From pattern to process: identifying predator–prey models from time-series data

Fitting nonlinear models to time-series is a technique of increasing importance in population ecology. In this article, we apply it to assess the importance of predator dependence in the predation process by comparing two alternative models of equal complexity (one with and one without predator dependence) to predator–prey time-series. Stochasticities in such data come from both observation error and process error. We consider how these errors must be taken into account in the fitting process, and we develop eight different model selection criteria. Applying these criteria to laboratory data on simple protozoan and arthropod predator–prey systems shows that little predator dependence is present, with one interesting exception. Field data are more ambiguous (either selection depends on the particular criterion or no significant differences can be detected), and we show that both models fit reasonably well. We conclude that, within our modeling framework, predator dependence is in general insignificant in simple systems in homogeneous environments. Relatively complex systems show significant predator dependence more often than simple ones but the data are also often inconclusive. The analysis of such systems should rely on several models to detect predictions that are sensitive to predator dependence and to direct further research if necessary. Received: July 13, 2000 / Accepted: September 25, 2001     

Slight phenotypic variation in predators and prey causes complex predator-prey oscillations

Predator-prey oscillations are expected to show a 1/4-phase lag between predator and prey. However, observed dynamics of natural or experimental predator-prey systems are often more complex. A striking but hardly studied example are sudden interruptions of classic 1/4-lag cycles with periods of antiphase oscillations, or periods without any regular predator-prey oscillations. These interruptions occur for a limited time before the system reverts to regular 1/4-lag oscillations, thus yielding intermittent cycles. Reasons for this behaviour are often difficult to reveal in experimental systems. Here we test the hypothesis that such complex dynamical behaviour may result from minor trait variation and trait adaptation in both the prey and predator, causing recurrent small changes in attack rates that may be hard to capture by empirical measurements. Using a model structure where the degree of trait variation in the predator can be explicitly controlled, we show that a very limited amount of adaptation resulting in 10–15% temporal variation in attack rates is already sufficient to generate these intermittent dynamics. Such minor variation may be present in experimental predator-prey systems, and may explain disruptions in regular 1/4-lag oscillations.     

Consequences of behavioral dynamics for the population dynamics of predator-prey systems with switching

This article explores how different mechanisms governing the rate of change of the predators preference alter the dynamics of predator-prey systems in which the predator exhibits positive frequency-dependent predation. The models assume that individuals of the predator species adaptively adjust a trait that determines their relative capture rates of each of two prey species. The resulting switching behavior does not instantaneously attain the optimum for current prey densities, but instead lags behind it. Several mechanisms producing such lags are discussed and modeled. In all cases examined, our question is whether a realistic behavioral lag can significantly change the dynamics of the system relative to an analogous case in which the predators switching is effectively instantaneous. We also explore whether increasing the rate parameters of dynamic models of behavior results in convergence to the population dynamics of analogous models with instantaneous switching, and whether different behavioral models produce similar population dynamics. The analysis concentrates on systems that undergo endogenously generated predator-prey cycles in the absence of switching behavior. The average densities and the nature of indirect interactions are often sensitive to the rate of behavioral change, and are often qualitatively different for different classes of behavioral models. Dynamics and average densities can be very sensitive to small changes in parameters of either the prey growth or predator switching functions. These differences suggest that an understanding of switching in natural systems will require research into the behavioral mechanisms that govern lags in the response of predator preference to changes in prey density.     

Evolutionary responses of foraging-related traits in unstable predator–prey systems

The evolutionary responses of predators to prey and of prey to predators are analysed using models for the dynamics of a quantitative trait that determines the capture rate of prey by an average searching predator. Unlike previous investigations, the analysis centres on models and/or parameter values for which the two-species equilibrium is locally unstable. The instability in some models is driven by the predators non-linear functional response to prey; in other models, the cycles are a direct consequence of evolutionary response to selection acting on the trait. When the values of predator and prey traits combine multiplicatively to determine the capture rate, the predators trait shows only a transient response to changes in the preys trait in stable systems. However, when the population densities exhibit sustained oscillations, predators often evolve an increased long-term mean capture rate in response to an increased prey escape ability. Under the multiplicative model, prey in stable systems always evolve increased escape ability in response to an increased predator capture a     

Density-dependent prey behaviours and mutable predator foraging modes induce Allee effects and over-prediction of prey mortality rates

