Identifying predator-prey processes from time-series

The functional response is a key element in predator-prey models as well as in food chains and food webs. Classical models consider it as a function of prey abundance only. However, many mechanisms can lead to predator dependence, and there is increasing evidence for the importance of this dependence. Identification of the mathematical form of the functional response from real data is therefore a challenging task. In this paper we apply model-fitting to test if typical ecological predator-prey time series data, which contain both observation error and process error, can give some information about the form of the functional response. Working with artificial data (for which the functional response is known) we will show that with moderate noise levels, identification of the model that generated the data is possible. However, the noise levels prevailing in real ecological time-series can give rise to wrong identifications. We will also discuss the quality of parameter estimation by fitting differential equations to such time-series.     

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     

Reconsidering the importance of the past in predator–prey models: both numerical and functional responses depend on delayed prey densities

We propose that delayed predator–prey models may provide superficially acceptable predictions for spurious reasons. Through experimentation and modelling, we offer a new approach: using a model experimental predator–prey system (the ciliates Didinium and Paramecium), we determine the influence of past-prey abundance at a fixed delay (approx. one generation) on both functional and numerical responses (i.e. the influence of present : past-prey abundance on ingestion and growth, respectively). We reveal a nonlinear influence of past-prey abundance on both responses, with the two responding differently. Including these responses in a model indicated that delay in the numerical response drives population oscillations, supporting the accepted (but untested) notion that reproduction, not feeding, is highly dependent on the past. We next indicate how delays impact short- and long-term population dynamics. Critically, we show that although superficially the standard (parsimonious) approach to modelling can reasonably fit independently obtained time-series data, it does so by relying on biologically unrealistic parameter values. By contrast, including our fully parametrized delayed density dependence provides a better fit, offering insights into underlying mechanisms. We therefore present a new approach to explore time-series data and a revised framework for further theoretical studies.     

Rapid prey evolution and the dynamics of two-predator food webs

Traits affecting ecological interactions can evolve on the same time scale as population and community dynamics, creating the potential for feedbacks between evolutionary and ecological dynamics. Theory and experiments have shown in particular that rapid evolution of traits conferring defense against predation can radically change the qualitative dynamics of a predator–prey food chain. Here, we ask whether such dramatic effects are likely to be seen in more complex food webs having two predators rather than one, or whether the greater complexity of the ecological interactions will mask any potential impacts of rapid evolution. If one prey genotype can be well-defended against both predators, the dynamics are like those of a predator–prey food chain. But if defense traits are predator-specific and incompatible, so that each genotype is vulnerable to attack by at least one predator, then rapid evolution produces distinctive behaviors at the population level: population typically oscillate in ways very different from either the food chain or a two-predator food web without rapid prey evolution. When many prey genotypes coexist, chaotic dynamics become likely. The effects of rapid evolution can still be detected by analyzing relationships between prey abundance and predator population growth rates using methods from functional data analysis.     

Simple testing procedures for the Holling type II model

In this paper, we consider predator–prey data that can be viewed as solutions to a planar system of ordinary differential equations (ODE) observed with random error. The ODE system admits a limit cycle, while the random error is supposed to act additively in the log-scale. One of the oldest such systems is Holling’s type II model. In spite of its simplicity, it is still very popular in data analyses, although more sophisticated models have been introduced in the literature. We propose a simple way of deciding whether a set of predator–prey pairs is indicative or not of a departure from this basic model by exploiting the geometric properties of the solution in the phase plane. To illustrate our method, we use simulated and real data.     

Predicting and testing functional responses: An example from a tardigrade–nematode system

Numerous studies have empirically measured consumer functional responses or theoretically developed response models, but whether these models can quantitatively predict observed data has hardly been tested. We perform such a test for the terrestrial predator–prey system Macrobiotus richtersi (Tardigrada)–Acrobeloides nanus (Nematoda). For two different size classes of A. nanus, we report a functional response as measured in the laboratory and quantitatively compare it to predictions of three models with different degrees of complexity: (1) the disc equation which does not include satiation effects; (2) the steady-state satiation (SSS) equation which assumes a constant level of predator satiation; and (3) the satiation model which accounts for prey depletion and increasing predator satiation over the course of the experiments. We parameterized these models with data that were measured independently of the functional response experiments. In both prey-size classes, the predictions of the satiation model matched the observations best, and the match came close to that of logistic regressions fitted to the observations. Thus, the parameterized satiation model seems to include the most important determinants of the functional response in our focal system. For understanding functional responses, we need more studies that compare data to independently derived model predictions.     

