species, spending a larger fraction (φ) of search time in an area as the relative prey abundance there increases. This always causes switching in the model, and changes a functional response from one that imposes a risk on the average prey that decreases with prey density in the direction of one that imposes an increasing risk. I discuss the conditions for a response that is density dependent, and those predatory attributes that make such a response more likely. Transit time between subhabitats always increases the density dependent effect, and is necessary for “system stability” in a Lotka-Volterra model with two prey species. Experiments have confirmed the model's basic assumption. General predators do not fit easily into classical predator-prey models of simple “closed” communities, and then the degree of density dependence of the functional response becomes a useful measure of a predator's short-term stabilizing effect on a prey species. The model demonstrates how spatial heterogeneity can be stabilizing.     

Use of prey hotspots by an avian predator: purposeful unpredictability?

The use of space by predators in relation to their prey is a poorly understood aspect of predator-prey interactions. Classic theory suggests that predators should focus their efforts on areas of abundant prey, that is, prey hotspots, whereas game-theoretical models of predator and prey movement suggest that the distribution of predators should match that of their prey's resources. If, however, prey are spatially anchored to one location and these prey have particularly strong antipredator responses that make them difficult to capture with frequent attacks, then predators may be forced to adopt alternative movement strategies to hunt behaviorally responsive prey. We examined the movement patterns of bird-eating sharp-shinned hawks (Accipiter striatus) in an attempt to shed light on hotspot use by predators. Our results suggest that these hawks do not focus on prey hotspots such as bird feeders but instead maintain much spatial and temporal unpredictability in their movements. Hawks seldom revisited the same area, and the few frequently used areas were revisited in a manner consistent with unpredictable returns, giving prey little additional information about risk.     

Effects of sterile insect releases on a population under predation or parasitism

A continuous-time differential equation model was constructed which describes the population dynamics of a predator prey system in which sterile prey are released in a program designed to eradicate or reduce the prey population. It was found that the dynamics of the system behave quite differently when predators are present. Two conditions were found which have differing implications for the control program. If the predators still exist when the wild prey population declines to extinction, then the SIRM is assisted by the predators, sometimes to a considereble extent. If the predators decline to extinction before the wild prey population goes extinct, then the predators may or may not assist the SIRM depending on the parameters of the system. If the predators do assist the SIRM, then a potentially dangerous situation exists in which an explosion of the prey population could occur after the predators go extinct. Predator polyphagy would probably minimize this danger of an explosion since it would stabilize the predator population.     

A comparison of foraging strategies in a patchy environment.

In this paper we compare foraging strategies that might be used by predators seeking prey in a patchy environment. The strategies differ in the extent to which predators aggregate in response to prey density. The approach to the comparison is suggested by the idea of evolutionarily stable strategies. A strategy is said to be evolutionarily stable if it cannot be invaded by another strategy. Thus we examine scenarios where a small number of individuals using one strategy are introduced into a situation where a large number of individuals using the other strategy are already present. However, our foraging models do not explicitly incorporate predator population dynamics, so we use net energy uptake as a surrogate for reproductive fitness. In cases where all of the patches visited by predators sustain prey populations, we find that for any pair of strategies one of them will have a higher net energy uptake than the other whether it is the resident or the introduced strain. However, which one is higher will typically depend on the total predator population, which is determined by the resident strain. If the predators leave prey densities high, the more aggregative strain will have the advantage. If the predators reduce prey densities to low levels the less aggregative strain will have the advantage. In cases where one strain of predators aggregates in response to prey density and the other does not, then there might be patches which do not contain prey but do contain (non-aggregating) predators. In those cases, there is the possibility that whichever strategy is used by the introduced strain will yield a higher energy uptake than that used by the resident strain. This suggests that if some patches are empty of prey then aggregative and non-aggregative strategies may be able to coexist.     

Predator-prey responses in an acarine system

Summary This study examines the responses of the predatory mite,Phytoseiulus persimilis, to the density and distribution of its prey,Tetranychus urticae. It is divided into three parts. Firstly, the functional responses of protonymph, deutonymph and adult females towards different prey stages are displayed. The great majority of the responses are of the type II form, and the variations in the values of attack ratea′ and handling timeT _h are discussed. Experiments are then described in which individual protonymph, deutonymph and adult female predators are presented with varying ratios of two prey age-classes (eggs and deutonymphs, and larvae and deutonymphs). Any observed preference for one of the prey stages is discussed in relation to the predicted preference on the basis of the separate functional response experiments. Finally, the response of different densities of adult female predators to a non-random distribution of deutonymph prey on bean leaflets is examined. The predators show a clear tendency to aggregate on the leaflets of high prey density, counteracted to some extent by interference increasing the probability of dispersal to other leaflets.     

