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.     

Testing for predator dependence in predator-prey dynamics: a non-parametric approach

The functional response is a key element in all predator-prey interactions. Although functional responses are traditionally modelled as being a function of prey density only, evidence is accumulating that predator density also has an important effect. However, much of the evidence comes from artificial experimental arenas under conditions not necessarily representative of the natural system, and neglecting the temporal dynamics of the organism (in particular the effects of prey depletion on the estimated functional response). Here we present a method that removes these limitations by reconstructing the functional response non-parametrically from predator-prey time-series data. This method is applied to data on a protozoan predator-prey interaction, and we obtain significant evidence of predator dependence in the functional response. A crucial element in this analysis is to include time-lags in the prey and predator reproduction rates, and we show that these delays improve the fit of the model significantly. Finally, we compare the non-parametrically reconstructed functional response to parametric forms, and suggest that a modified version of the Hassell-Varley predator interference model provides a simple and flexible function for theoretical investigation and applied modelling.     

Census error and the detection of density dependence

1. Studies aiming to identify the prevalence and nature of density dependence in ecological populations have often used statistical analysis of ecological time-series of population counts. Such time-series are also being used increasingly to parameterize models that may be used in population management. 2. If time-series contain measurement errors, tests that rely on detecting a negative relationship between log population change and population size are biased and prone to spuriously detecting density dependence (Type I error). This is because the measurement error in density for a given year appears in the corresponding change in population density, with equal magnitude but opposite sign. 3. This effect introduces bias that may invalidate comparisons of ecological data with density-independent time-series. Unless census error can be accounted for, time-series may appear to show strongly density-dependent dynamics, even though the density-dependent signal may in reality be weak or absent. 4. We distinguish two forms of census error, both of which have serious consequences for detecting density dependence. 5. First, estimates of population density are based rarely on exact counts, but on samples. Hence there exists sampling error, with the level of error depending on the method employed and the number of replicates on which the population estimate is based. 6. Secondly, the group of organisms measured is often not a truly self-contained population, but part of a wider ecological population, defined in terms of location or behaviour. Consequently, the subpopulation studied may effectively be a sample of the population and spurious density dependence may be detected in the dynamics of a single subpopulation. In this case, density dependence is detected erroneously, even if numbers within the subpopulation are censused without sampling error. 7. In order to illustrate how process variation and measurement error may be distinguished we review data sets (counts of numbers of birds by single observers) for which both census error and long-term variance in population density can be estimated. 8. Tests for density dependence need to obviate the problem that measured population sizes are typically estimates rather than exact counts. It is possible that in some cases it may be possible to test for density dependence in the presence of unknown levels of census error, for example by uncovering nonlinearities in the density response. However, it seems likely that these may lack power compared with analyses that are able to explicitly include census error and we review some recently developed methods.     

The nature of predation: prey dependent, ratio dependent or neither?

To describe a predator-prey relationship, it is necessary to specify the rate of prey consumption by an average predator. This functional response largely determines dynamic stability, responses to environmental influences and the nature of indirect effects in the food web containing the predator-prey pair. Nevertheless, measurements of functional responses in nature are quite rare. Recently, much work has been devoted to comparing two idealized forms of the functional response: prey dependent and ratio dependent. Although we agree that predator abundance often affects the consumption rate of individual predators, this phenomenon requires more attention. Disagreement remains over which of the two idealized responses serves as a better starting point in building models when data on predator dependence are absent.     

Temperature dependence of the functional response

The Arrhenius equation has emerged as the favoured model for describing the temperature dependence of consumption in predator-prey models. To examine the relevance of this equation, we undertook a meta-analysis of published relationships between functional response parameters and temperature. We show that, when plotted in lin-log space, temperature dependence of both attack rate and maximal ingestion rate exhibits a hump-shaped relationship and not a linear one as predicted by the Arrhenius equation. The relationship remains significantly downward concave even when data from temperatures above the peak of the hump are discarded. Temperature dependence is stronger for attack rate than for maximal ingestion rate, but the thermal optima are not different. We conclude that the use of the Arrhenius equation to describe consumption in predator-prey models requires the assumption that temperatures above thermal optima are unimportant for population and community dynamics, an assumption that is untenable given the available data.     