A spatial theory for characterizing predator–multiprey interactions in heterogeneous landscapes

Trophic interactions in multiprey systems can be largely determined by prey distributions. Yet, classic predator–prey models assume spatially homogeneous interactions between predators and prey. We developed a spatially informed theory that predicts how habitat heterogeneity alters the landscape-scale distribution of mortality risk of prey from predation, and hence the nature of predator interactions in multiprey systems. The theoretical model is a spatially explicit, multiprey functional response in which species-specific advection–diffusion models account for the response of individual prey to habitat edges. The model demonstrates that distinct responses of alternative prey species can alter the consequences of conspecific aggregation, from increasing safety to increasing predation risk. Observations of threatened boreal caribou, moose and grey wolf interacting over 378 181 km² of human-managed boreal forest support this principle. This empirically supported theory demonstrates how distinct responses of apparent competitors to landscape heterogeneity, including to human disturbances, can reverse density dependence in fitness correlates.     

The role of noise in a predator–prey model with Allee effect

The existence and implications of alternative stable states in ecological systems have been investigated extensively within deterministic models. However, it is known that natural systems are undeniably subject to random fluctuations, arising from either environmental variability or internal effects. Thus, in this paper, we study the role of noise on the pattern formation of a spatial predator–prey model with Allee effect. The obtained results show that the spatially extended system exhibits rich dynamic behavior. More specifically, the stationary pattern can be induced to be a stable target wave when the noise intensity is small. As the noise intensity is increased, patchy invasion emerges. These results indicate that the dynamic behavior of predator–prey models may be partly due to stochastic factors instead of deterministic factors, which may also help us to understand the effects arising from the undeniable susceptibility to random fluctuations of real ecosystems.     

Trophic cascades alter eco-evolutionary dynamics and body size evolution

Trait evolution in predator–prey systems can feed back to the dynamics of interacting species as well as cascade to impact the dynamics of indirectly linked species (eco-evolutionary trophic cascades; EETCs). A key mediator of trophic cascades is body mass, as it both strongly influences and evolves in response to predator–prey interactions. Here, we use Gillespie eco-evolutionary models to explore EETCs resulting from top predator loss and mediated by body mass evolution. Our four-trophic-level food chain model uses allometric scaling to link body mass to different functions (ecological pleiotropy) and is realistically parameterized from the FORAGE database to mimic the parameter space of a typical freshwater system. To track real-time changes in selective pressures, we also calculated fitness gradients for each trophic level. As predicted, top predator loss generated alternating shifts in abundance across trophic levels, and, depending on the nature and strength in changes to fitness gradients, also altered trajectories of body mass evolution. Although more distantly linked, changes in the abundance of top predators still affected the eco-evolutionary dynamics of the basal producers, in part because of their relatively short generation times. Overall, our results suggest that impacts on top predators can set off transient EETCs with the potential for widespread indirect impacts on food webs.     

Temperature alters food web body-size structure

The increased temperature associated with climate change may have important effects on body size and predator–prey interactions. The consequences of these effects for food web structure are unclear because the relationships between temperature and aspects of food web structure such as predator–prey body-size relationships are unknown. Here, we use the largest reported dataset for marine predator–prey interactions to assess how temperature affects predator–prey body-size relationships among different habitats ranging from the tropics to the poles. We found that prey size selection depends on predator body size, temperature and the interaction between the two. Our results indicate that (i) predator–prey body-size ratios decrease with predator size at below-average temperatures and increase with predator size at above-average temperatures, and (ii) that the effect of temperature on predator–prey body-size structure will be stronger at small and large body sizes and relatively weak at intermediate sizes. This systematic interaction may help to simplify forecasting the potentially complex consequences of warming on interaction strengths and food web stability.     