1. Predator–prey models are often used to represent consumptive interactions between species but, typically, are derived using simple experimental systems with little plasticity in prey or predator behaviours. However, many prey and predators exhibit a broad suite of behaviours. Here, we experimentally tested the effect of density-dependent prey and predator behaviours on per capita relative mortality rates using Florida bass (Micropterus floridanus) consuming juvenile Bluegill (Lepomis macrochirus).
2. Experimental ponds were stocked with a factorial design of low, medium, and high prey and predator densities. Prey mortality, prey–predator behaviours, and predator stomach contents were recorded over or after 7 days. We assumed the mortality dynamics followed foraging arena theory. This pathologically flexible predator–prey model separates prey into invulnerable and vulnerable pools where predators can consume prey in the latter. As this approach can represent classic Lotka–Volterra and ratio-dependent dynamics, we fit a foraging arena predator–prey model to the number of surviving prey.
3. We found that prey exhibited density-dependent prey behaviours, hiding at low densities, shoaling at medium densities, and using a provided refuge at high densities. Predators exhibited ratio-dependent behaviours, using an ambush foraging mode when one predator was present, hiding in the shadows at low prey–high predator densities, and shoaling at medium and high prey–high predator densities. The foraging arena model predicted the mortality rates well until the high prey–high predator treatment where group vigilance prey behaviours occurred and predators probably interfered with one another resulting in the model predicting higher mortality than observed.
4. This is concerning given the ubiquity of predator–prey models in ecology and natural resource management. Furthermore, as Allee effects engender instability in population regulation, it could lead to inaccurate predictions of conservation status, population rebuilding or harvest rates.

     

A synthetic Escherichia coli predator–prey ecosystem

We have constructed a synthetic ecosystem consisting of two Escherichia coli populations, which communicate bi‐directionally through quorum sensing and regulate each other's gene expression and survival via engineered gene circuits. Our synthetic ecosystem resembles canonical predator–prey systems in terms of logic and dynamics. The predator cells kill the prey by inducing expression of a killer protein in the prey, while the prey rescue the predators by eliciting expression of an antidote protein in the predator. Extinction, coexistence and oscillatory dynamics of the predator and prey populations are possible depending on the operating conditions as experimentally validated by long‐term culturing of the system in microchemostats. A simple mathematical model is developed to capture these system dynamics. Coherent interplay between experiments and mathematical analysis enables exploration of the dynamics of interacting populations in a predictable manner.     

Habitat choice in predator-prey systems: spatial instability due to interacting adaptive movements

The role of habitat choice behavior in the dynamics of predator-prey systems is explored using simple mathematical models. The models assume a three-species food chain in which each population is distributed across two or more habitats. The predator and prey adjust their locations dynamically to maximize individual per capita growth, while the prey's resource has a low rate of random movement. The two consumer species have Type II functional responses. For many parameter sets, the populations cycle, with predator and prey "chasing" each other back and forth between habitats. The cycles are driven by the aggregation of prey, which is advantageous because the predator's saturating functional response induces a short-term positive density dependence in prey fitness. The advantage of aggregation in a patch is only temporary because resources are depleted and predators move to or reproduce faster in the habitat with the largest number of prey, perpetuating the cycle. Such spatial cycling can stabilize population densities and qualitatively change the responses of population densities to environmental perturbations. These models show that the coupled processes of moving to habitats with higher fitness in predator and prey may often fail to produce ideal free distributions across habitats.     

Optimal-foraging predator favors commensalistic Batesian mimicry

Background

Mimicry, in which one prey species (the Mimic) imitates the aposematic signals of another prey (the Model) to deceive their predators, has attracted the general interest of evolutionary biologists. Predator psychology, especially how the predator learns and forgets, has recently been recognized as an important factor in a predator–prey system. This idea is supported by both theoretical and experimental evidence, but is also the source of a good deal of controversy because of its novel prediction that in a Model/Mimic relationship even a moderately unpalatable Mimic increases the risk of the Model (quasi-Batesian mimicry).

Methodology/Principal Findings

We developed a psychology-based Monte Carlo model simulation of mimicry that incorporates a “Pavlovian” predator that practices an optimal foraging strategy, and examined how various ecological and psychological factors affect the relationships between a Model prey species and its Mimic. The behavior of the predator in our model is consistent with that reported by experimental studies, but our simulation''s predictions differed markedly from those of previous models of mimicry because a more abundant Mimic did not increase the predation risk of the Model when alternative prey were abundant. Moreover, a quasi-Batesian relationship emerges only when no or very few alternative prey items were available. Therefore, the availability of alternative prey rather than the precise method of predator learning critically determines the relationship between Model and Mimic. Moreover, the predation risk to the Model and Mimic is determined by the absolute density of the Model rather than by its density relative to that of the Mimic.