Modelling Size Structured Food Webs Using a Modified Niche Model with Two Predator Traits

The structure of food webs is frequently described using phenomenological stochastic models. A prominent example, the niche model, was found to produce artificial food webs resembling real food webs according to a range of summary statistics. However, the size structure of food webs generated by the niche model and real food webs has not yet been rigorously compared. To fill this void, I use a body mass based version of the niche model and compare prey-predator body mass allometry and predator-prey body mass ratios predicted by the model to empirical data. The results show that the model predicts weaker size structure than observed in many real food webs. I introduce a modified version of the niche model which allows to control the strength of size-dependence of predator-prey links. In this model, optimal prey body mass depends allometrically on predator body mass and on a second trait, such as foraging mode. These empirically motivated extensions of the model allow to represent size structure of real food webs realistically and can be used to generate artificial food webs varying in several aspects of size structure in a controlled way. Hence, by explicitly including the role of species traits, this model provides new opportunities for simulating the consequences of size structure for food web dynamics and stability.     

Matching the forecast horizon with the relevant spatial and temporal processes and data sources

Most phenomenological, statistical models used to generate ecological forecasts take either a time-series approach, based on long-term data from one location, or a space-for-time approach, based on data describing spatial patterns across environmental gradients. However, the magnitude and even the sign of environment–response relationships detected using these two approaches often differs, leading to contrasting predictions about responses to future environmental change. Here we consider how the forecast horizon determines whether more accurate predictions come from the time-series approach, the space-for-time approach or a combination of the two. As proof of concept, we use simulated case studies to show that forecasts for short and long forecast horizons need to focus on different ecological processes, which are reflected in different kinds of data. First, we simulated population or community dynamics under stationary temperature using two simple, mechanistic models. Second, we fit statistical models to the simulated data using a time-series approach, a space-for-time approach or a weighted average. We then forecast the response to a temperature increase using the statistical models, and compared these forecasts to temperature effects simulated by the mechanistic models. We found that the time-series approach made accurate short-term predictions because it captured initial conditions and effects of fast processes such as birth and death. The space-for-time approach made more accurate long-term predictions because it better captured the influence of slower processes such as evolutionary and ecological selection. The weighted average made accurate predictions at all time scales, including intermediate time-scales where the other two approaches performed poorly. A weighted average of time-series and space-for-time approaches shows promise, but making this weighted model operational will require new research to predict the rate at which slow processes begin to influence dynamics.     

Oscillations and chaos behind predator-prey invasion: mathematical artifact or ecological reality?

A constant dilemma in theoretical ecology is knowing whether model predictions corrspond to real phenomena or whether they are artifacts of the modelling framework. The frequent absence of detailed ecological data against which models can be tested gives this issue particular importance. We address this question in the specific case of invasion in a predator-prey system with oscillatory population kinetics, in which both species exhibit local random movement. Given only these two basic qualitative features, we consider whether we can deduce any properties of the behaviour following invasion. To do this we study four different types of mathematical model, which have no formal relationship, but which all reflect our two qualitative ingredients. The models are: reaction-diffusion equations, coupled map lattices, deterministic cellular automata, and integrodifference equations. We present results of numerical simulations of the invasion of prey by predators for each model, and show that although there are certain differences, the main qualitative features of the behaviour behind invasion are the same for all the models. Specifically, there are either irregular spatiotemporal oscillations behind the invasion, or regular spatiotemporal oscillations with the form of a periodic travelling ''wake'', depending on parameter values. The observation of this behaviour in all types of model strongly suggests that it is a direct consequence of our basic qualitative assumptions, and as such is an ecological reality which will always occur behind invasion in actual oscillatory predator-prey systems.     

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     

Sensitivity Analysis of Reactive Ecological Dynamics

Ecological systems with asymptotically stable equilibria may exhibit significant transient dynamics following perturbations. In some cases, these transient dynamics include the possibility of excursions away from the equilibrium before the eventual return; systems that exhibit such amplification of perturbations are called reactive. Reactivity is a common property of ecological systems, and the amplification can be large and long-lasting. The transient response of a reactive ecosystem depends on the parameters of the underlying model. To investigate this dependence, we develop sensitivity analyses for indices of transient dynamics (reactivity, the amplification envelope, and the optimal perturbation) in both continuous- and discrete-time models written in matrix form. The sensitivity calculations require expressions, some of them new, for the derivatives of equilibria, eigenvalues, singular values, and singular vectors, obtained using matrix calculus. Sensitivity analysis provides a quantitative framework for investigating the mechanisms leading to transient growth. We apply the methodology to a predator-prey model and a size-structured food web model. The results suggest predator-driven and prey-driven mechanisms for transient amplification resulting from multispecies interactions.     

Ratio-Dependent Predator-Prey Models of Interacting Populations

Ratio-dependent predator-prey models are increasingly favored by both the theoretical and experimental ecologists as a more suitable alternative to describe predator-prey interactions when the predators hunt seriously. In this article, the classical Bazykin’s model is modified with ratio-dependent functional response. Stability and bifurcation situations of the system are observed. Since the ratio-dependent model always has difficult dynamics in the vicinity of the origin, the analytical behavior of the system near origin is studied completely. It is found that paradox of enrichment can happen to this system under certain parameter values, although the functional response is ratio-dependent. The parametric space for Turing spatial structure is determined. We also conclude that competition among the predator population might be beneficial for predator species under certain circumstances. Finally, ecological interpretations of our results are presented in the discussion section.     