Dynamics of predator and modular prey: effects of module consumption on stability of prey–predator system

In traditional models of predator–prey population dynamics, it is usually assumed that consumed prey are immediately removed from the population. However, in plant–herbivore interactions, damaged plants are generally alive after attacks by herbivores. This can result in successive or simultaneous attacks by multiple predators on a single prey item (here, the term prey is expanded to include plants). We constructed a mathematical model with two time scales, taking into account predation processes within a generation, considering post‐predation survival and the modularity of prey. We assumed that a prey item can be divided into modules and that it can be fed on by multiple predators or parasitized by multiple parasites at the same time. The model includes two essential factors: the modularity of prey for predators (n) and the detaching/attaching ratio of predators to prey (ε). Based on the formulae, we revealed a general property of realistic dynamics in plant–herbivore and host–parasite interactions. The analysis showed that the model could be approximated by models with the type I, type II or Beddington–DeAngelis functional responses by taking appropriate limits to the situations. When modularity is low or the detaching/attaching ratio is high, population dynamics tend to be stabilized. These stabilizing effects may be related to interference competition among predator individuals or increases in free prey modules and free predator individuals. In the limit of high modularity, the ratio of the attached prey modules to the total prey modules becomes negligible and the dynamics tend to be destabilized. However, if quantity and quality of prey modules are negatively correlated, the equilibrium is likely to be stabilized at high modularity as long as it remains feasible. These results suggest that considering post‐predation survival and modularity of prey is crucial to understand predator–prey interactions.     

Impacts of predation on dynamics of age-structured prey: Allee effects and multi-stability

With a series of mathematical models, we explore impacts of predation on a prey population structured into two age classes, juveniles and adults, assuming generalist, age-specific predators. Predation on any age class is either absent, or represented by types II or III functional responses, in various combinations. We look for Allee effects or more generally for multiple stable steady states in the prey population. One of our key findings is the occurrence of a predator pit (low-density ??refuge?? state of prey induced by predation; the chance of escaping predation thus increases both below and above an intermediate prey density) when only one age class is consumed and predators use a type II functional response ??this scenario is known to occur for an unstructured prey consumed via a type III functional response and can never occur for an unstructured prey consumed via a type II one. In the case where both age classes are consumed by type II generalist predators, an Allee effect occurs frequently, but some parameters give also rise to a predator pit and even three stable equilibria (one extinction equilibrium and two positive ones??Allee effect and predator pit combined). Multiple positive stable equilibria are common if one age class is consumed via a type II functional response and the other via a type III functional response??here, in addition to the behaviours mentioned above one may even observe three stable positive equilibria????double?? predator pit. Some of these results are discussed from the perspective of population management.     

Integrating disparate datasets to model the functional response of a marine predator: A case study of harbour porpoises in the southern North Sea

1. Quantifying consumption and prey choice for marine predator species is key to understanding their interaction with prey species, fisheries, and the ecosystem as a whole. However, parameterizing a functional response for large predators can be challenging because of the difficulty in obtaining the required data on predator diet and on the availability of multiple prey species.
2. This study modeled a multi‐species functional response (MSFR) to describe the relationship between consumption by harbour porpoises (Phocoena phocoena) and the availability of multiple prey species in the southern North Sea. Bayesian methodology was employed to estimate MSFR parameters and to incorporate uncertainties in diet and prey availability estimates. Prey consumption was estimated from stomach content data from stranded harbour porpoises. Prey availability to harbour porpoises was estimated based on the spatial overlap between prey distributions, estimated from fish survey data, and porpoise foraging range in the days prior to stranding predicted from telemetry data.
3. Results indicated a preference for sandeels in the study area. Prey switching behavior (change in preference dependent on prey abundance) was confirmed by the favored type III functional response model. Variation in the size of the foraging range (estimated area where harbour porpoises could have foraged prior to stranding) did not alter the overall pattern of the results or conclusions.
4. Integrating datasets on prey consumption from strandings, predator foraging distribution using telemetry, and prey availability from fish surveys into the modeling approach provides a methodological framework that may be appropriate for fitting MSFRs for other predators.

