Predation across spatial scales in heterogeneous environments

A hierarchy of scales is introduced to the spatially heterogeneous Lotka-Volterra predator-prey diffusion model, and its effects on the model's spatial and temporal behavior are studied. When predators move on a large scale relative to prey, local coupling of the predator-prey interaction is replaced by global coupling. Prey with low dispersal ability become narrowly confined to the most productive habitats, strongly amplifying the underlying spatial pattern of the environment. As prey diffusion rate increases, the prey distribution spreads out and predator abundance declines. The model retains neutrally stable Lotka-Volterra temporal dynamics: different scales of predator and prey dispersal do not stabilize the interaction. The model predicts that, for prey populations that are limited by widely ranging predators, species with low dispersal ability should be restricted to discrete high density patches, and those with greater mobility should be more uniformly distributed at lower density.     

Scale dependent effects of native prey diversity,prey biomass and natural disturbance on the invasion success of an exotic predator

There is growing support for the general notion that the drivers of invasion success often shift from biotic to abiotic factors with increasing spatial scale. Most of this research, however, has been conducted on a single trophic level; i.e. it has primarily looked at how the diversity of native competitors may influence invasion success. Less attention has been paid to understanding how native prey diversity may influence the invasion success of exotic predators and whether such biotic factors are scale-dependent. We used a hierarchical spatial survey of 17 stream communities to test whether native prey diversity, along with native prey biomass, algal resource abundance and annual stream discharge, influenced the abundance of an exotic crayfish predator, and whether the importance of these factors were scale-dependent. We used a hierarchical generalized linear model to evaluate the influence of these community and stream characteristics on exotic crayfish abundance at both the transect scale (1 m²) and the stream scale (400 m²). Our results indicated that at the stream scale, high stream discharge significantly limited invader abundance. However, at the smaller transect scale, native prey biomass was a significant driver of invasion success and positively correlated with invader abundance. We suggest that our results add to the emerging pattern that abiotic processes are stronger determinants of invasion success at large spatial scales, whereas biotic processes become more important with decreasing spatial scale. However, for predator invasions, prey biomass, not prey diversity may be a more important for driver of invasion success at small spatial scales.     

An emerging infectious disease triggering large-scale hyperpredation

Hyperpredation refers to an enhanced predation pressure on a secondary prey due to either an increase in the abundance of a predator population or a sudden drop in the abundance of the main prey. This scarcely documented mechanism has been previously studied in scenarios in which the introduction of a feral prey caused overexploitation of native prey. Here we provide evidence of a previously unreported link between Emergent Infectious Diseases (EIDs) and hyperpredation on a predator-prey community. We show how a viral outbreak caused the population collapse of a host prey at a large spatial scale, which subsequently promoted higher-than-normal predation intensity on a second prey from shared predators. Thus, the disease left a population dynamic fingerprint both in the primary host prey, through direct mortality from the disease, and indirectly in the secondary prey, through hyperpredation. This resulted in synchronized prey population dynamics at a large spatio-temporal scale. We therefore provide evidence for a novel mechanism by which EIDs can disrupt a predator-prey interaction from the individual behavior to the population dynamics. This mechanism can pose a further threat to biodiversity through the human-aided disruption of ecological interactions at large spatial and temporal scales.     

Detecting predator–prey relationships in the sea

Understanding the strength and diversity of predator‐prey interactions among species is essential to understand ecosystem consequences of population‐level variation. Directly quantifying the predatory behaviour of wild fishes at large spatial scales (>100 m) in the open sea is fraught with difficulties. To date the only empirical approach has been to search for correlations in the abundance of predators and their putative prey. As an example we use this approach to search for predators of the keystone crown‐of‐thorns starfish. We show that this approach is unlikely to detect predator–prey linkages because the theoretical relationship is non‐linear, resulting in multiple possible prey responses for single given predator abundance. Instead we suggest some indication of the strength and ecosystem importance of a predator–prey relationship can be gained by using the abundance of both predators and their putative prey to parameterize functional response models.     

Individual‐based models of cod movement and population dynamics

Understanding the strength and diversity of predator‐prey interactions among species is essential to understand ecosystem consequences of population‐level variation. Directly quantifying the predatory behaviour of wild fishes at large spatial scales (>100 m) in the open sea is fraught with difficulties. To date the only empirical approach has been to search for correlations in the abundance of predators and their putative prey. As an example we use this approach to search for predators of the keystone crown‐of‐thorns starfish. We show that this approach is unlikely to detect predator–prey linkages because the theoretical relationship is non‐linear, resulting in multiple possible prey responses for single given predator abundance. Instead we suggest some indication of the strength and ecosystem importance of a predator–prey relationship can be gained by using the abundance of both predators and their putative prey to parameterize functional response models.     