Predator functional response changed by induced defenses in prey

Functional responses play a central role in the nature and stability of predator-prey population dynamics. Here we investigate how induced defenses affect predator functional responses. In experimental communities, prey (Paramecium) expressed two previously undocumented inducible defenses--a speed reduction and a width increase--in response to nonlethal exposure to predatory Stenostomum. Nonlethal exposure also changed the shape of the predator's functional response from Type II to Type III, consistent with changes in the density dependence of attack rates. Handling times were also affected by prey defenses, increasing at least sixfold. These changes show that induced changes in prey have a real defensive function. At low prey densities, induction led to lower attack success; at high prey densities, attack rates were actually higher for induced prey. However, induction increased handling times sufficiently that consumption rates of defended prey were lower than those of undefended prey. Modification of attack rate and handling time has important potential consequences for population dynamics; Type III functional responses can increase the stability of population dynamics and persistence because predation on small populations is low, allowing a relict population to survive. Simulations of a predator-prey population dynamic model revealed the stabilizing potential of the Type III response.     

Emergence of Holling type III zooplankton functional response: Bringing together field evidence and mathematical modelling

Food-web population models are rather sensitive to parameterization of functional response in predation terms. Theoretical studies predict enhancing of ecosystems’ stability for a functional response of sigmoid type (Holling type III). The choice of a correct type of response is especially important for modelling outcome of grazing control of algal blooms by zooplankton in nutrient-rich ecosystems. Extensive experiments on zooplankton feeding in laboratories show non-sigmoid nature of response for most herbivorous zooplankton species. As a consequence, there is a strong opinion in literature that the implementation of Holling III type grazing in plankton models is biologically meaningless. I argue, however, that such an ‘evident’ claim might be wrong and sigmoid functional responses in real plankton communities would emerge more often than was suggested earlier. Especially, this concerns plankton models without vertical resolution, which ignore heterogeneity in vertical distribution of species. Having conducted extensive literature search of data on zooplankton feeding in situ, I show that vertical heterogeneity in food distribution as well as active food searching behaviour of zooplankton can modify the type of functional response. In particular, the rate of food intake by the whole zooplankton population in the column, as a function of total amount of food, often exhibits a sigmoid behaviour, instead of a non-sigmoid one postulated previously based on laboratory experiments. This conceptual discrepancy is due to the ability of zooplankton to feed mostly in layers with high algal density. I propose a generic model explaining the observed alteration of type between overall and local functional responses. I show that emergence of Holling type III in plankton systems is due to mechanisms different from those well known in the ecological literature (e.g. food search learning, existence of alternative food, refuge for prey).     

Functional responses modified by predator density

Realistic functional responses are required for accurate model predictions at the community level. However, controversy remains regarding which types of dependencies need to be included in functional response models. Several studies have shown an effect of very high predator densities on per capita predation rates, but it is unclear whether this predator dependence is also important at low predator densities. We fit integrated functional response models to predation data from 4-h experiments where we had varied both predator and prey densities. Using an information theoretic approach we show that the best-fit model includes moderate predator dependence, which was equally strong even at low predator densities. The best fits of Beddington–DeAngelis and Arditi–Akçakaya functional responses were closely followed by the fit of the Arditi–Ginzburg model. A Holling type III functional response did not describe the data well. In addition, independent behavioral observations revealed high encounter rates between predators. We quantified the number of encounters between predators and the time the focal predator spent interacting with other individuals per encounter. This time “wasted” on conspecifics reduced the total time available for foraging and may therefore account for lower predation rates at higher predator densities. Our findings imply that ecological theory needs to take realistic levels of predator dependence into account.     

Predator-prey interactions in natural communities.

The familiar concepts of harvest and yield are developed for the purpose of describing predator-prey interactions in a community context. In this regard the functional response (appropriate for one predator-one prey systems) is replaced by a community harvest function. Conditions for the stability of an ecological community are obtained. Exploring the dynamics of predator-prey interactions within this framework leads to new interpretations of other dynamical models such as the Lotka-Volterra model. The concept of a community moving attractor point is introduced in order to describe the changes in all populations over time.     