     

Modeling changes in predator functional response to prey across spatial scales

Extrapolation of predator functional responses from laboratory observations to the field is often necessary to predict predation rates and predator-prey dynamics at spatial and temporal scales that are difficult to observe directly. We use a spatially explicit individual-based model to explore mechanisms behind changes in functional responses when the scale of observation is increased. Model parameters were estimated from a predator-prey system consisting of the predator Delphastus catalinae (Coleoptera: Coccinellidae) and Bemisia tabaci biotype B (Hemiptera: Aleyrodidae) on tomato plants. The model explicitly incorporates prey and predator distributions within single plants, the search behavior of predators within plants, and the functional response to prey at the smallest scale of interaction (within leaflets) observed in the laboratory. Validation revealed that the model is useful in scaling up from laboratory observations to predation in whole tomato plants of varying sizes. Comparing predicted predation at the leaflet scale, as observed in laboratory experiments, with predicted predation on whole plants revealed that the predator functional response switches from type II within leaflets to type III within whole plants. We found that the magnitude of predation rates and the type of functional response at the whole plant scale are modulated by (1) the degree of alignment between predator and prey distributions and (2) predator foraging behavior, particularly the effect of area-concentrated search within plants when prey population density is relatively low. The experimental and modeling techniques we present could be applied to other systems in which active predators prey upon sessile or slow-moving species.     

The Effects of Prey Patchiness, Predator Aggregation, and Mutual Interference on the Functional Response of Phytoseiulus persimilis Feeding on Tetranychus urticae (Acari: Phytoseiidae, Tetranychidae)

The spatial distributions of two-spotted spider mites Tetranychus urticae and their natural enemy, the phytoseiid predator Phytoseiulus persimilis, were studied on six full-grown cucumber plants. Both mite species were very patchily distributed and P. persimilis tended to aggregate on leaves with abundant prey. The effects of non-homogenous distributions and degree of spatial overlap between prey and predators on the per capita predation rate were studied by means of a stage-specific predation model that averages the predation rates over all the local populations inhabiting the individual leaves. The empirical predation rates were compared with predictions assuming random predator search and/or an even distribution of prey. The analysis clearly shows that the ability of the predators to search non-randomly increases their predation rate. On the other hand, the prey may gain if it adopts a more even distribution when its density is low and a more patchy distribution when density increases. Mutual interference between searching predators reduces the predation rate, but the effect is negligible. The stage-specific functional response model was compared with two simpler models without explicit stage structure. Both unstructured models yielded predictions that were quite similar to those of the stage-structured model.     

A mechanistic model for partial preferences

Classic prey optimal foraging model assumes that individual predators are globally omniscient; that is, they have exact knowledge of prey population densities in the environment. This study examines a spatially explicit individual-based model of a one-predator two-prey system where individual predators are assumed to be omniscient only locally, i.e., to know prey population densities only in the range of their perception. Due to local variations in prey numbers, the probability of acceptance of less profitable prey shifts from the zero-one rule to a gradually decreasing function, for which an explicit formula is derived, giving way to partial preferences. A corresponding predator functional response to more profitable prey is shown to have a sigmoid-like form.     

The allometry of prey preferences

The distribution of weak and strong non-linear feeding interactions (i.e., functional responses) across the links of complex food webs is critically important for their stability. While empirical advances have unravelled constraints on single-prey functional responses, their validity in the context of complex food webs where most predators have multiple prey remain uncertain. In this study, we present conceptual evidence for the invalidity of strictly density-dependent consumption as the null model in multi-prey experiments. Instead, we employ two-prey functional responses parameterised with allometric scaling relationships of the functional response parameters that were derived from a previous single-prey functional response study as novel null models. Our experiments included predators of different sizes from two taxonomical groups (wolf spiders and ground beetles) simultaneously preying on one small and one large prey species. We define compliance with the null model predictions (based on two independent single-prey functional responses) as passive preferences or passive switching, and deviations from the null model as active preferences or active switching. Our results indicate active and passive preferences for the larger prey by predators that are at least twice the size of the larger prey. Moreover, our approach revealed that active preferences increased significantly with the predator-prey body-mass ratio. Together with prior allometric scaling relationships of functional response parameters, this preference allometry may allow estimating the distribution of functional response parameters across the myriads of interactions in natural ecosystems.     