What causes outbreaks of the gypsy moth in North America?

The gypsy moth has been present in North America for more than 100 years, and in many of the areas where it has become established outbreaks occur with varying degrees of periodicity. There also exists extensive spatial synchrony in the onset of outbreaks over large geographic regions. Density-dependent mortality clearly limits high-density populations, but there is little evidence for strong regulation of low-density populations. Predation by small mammals appears to be the major source of mortality affecting low-density populations, but because these are generalist predators and gypsy moths are a less preferred food item, mammals do not appear to regulate populations in a density-dependent fashion. Instead, predation levels appear to be primarily determined by small mammal abundance, which is in turn closely linked to the production of acorns that are a major source of food for overwintering predator populations. Mast production by host oak trees is typically variable among years, but considerable spatial synchrony in masting exists over large geographic areas. Thus, it appears that the temporal and spatial patterns of mast production may be responsible for the episodic and spatially synchronous behavior of gypsy moth outbreaks in North America. This multitrophic relationship among mast, predators, and gypsy moths represents a very different explanation of forest insect outbreak dynamics than the more widely applied theories based upon predator–prey cycles or feedbacks with host foliage quality. Received: September 8, 1999 / Accepted: September 20, 2000     

Evolutionary ecology of movement by predators and prey

An essential key to explaining the mechanistic basis of ecological patterns lies in understanding the consequences of adaptive behavior for distributions and abundances of organisms. We developed a model that simultaneously incorporates (a) ecological dynamics across three trophic levels and (b) evolution of behaviors via the processes of mutation, selection, and drift in populations of variable, unique individuals. Using this model to study adaptive movements of predators and prey in a spatially explicit environment produced a number of unexpected results. First, even though predators and prey had limited information and sometimes moved in the “wrong” direction, evolved movement mechanisms allowed them to achieve average spatial distributions approximating optimal, ideal free distributions. Second, predators’ demographic parameters had marked, nonlinear effects on the evolution of movement mechanisms in the prey: As the predator mortality rate was increased past a critical point, prey abruptly shifted from making very frequent movements away from predators to making infrequent movements mainly in response to resources. Third, time series analyses revealed that adaptive, conditional movements coupled ecological dynamics across species and space. Our results provide general predictions, heretofore lacking, about how predators and prey should respond to one another on both ecological and evolutionary time scales.     

Spatial, temporal and habitat-related variation in the abundance of large predatory fish at One Tree Reef, Australia

 Patterns of abundance of large piscivorous fish (>200 mm TL) were documented at two spatial and four temporal scales within the main lagoon of One Tree Reef on Australia’s Great Barrier Reef. Grouper (Serranidae), snapper (Lutjanidae) and wrasses (Labridae) were the most abundant large piscivores. On a large scale (hundreds of metres), patterns of predator abundance were consistently greater on the inner edge than centre of the lagoon over a range of temporal scales: days, weeks, months and years. On a small spatial scale (tens of metres), the abundance of large predatory fish was patchy. At both spatial scales, fish were consistently aggregated in particular areas and associated with specific structural features of the reef habitat. Predator abundance was high where live corals were predominant and the topography was more complex. Hence, predation pressure and its potential effects on the distribution and abundance of prey populations, both in time and space, may vary greatly within lagoonal environments. Accepted: 25 May 1997     

Local spatial structure and predator-prey dynamics: counterintuitive effects of prey enrichment

The Lotka-Volterra predator-prey model with prey density dependence shows the final prey density to be independent of its vital rates. This result assumes the community to be well mixed so that encounters between predators and prey occur as a product of the landscape densities, yet empirical evidence suggests that over small spatial scales this may not be the normal pattern. Starting from an individual-based model with neighborhood interactions and movements, a deterministic approximation is derived, and the effect of local spatial structure on equilibrium densities is investigated. Incorporating local movements and local interactions has important consequences for the community dynamics. Now the final prey density is very much dependent on its birth, death, and movement rates and in ways that seem counterintuitive. Increasing prey fecundity or mobility and decreasing the coefficient of competition can all lead to decreases in the final density of prey if the predator is also relatively immobile. However, analysis of the deterministic approximation makes the mechanism for these results clear; each of these changes subtly alters the emergent spatial structure, leading to an increase in the predator-prey spatial covariance at short distances and hence to a higher predation pressure on the prey.     