Linking saturation,stability and sustainability in food webs with observed equilibrium structure

Stability of a dynamic equilibrium in a predator-prey system depends both on the type of functional response and on the point of equilibrium on the response curve. Saturation effects from Holling type II responses are known to destabilise prey populations, while a type III (sigmoid) response curve has been shown to provide stability at lower levels of saturation. These effects have also been shown in multi-trophic model systems. However, stability analyses of observed equilibria in real complex ecosystems have as yet not assumed non-linear functional responses. Here, we evaluate the implications of saturation in observed balanced material-flow structures, for system stability and sustainability. We first make the effects of the non-linear functional responses on the interaction strengths in a food web transparent by expressing the elements of Jacobian ‘community’ matrices for type II and III systems as simple functions of their linear (type I) counterparts. We then determine the stability of the systems and distinguish two critical saturation levels: (1) a level where the system is just as stable as a type I system and (2) a level above which the system cannot be stable unless it is subsidised, separating a stable materially sustainable regime from an unsustainable one. We explain the stabilising and destabilising effects in terms of the feedbacks in the systems. The results shed light on the robustness of observed patterns of interaction strengths in complex food webs and suggest the implausibility of saturation playing a significant role in the equilibrium dynamics of sustainable ecosystems.     

Stability of complex food webs: resilience, resistance and the average interaction strength

In the face of stochastic climatic perturbations, the overall stability of an ecosystem will be determined by the balance between its resilience and its resistance, but their relative importance is still unknown. Using aquatic food web models we study ecosystem stability as a function of food web complexity. We measured three dynamical stability properties: resilience, resistance, and variability. Specifically, we evaluate how a decrease in the strength of predator-prey interactions with food web complexity, reflecting a decrease in predation efficiency with the number of prey per predator, affects the overall stability of the ecosystem. We find that in mass conservative ecosystems, a lower interaction strength slows down the mass cycling rate in the system and this increases its resistance to perturbations of the growth rate of primary producers. Furthermore, we show that the overall stability of the food webs is mostly given by their resistance, and not by their resilience. Resilience and resistance display opposite trends, although they are shown not to be simply opposite concepts but rather independent properties. The ecological implication is that weaker predator-prey interactions in closed ecosystems can stabilize food web dynamics by increasing its resistance to climatic perturbations.     

Does mutual interference always stabilize predator–prey dynamics? A comparison of models

Based on a qualitative analysis of ODE systems, the dynamic properties of alternative predator-prey models with predator-dependent functional response have been compared in order to study the role that predator interference plays in the stabilisation of trophic systems. The models considered for interference have different mathematical expressions and different conceptual foundations. Despite these differences, they give essentially the same qualitative results: when interference is low, increasing it has a positive effect on asymptotic stability and thus on the resilience of the biological system. When it is high, it is the contrary (with logistic prey growth, increasing the interference parameter ensures stability but leads to very small predator densities). Possible consequences on the evolution of the interference level in real ecosystems are discussed.     

Modelling the ecological-functional diversification of marine Metazoa on geological time scales

The ecological traits and functional capabilities of marine animals have changed significantly since their origin in the late Precambrian. These changes can be analysed quantitatively using multi-dimensional parameter spaces in which the ecological lifestyles of species are represented by particular combinations of parameter values. Here, we present models that describe the filling of this multi-dimensional 'ecospace' by ecological lifestyles during metazoan diversification. These models reflect varying assumptions about the processes that drove ecological diversification; they contrast diffusive expansion with driven expansion and niche conservatism with niche partitioning. Some models highlight the importance of interactions among organisms (ecosystem engineering and predator-prey escalation) in promoting new lifestyles or eliminating existing ones. These models reflect processes that were not mutually exclusive; rigorous analyses will continue to reveal their applicability to episodes in metazoan history.     

Shift in a Large River Fish Assemblage: Body-Size and Trophic Structure Dynamics

As the intensity and speed of environmental change increase at both local and global scales it is imperative that we gain a better understanding of the ecological implications of community shifts. While there has been substantial progress toward understanding the drivers and subsequent responses of community change (e.g. lake trophic state), the ecological impacts of food web changes are far less understood. We analyzed Wabash River fish assemblage data collected from 1974-2008, to evaluate temporal variation in body-size structure and functional group composition. Two parameters derived from annual community size-spectra were our major response variables: (1) the regression slope is an index of ecological efficiency and predator-prey biomass ratios, and (2) spectral elevation (regression midpoint height) is a proxy for food web capacity. We detected a large assemblage shift, over at least a seven year period, defined by dramatic changes in abundance (measured as catch-per-unit-effort) of the dominant functional feeding groups among two time periods; from an assemblage dominated by planktivore-omnivores to benthic invertivores. There was a concurrent increase in ecological efficiency (slopes increased over time) following the shift associated with an increase in large-bodied low trophic level fish. Food web capacity remained relatively stable with no clear temporal trends. Thus, increased ecological efficiency occurred simultaneous to a compensatory response that shifted biomass among functional feeding groups.