Conclusions/Significance

Although these predictions are counterintuitive, they can explain various kinds of data that have been offered in support of competitive theories. Our model results suggest that to understand mimicry in nature it is important to consider the likely presence of alternative prey and the possibility that predation pressure is not constant.     

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.     

Influence of predator‐specific defense adaptation on intraguild predation

The persistence of intraguild predation (IGP), the prey–predator interaction between competing species, is puzzling because simple IGP models readily predict species extinction. In this study, we explored a mathematical model incorporating predator‐specific defense adaptation of basal prey against intraguild prey and intraguild predator. The model explicitly described the dynamics of the defense effort against each predator under the assumption that anti‐predator defense was associated with reducing effort allocated to reproduction. The model predicted that defense adaptation (i.e. the ability to reallocate defense effort) would facilitate coexistence, particularly when system productivity is high; at low productivity, coexistence would be facilitated or inhibited depending on initial effort allocation prior to defense adaptation. In addition, we found that three‐species dynamics became more stable at higher adaptation rates. The results suggest that common behavioral changes, such as predator‐specific defense adaptation, have significant implications for the community structure and dynamics of IGP systems.     

Stochastic predation exposes prey to predator pits and local extinction

Understanding how predators affect prey populations is a fundamental goal for ecologists and wildlife managers. A well-known example of regulation by predators is the predator pit, where two alternative stable states exist and prey can be held at a low density equilibrium by predation if they are unable to pass the threshold needed to attain a high density equilibrium. While empirical evidence for predator pits exists, deterministic models of predator–prey dynamics with realistic parameters suggest they should not occur in these systems. Because stochasticity can fundamentally change the dynamics of deterministic models, we investigated if incorporating stochasticity in predation rates would change the dynamics of deterministic models and allow predator pits to emerge. Based on realistic parameters from an elk–wolf system, we found predator pits were predicted only when stochasticity was included in the model. Predator pits emerged in systems with highly stochastic predation and high carrying capacities, but as carrying capacity decreased, low density equilibria with a high likelihood of extinction became more prevalent. We found that incorporating stochasticity is essential to fully understand alternative stable states in ecological systems, and due to the interaction between top–down and bottom–up effects on prey populations, habitat management and predator control could help prey to be resilient to predation stochasticity.     

Transient dynamics limit the effectiveness of keystone predation in bringing about coexistence

We analyze the transient dynamics of simple models of keystone predation, in which a predator preferentially consumes the dominant of two (or more) competing prey species. We show that coexistence is unlikely in many systems characterized both by successful invasion of either prey species into the food web that lacks it and by a stable equilibrium with high densities of all species. Invasion of the predator-resistant consumer species often causes the resident, more vulnerable prey to crash to such low densities that extinction would occur for many realistic population sizes. Subsequent transient cycles may entail very low densities of the predator or of the initially successful invader, which may also preclude coexistence of finite populations. Factors causing particularly low minimum densities during the transient cycles include biotic limiting resources for the prey, limited resource partitioning between the prey, a highly efficient predator with relatively slow dynamics, and a vulnerable prey whose population dynamics are rapid relative to the less vulnerable prey. Under these conditions, coexistence of competing prey via keystone predation often requires that the prey's competitive or antipredator characteristics fall within very narrow ranges. Similar transient crashes are likely to occur in other food webs and food web models.     

Prey persistence and abundance in systems with intraguild predation and type-2 functional responses

The apparent prevalence of intraguild predation in productive environments has been regarded as puzzling because some simple models suggest that the intraguild prey species is often either reduced in abundance or driven extinct at high resource productivity. While various theoretical mechanisms that avoid this prediction have been uncovered, they have often been viewed as being narrowly applicable. This article examines the fate of the intraguild prey in models in which consumer species may have type-2 functional responses; these are usually characterized by sustained fluctuations in population density at high enough resource productivities. The models also include adaptive, but imperfect diet choice by the top predator. We concentrate on two situations: (1) the prey exhibits less saturation in its functional response to the resource than does the predator and (2) the predator is unable to persist on the basal resource alone. The reasons given by previous studies for discounting these cases are re-examined. The present analysis shows that prey abundance often increases with increasing productivity in both cases, as does the range of prey parameters that allows prey persistence. It is also possible for the prey to coexist with the predator in spite of having a larger equilibrium requirement for the resource. Different assumptions about the dynamics of diet choice can have a large impact on population responses to enrichment. We argue that the persistence and/or increase in abundance of intraguild prey at higher productivity should not be regarded as puzzling because observations are consistent with a range of theoretical models that reflect commonly observed mechanisms.     