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.     

Lyapunov functions for Lotka–Volterra predator–prey models with optimal foraging behavior

 The theory of optimal foraging predicts abrupt changes in consumer behavior which lead to discontinuities in the functional response. Therefore population dynamical models with optimal foraging behavior can be appropriately described by differential equations with discontinuous right-hand sides. In this paper we analyze the behavior of three different Lotka–Volterra predator–prey systems with optimal foraging behavior. We examine a predator–prey model with alternative food, a two-patch model with mobile predators and resident prey, and a two-patch model with both predators and prey mobile. We show that in the studied examples, optimal foraging behavior changes the neutral stability intrinsic to Lotka–Volterra systems to the existence of a bounded global attractor. The analysis is based on the construction and use of appropriate Lyapunov functions for models described by discontinuous differential equations. Received: 23 March 1999     

A stochastic foraging model with predator training effects: I. Functional response,switching, and run lengths

Training effects are changes in a predator's behavior while it is searching for the next prey to eat which are caused by the predator's experience with the last prey encountered. A stochastic foraging model is formulated incorporating several specific types of training effects, and their impact on the functional response shape, switching, and mean prey run lengths is evaluated. The main result is that training effects such as search image formation can cause sigmoid functional responses and switching, and can result in runs of prey captures longer than expected in the absence of training.     

Extinction of top-predator in a three-level food-chain model

 In this paper we extend the Lyapunov functions, constructed by A. Ardito and P. Ricciardi for predator–prey system [1], to the three level food chain models. We first consider a general three-level food-chain model. A criterion for the extinction of top predator will be given. Then we restrict our attentions to the case in which the prey is of logistic growth and predators have Holling’s type II functional responses. Received: 10 October 1997     

Bayesian Inference for Functional Response in a Stochastic Predator–Prey System

We present a Bayesian method for functional response parameter estimation starting from time series of field data on predator–prey dynamics. Population dynamics is described by a system of stochastic differential equations in which behavioral stochasticities are represented by noise terms affecting each population as well as their interaction. We focus on the estimation of a behavioral parameter appearing in the functional response of predator to prey abundance when a small number of observations is available. To deal with small sample sizes, latent data are introduced between each pair of field observations and are considered as missing data. The method is applied to both simulated and observational data. The results obtained using different numbers of latent data are compared with those achieved following a frequentist approach. As a case study, we consider an acarine predator–prey system relevant to biological control problems.     

Spatiotemporal patterns provoked by environmental variability in a predator–prey model

The emergence of spatiotemporal patterns in the distribution of species is one of the most striking phenomena in ecology and nonlinear science. Since it is known that spatial inhomogeneities can significantly affect the dynamics of ecological populations, in the present paper we investigate the impact of environmental variability on the formation of patterns in a spatially extended predator–prey model. In particular, we utilize a predator–prey system with a Holling III functional response and introduce random spatial variations of the kinetic parameter signifying the intrinsic growth rate of the prey, reflecting the impact of a heterogeneous environment. Our results reveal that in the proximity of the Hopf bifurcation environmental variability is able to provoke pattern formation, whereby the coherence of the patterns exhibits a resonance-like dependence on the variability strength. Furthermore, we show that the phenomenon can only be observed if the spatial heterogeneities exhibit large enough regions with high growth rates of the prey. Our findings thus indicate that variability could be an essential pattern formation mechanism of the populations.     