Evaluating the effects of large marine predators on mobile prey behavior across subtropical reef ecosystems

The indirect effect of predators on prey behavior, recruitment, and spatial relationships continues to attract considerable attention. However, top predators like sharks or large, mobile teleosts, which can have substantial top–down effects in ecosystems, are often difficult to study due to their large size and mobility. This has created a knowledge gap in understanding how they affect their prey through nonconsumptive effects. Here, we investigated how different functional groups of predators affected potential prey fish populations across various habitats within Biscayne Bay, FL. Using baited remote underwater videos (BRUVs), we quantified predator abundance and activity as a rough proxy for predation risk and analyzed key prey behaviors across coral reef, sea fan, seagrass, and sandy habitats. Both predator abundance and prey arrival times to the bait were strongly influenced by habitat type, with open homogenous habitats receiving faster arrival times by prey. Other prey behaviors, such as residency and risk‐associated behaviors, were potentially driven by predator interaction. Our data suggest that small predators across functional groups do not have large controlling effects on prey behavior or stress responses over short temporal scales; however, habitats where predators are more unpredictable in their occurrence (i.e., open areas) may trigger risk‐associated behaviors such as avoidance and vigilance. Our data shed new light on the importance of habitat and context for understanding how marine predators may influence prey behaviors in marine ecosystems.     

Inside‐container effects drive mosquito community structure in Brazilian Atlantic forest

Metacommunity theory is a convenient framework in which to investigate how local communities linked by dispersal influence patterns of species distribution and abundance across large spatial scales. For organisms with complex life cycles, such as mosquitoes, different pressures are expected to act on communities due to behavioral and ecological partitioning of life stages. Adult females select habitats for oviposition, and resulting offspring are confined to that habitat until reaching adult stages capable of flight; outside‐container effects (OCE) (i.e., spatial factors) are thus expected to act more strongly on species distributions as a function of adult dispersal capability, which should be limited by geographic distances between sites. However, larval community dynamics within a habitat are influenced by inside‐container effects (ICE), mainly interactions with conspecifics and heterospecifics (e.g., through effects of competition and predation). We used a field experiment in a mainland‐island scenario to assess whether environmental, spatial, and temporal factors influence mosquito prey and predator distributions and abundances across spatial scales: within‐site, between‐site, and mainland‐island. We also evaluated whether predator abundances inside containers play a stronger role in shaping mosquito prey community structure than do OCE (e.g., spatial and environmental factors). Temporal influence was more important for predators than for prey mosquito community structure, and the changes in prey mosquito species composition over time appear to be driven by changes in predator abundances. There was a negligible effect of spatial and environmental factors on mosquito community structure, and temporal effects on mosquito abundances and distributions appear to be driven by changes in abundance of the dominant predator, perhaps because ICE are stronger than OCE due to larval habitat restriction, or because adult dispersal is not limited at the chosen spatial scales.     

Lack of spatial coherence of predators with prey: a bioenergetic explanation for Atlantic cod feeding on capelin

We tested two biologically based predictions that potentially influence scales of spatial association between Atlantic cod, Gadus morhua , and prey populations of capelin, Mallotus vilhsus . If cod aggregate in response to concentrations of prey, then spatial association (coherence) between capelin and cod was predicted to peak at the scale of maximum capelin spatial variance. If capelin-cod coherence did not match the scale of maximum prey spatial variability, then capelin-cod coherence was predicted to peak at the spatial scale that maximizes net energetic benefit to the predator. Contrary to predictions, we found no evidence of aggregative responses of cod to capelin over resolution scales of 20 m to 10 km. This result was observed consistently at the temporal scale of a single transect ( c . 1 h duration) and at the scale of averaged transects ( c . 2 weeks duration). Estimates of cod foraging energetics showed that they were not constrained by physiology to aggregate relative to capelin at any scale less than 10 km. A net energetic gain of 478 to 784 kJ would result if a 44 cm, 752 g cod consumed a ration of eight to 12 capelin over a period of 58 h. Energetic calculations included costs of egestion and excretion (317 to 476 kJ), maintenance (58 kJ), digestion (125 to 188 kJ), and continuous swimming during ration assimilation (79 kJ). During this period, a 44 cm cod could travel over 38 km swimming at 1 b.l. s⁻¹. Foraging cod are virtually certain to encounter capelin over this distance based on the abundance of pre-spawning capelin present in coastal bays during the spawning season. This study illustrates that aggregative responses of predators do no occur at all scales and possibly occur over a very limited range of scales.     