Dynamics of a intraguild predation model with generalist or specialist predator

Intraguild predation (IGP) is a combination of competition and predation which is the most basic system in food webs that contains three species where two species that are involved in a predator/prey relationship are also competing for a shared resource or prey. We formulate two intraguild predation (IGP: resource, IG prey and IG predator) models: one has generalist predator while the other one has specialist predator. Both models have Holling-Type I functional response between resource-IG prey and resource-IG predator; Holling-Type III functional response between IG prey and IG predator. We provide sufficient conditions of the persistence and extinction of all possible scenarios for these two models, which give us a complete picture on their global dynamics. In addition, we show that both IGP models can have multiple interior equilibria under certain parameters range. These analytical results indicate that IGP model with generalist predator has “top down” regulation by comparing to IGP model with specialist predator. Our analysis and numerical simulations suggest that: (1) Both IGP models can have multiple attractors with complicated dynamical patterns; (2) Only IGP model with specialist predator can have both boundary attractor and interior attractor, i.e., whether the system has the extinction of one species or the coexistence of three species depending on initial conditions; (3) IGP model with generalist predator is prone to have coexistence of three species.     

Dynamic versus instantaneous models of diet choice

We investigate the dynamics of a series of two-prey-one-predator models in which the predator exhibits adaptive diet choice based on the different energy contents and/or handling times of the two prey species. The predator is efficient at exploiting its prey and has a saturating functional response; these two features combine to produce sustained population cycles over a wide range of parameter values. Two types of models of behavioral change are compared. In one class of models ("instantaneous choice"), the probability of acceptance of the poorer prey by the predator instantaneously approximates the optimal choice, given current prey densities. In the second class of models ("dynamic choice"), the probability of acceptance of the poorer prey is a dynamic variable, which begins to change in an adaptive direction when prey densities change but which requires a finite amount of time to approach the new optimal behavior. The two types of models frequently predict qualitatively different population dynamics of the three-species system, with chaotic dynamics and complex cycles being a common outcome only in the dynamic choice models. In dynamic choice models, factors that reduce the rate of behavioral change when the probability of accepting the poorer prey approaches extreme values often produce complex population dynamics. Instantaneous and dynamic models often predict different average population densities and different indirect interactions between prey species. Alternative dynamic models of behavior are analyzed and suggest, first, that instantaneous choice models may be good approximations in some circumstances and, second, that different types of dynamic choice models often lead to significantly different population dynamics. The results suggest possible behavioral mechanisms leading to complex population dynamics and highlight the need for more empirical study of the dynamics of behavioral change.     

The stability of ecosystems: A brief overview of the paradox of enrichment

In theory, enrichment of resource in a predator-prey model leads to destabilization of the system,thereby collapsing the trophic interaction,a phenomenon referred to as "the paradox of enrichment". After it was first pro posed by Rosenzweig (1971), a number of subsequent studies were carried out on this dilemma over many decades. In this article, we review these theoretical and experimental works and give a brief overview of the proposed solutions to the paradox. The mechanisms that have been discussed are modifications of simple predator -prey models in the presence of prey that is inedible, invulnerable, unpalatable and toxic. Another class of mechanisms includes an incorporation of a ratio-dependent functional form,inducible defence of prey and density-dependent mortality of the predator. Moreover, we find a third set of explanations based on complex population dynamics including chaos in space and time. We conclude that,although any one of the various mechanisms proposed so far might potentially prevent destabilization of the predator-prey dynamics following enrichment, in nature different mechanisms may combine to cause stability, even when a system is enriched. The exact mechanisms,which may differ among systems,need to be disentangled through extensive field studies and laboratory experiments coupled with realistic theoretical models.     