Dynamics of additional food provided predator–prey system with mutually interfering predators

Use of additional/alternative food source to predators is one of the widely recognised practices in the field of biological control. Both theoretical and experimental works point out that quality and quantity of additional food play a vital role in the controllability of the pest. Theoretical studies carried out previously in this direction indicate that incorporating mutual interference between predators can stabilise the system. Experimental evidence also point out that mutual interference between predators can affect the outcome of the biological control programs. In this article dynamics of additional food provided predator–prey system in the presence of mutual interference between predators has been studied. The mutual interference between predators is modelled using Beddington–DeAngelis type functional response. The system analysis highlights the role of mutual interference on the success of biological control programs when predators are provided with additional food. The model results indicate the possibility of stable coexistence of predators with low prey population levels. This is in contrast to classical predator–prey models wherein this stable co-existence at low prey population levels is not possible. This study classifies the characteristics of biological control agents and additional food (of suitable quality and quantity), permitting the eco-managers to enhance the success rate of biological control programs.     

Microgeographic divergence of functional responses among salamanders under antagonistic selection from apex predators

A predator''s functional response determines predator–prey interactions by describing the relationship between the number of prey available and the number eaten. Its shape and parameters fundamentally govern the dynamic equilibrium of predator–prey interactions and their joint abundances. Yet, estimates of these key parameters generally assume stasis in space and time and ignore the potential for local adaptation to alter feeding responses and the stability of trophic dynamics. Here, we evaluate if functional responses diverge among populations of spotted salamander (Ambystoma maculatum) larvae that face antagonistic selection on feeding strategies based on their own risk of predation. Common garden experiments revealed that spotted salamander from ponds with varying predation risks differed in their functional responses, suggesting an evolutionary response. Applying mechanistic equations, we discovered that the combined changes in attack rates, handling times and shape of the functional response enhanced feeding rate in environments with high densities of gape-limited predators. We suggest how these parameter changes could alter community equilibria and other emergent properties of food webs. Community ecologists might often need to consider how local evolution at fine scales alters key relationships in ways that alter local diversity patterns, food web dynamics, resource gradients and community responses to disturbance.     

The effect of dilution on eco‐evolutionary dynamics of experimental microbial communities

Changing environmental conditions can infer structural modifications of predator‐prey communities. New conditions often increase mortality which reduces population sizes. Following this, predation pressure may decrease until populations are dense again. Dilution may thus have substantial impact not only on ecological but also on evolutionary dynamics because it amends population densities. Experimental studies, in which microbial populations are maintained by a repeated dilution into fresh conditions after a certain period, are extensively used approaches allowing us to obtain mechanistic insights into fundamental processes. By design, dilution, which depends on transfer volume (modifying mortality) and transfer interval (determining the time of interaction), is an inherent feature of these experiments, but often receives little attention. We further explore previously published data from a live predator‐prey (bacteria and ciliates) system which investigated eco‐evolutionary principles and apply a mathematical model to predict how various transfer volumes and transfer intervals would affect such an experiment. We find not only the ecological dynamics to be modified by both factors but also the evolutionary rates to be affected. Our work predicts that the evolution of the anti‐predator defense in the bacteria, and the evolution of the predation efficiency in the ciliates, both slow down with lower transfer volume, but speed up with longer transfer intervals. Our results provide testable hypotheses for future studies of predator‐prey systems, and we hope this work will help improve our understanding of how ecological and evolutionary processes together shape composition of microbial communities.     

Simple order of prey preference technique for modelling the predator functional response

The predator functional response to several prey types and densities may be conceptualized as a multi-dimensional version of the one-dimensional Holling functional-response curves; however, this empirical approach requires inordinate amounts of data to develop and test. A simulation method of modelling this functional response is to consider the behavior of a predator faced with the choice of several prey types. In this model, when all prey are available the predator’s selection will depend on the absolute abundance of the most-preferred prey type, irrespective of the abundances of the less-preferred prey types. Consequently, the predator will consume only the most-preferred prey types while that type is available in sufficient numbers. When abundance of the most-preferred type declines below a certain level, the predator will begin to include in its diet the second-most-preferred prey type along with the most-preferred prey type. This order-of-preference technique holds up well when the model is compared to population data fromOligonychus pratensis (Acarina: Tetranychidae)/Neoseiulus fallacis (Acarina: Phytoseiidae), and is consistent with optimal foraging theory. Implementation is simple, and the data requirements are reduced to determining the predator’s order of preference and normalizing the nutritional values of the prey types to a single type.