Body masses,functional responses and predator–prey stability

The stability of ecological communities depends strongly on quantitative characteristics of population interactions (type‐II vs. type‐III functional responses) and the distribution of body masses across species. Until now, these two aspects have almost exclusively been treated separately leaving a substantial gap in our general understanding of food webs. We analysed a large data set of arthropod feeding rates and found that all functional‐response parameters depend on the body masses of predator and prey. Thus, we propose generalised functional responses which predict gradual shifts from type‐II predation of small predators on equally sized prey to type‐III functional‐responses of large predators on small prey. Models including these generalised functional responses predict population dynamics and persistence only depending on predator and prey body masses, and we show that these predictions are strongly supported by empirical data on forest soil food webs. These results help unravelling systematic relationships between quantitative population interactions and large‐scale community patterns.     

Salamander evolution across a latitudinal cline in gape-limited predation risk

General predictions of community dynamics require that insights derived from local habitats can be scaled up to explain phenomena across geographic scales. Across these larger spatial extents, adaptation can play an increasing role in determining the outcome of species interactions. If local adaptation is common, then our ability to generalize measures of species interaction strength across communities will be limited without an additional understanding of the genetic variation underlying interaction traits. In the context of predator–prey interactions, prey individuals commonly are expected to reduce risky foraging behaviors and subsequent growth under predation threat. However, rapid growth into a large body size can defend against gape-limited predators, creating a tradeoff between increased predation risk due to elevated foraging activity and decreased predation risk due to large size. Here I combine field observations, natural selection experiments, and common garden assays to understand potential adaptations of spotted salamander Ambystoma maculatum larvae to gape-limited and gape-unconstrained predators. Field observations and natural selection trials suggested antagonistic selection on prey body size among ponds dominated by gape-limited predator salamanders A. opacum and gape-unconstrained beetle larvae Dytiscus . In common garden experiments, prey from sites with high gape-limited predation risk grew larger than those from other sites, suggesting the evolution of rapid growth into a prey size refuge. Larvae from all sites grew to a large size when exposed to the gape-limited N. viridescens predator's kairomones. Hence, induced rapid growth into a size refuge may be an adaptive response to gape-limited predation risk. Results point to an important role for cross-community generalizations based on functional classifications of predators by their gape constraints and inter-site genetic variation in prey growth rates and behaviors.     

Bistability and limit cycles in generalist predator–prey dynamics

Since generalist predators feed on a variety of prey species they tend to persist in an ecosystem even if one particular prey species is absent. Predation by generalist predators is typically characterized by a sigmoidal functional response, so that predation pressure for a given prey species is small when the density of that prey is low. Many mathematical models have included a sigmoidal functional response into predator–prey equations and found the dynamics to be more stable than for a Holling type II functional response. However, almost none of these models considers alternative food sources for the generalist predator. In particular, in these models, the generalist predator goes extinct in the absence of the one focal prey. We model the dynamics of a generalist predator with a sigmoidal functional response on one dynamic prey and fixed alternative food source. We find that the system can exhibit up to six steady states, bistability, limit cycles and several global bifurcations.     

Functional feeding responses of piscivorous fishes from the northeast US continental shelf

The functional feeding response forms of piscivorous fishes used in multispecies and ecosystem modeling have been questioned because they were mostly conjectural or solely based on laboratory studies. Here, we investigate the functional feeding response of seven species of piscivorous fishes on four species of their prey from the northeast US continental shelf using field data that spans 30 years. Our study confirmed that Holling’s types II and III functional responses are the most common functional responses for piscivorous fishes in this region. However, our analyses also revealed that differences exist between piscivorous fishes’ functional responses, and, therefore, combining functional responses of piscivores is probably not appropriate in multispecies and ecosystem modeling. In the absence of specific predator–prey functional responses, we suggest that, for cruising, actively attacking predators, a type II functional response is slightly preferable; for a sedentary, ambush predator, a type III functional response is slightly preferable; at low prey densities for a generic fish predator, a type III functional response should be used; and at moderate to high prey densities, either should work sufficiently. Because we have shown that the functional response of a particular predator to individual prey species varies, these relationships must be further evaluated as we continue to develop and employ multispecies and ecosystem modeling.