Impacts of Biotic Resource Enrichment on a Predator–Prey Population

The environmental carrying capacity is usually assumed to be fixed quantity in the classical predator–prey population growth models. However, this assumption is not realistic as the environment generally varies with time. In a bid for greater realism, functional forms of carrying capacities have been widely applied to describe varying environments. Modelling carrying capacity as a state variable serves as another approach to capture the dynamical behavior between population and its environment. The proposed modified predator–prey model is based on the ratio-dependent models that have been utilized in the study of food chains. Using a simple non-linear system, the proposed model can be linked to an intra-guild predation model in which predator and prey share the same resource. Distinct from other models, we formulate the carrying capacity proportional to a biotic resource and both predator and prey species can directly alter the amount of resource available by interacting with it. Bifurcation and numerical analyses are presented to illustrate the system’s dynamical behavior. Taking the enrichment parameter of the resource as the bifurcation parameter, a Hopf bifurcation is found for some parameter ranges, which generate solutions that posses limit cycle behavior.     

A spatial theory for emergent multiple predator–prey interactions in food webs

Predator–prey interaction is inherently spatial because animals move through landscapes to search for and consume food resources and to avoid being consumed by other species. The spatial nature of species interactions necessitates integrating spatial processes into food web theory and evaluating how predators combine to impact their prey. Here, we present a spatial modeling approach that examines emergent multiple predator effects on prey within landscapes. The modeling is inspired by the habitat domain concept derived from empirical synthesis of spatial movement and interactions studies. Because these principles are motivated by synthesis of short‐term experiments, it remains uncertain whether spatial contingency principles hold in dynamical systems. We address this uncertainty by formulating dynamical systems models, guided by core habitat domain principles, to examine long‐term multiple predator–prey spatial dynamics. To describe habitat domains, we use classical niche concepts describing resource utilization distributions, and assume species interactions emerge from the degree of overlap between species. The analytical results generally align with those from empirical synthesis and present a theoretical framework capable of demonstrating multiple predator effects that does not depend on the small spatial or temporal scales typical of mesocosm experiments, and help bridge between empirical experiments and long‐term dynamics in natural systems.     

Toward the Quantification of Predation with Predator Gut Immunoassays: A New Approach Integrating Functional Response Behavior

Immunological methods have been widely used to identify key predator species and qualitatively evaluate predation of target prey. However, despite the quantitative nature of many immunoassays, the translation to number of prey attacked has been problematic because of the many factors that confound interpretation of the strength of the immunoassay response. We developed a new predation model that couples the proportion of predators positive for prey remains determined by enzyme-linked immunosorbent assay (ELISA), predator density, and predator functional response to prey density for estimating total prey attacked. We used single cotton plant arenas in the greenhouse to develop functional response models for two generalist predators, Geocoris punctipes (Say) and Orius insidiosus (Say), preying on Pectinophora gossypiella (Saunders) eggs. The model was validated and compared with other immunologically based predation models in multiple plant/multiple predator arenas. Our predation model was relatively accurate in predicting the total number of prey attacked by both predator species and was a significant improvement over previous models that rely on simple assumptions regarding predator attack rates. The model also improves the predictive capacity of the functional response model alone by correcting for the number of predators actually consuming prey. Sensitivity analyses indicated that model performance was most sensitive to accurate measurement of input variables such as temperature and the proportion of individuals positive for prey antigens by ELISA and less sensitive to changes in estimates of prey density. Accurate estimation of the functional response parameters is also important, especially for the behavioral parameter defining the decline in plant leaf area searched with increases in prey density. Limitations of the model and application to the field are discussed.     

The evolution of traits affecting resource acquisition and predator vulnerability: character displacement under real and apparent competition

This article investigates some simple models of the evolutionary interaction between two prey species that share a common resource and a common predator. Each prey species is characterized by a trait that determines both the rate of resource capture and vulnerability to a predator. In a simple model of a three-species food chain, such traits usually increase in response to an imposed reduction in resource density. When the per capita growth rates of each of two prey species depend linearly on resource density, such traits will change in opposite directions when the two prey come into sympatry. In addition, the ratio of the effect of the predator on prey fitness to the effect of the resource on prey fitness will diverge from the corresponding ratio in a second prey species when those species coexist in sympatry. These simple predictions need not hold under several alternative assumptions, which may be more common in biological systems. Parallel changes in sympatry may occur if the relationship between resource consumption and prey growth is nonlinear, if the prey species have partial overlap in the set of resources used or in the set of predators that consume them, or if prey experience direct intraspecific competition. The responses to a second prey can also differ significantly from those predicted by the simplest model if separate traits affect vulnerability to predators and resource acquisition rate. It is important to determine whether examples of character displacement previously interpreted as responses to competition for resources might also reflect responses to altered predation risks in sympatry.