Prey-Dependent Mortality Rate: A Critical Parameter in Microbial Models

Protozoa are key components of a wide range of ecosystems, but ecological models that incorporate these microbes often suffer from poor parameterisation, specifically of top-level predator loss rates. We (1) suggest that top-level predator mortality is prey-dependent, (2) provide a novel approach to assess this response, and (3) illustrate the ecological relevance of these findings. Ciliates, Paramecium caudatum (prey) and Didinium nasutum (predator), were used to evaluate predator mortality at varying prey levels. To assess mortality, multiple (>100) predators were individually examined (in 2-ml wells), daily (for 3 days), between 0 and 120 preys ml⁻¹. Data were used to determine non-linear mortality and growth responses over a range of prey abundances. The responses, plus literature data were then used to parameterise a predator–prey model, based on the Rosenzweig–MacArthur structure. The model assessed the impact of variable and three levels of constant (high, average and low) mortality rates on P. caudatum–D. nasutum population dynamics. Our method to determine variable mortality rate revealed a strong concave decline in mortality with increasing prey abundance. The model indicated: (1) high- and low-constant mortality rates yielded dynamics that deviate substantially from those obtained from a variable rate; (2) average mortality rate superficially produced dynamics similar to the variable rate, but there were differences in the period of predator–prey cycles, and the lowest abundance of prey and predators (by ~2 orders of magnitude). The differences between incorporating variable and constant mortality rate indicate that including a variable rate could substantially improve microbial-based ecological models.     

Alteration of sensory abilities regulates the spatial scale of nonlethal predator effects

Many studies have shown that nonlethal predator effects such as trait-mediated interactions (TMIs) can have significant impacts on the structure and function of communities, but the role that environmental conditions play in modulating the scale and magnitude of these effects has not been carefully investigated. TMIs occur when prey exhibit behavioral or physiological responses to predators and may be more prevalent when abiotic conditions increase prey reactions to consumers. The purpose of this study was to determine if turbulence would alter the distance over which prey in aquatic systems respond to chemical cues emitted by predators in nature, thus changing the scales over which nonlethal predator effects occur. Using hard clams and blue crabs as a model predator–prey system, we investigated the effects of turbulence on clam reactive distance to predatory blue crabs in the field. Results suggest that turbulence diminishes clam reactions to predators and that the environmental context must be considered when predicting the extent of indirect predator effects in natural systems.     

Carbon dioxide effects on stomatal responses to the environment and water use by crops under field conditions

The structural complexity of habitats has been espoused as an important factor influencing natural-enemy abundance and food-web dynamics in invertebrate-based communities, but a rigorous synthesis of published studies has not heretofore been conducted. We performed a meta-analytical synthesis of the density response of natural enemies (invertebrate predators and parasitoids) to experimental increases and decreases in the structural complexity of their habitats using data from 43 published studies, reporting 62 independent taxa. Studies varied in structural complexity at two spatial scales (habitat and within-plant architecture) and comprised a diverse array of natural-enemy taxa (natural-enemy assemblage at large, the entire spider assemblage, hunting spiders, web-building spiders, mites, hemipterans, coccinellid beetles, carabid beetles, ants, and parasitoids). For all taxa combined, increasing habitat structure resulted in a large and significant increase in natural enemy abundance. Similarly, decreasing habitat structure significantly diminished natural enemy abundance. Separate meta-analyses at two spatial scales (habitat and within-plant architecture) found that increasing habitat complexity resulted in significant increases in abundance. In particular, manipulating levels of detritus at the habitat spatial scale had the strongest effect on natural enemy abundance. In general, most guilds of natural enemies were significantly affected when the structural complexity of the habitat was altered. Seven of nine natural enemy guilds were more abundant under conditions of increased habitat complexity, with hunting spiders and web-building spiders showing the strongest response followed by hemipterans, mites, and parasitoids. Spiders in particular were negatively affected when habitat structure was simplified. The mechanisms underlying the accumulation of natural enemies in complex-structured habitats are poorly known. However, refuge from intraguild predation, more effective prey capture, and access to alternative resources (alternative prey, pollen, or nectar), are possible candidates. Our analysis was unable to confirm that predators aggregate in complex-structured habitats because prey (mostly herbivores) are more abundant there. The results of this meta-analysis support the view that basal resources mediate top-down impacts on herbivores, and provide encouragement that manipulations of habitat complexity can be made in agroecosystems that will enhance the effectiveness of the natural enemy complex for more effective pest suppression.Electronic Supplementary Material Supplementary material is available in the online version of this article at     

Mechanisms of coexistence in diverse herbivore–carnivore assemblages: demographic,temporal and spatial heterogeneities affecting prey vulnerability

Simple models coupling the dynamics of single predators to single prey populations tend to generate oscillatory dynamics of both predator and prey, or extirpation of the prey followed by that of the predator. In reality, such oscillatory dynamics may be counteracted by prey refugia or by opportunities for prey switching by the predator in multi‐prey assemblages. How these mechanisms operate depends on relative prey vulnerability, a factor ignored in simple interactive models. I outline how compositional, temporal, demographic and spatial heterogeneities help explain the contrasting effects of top predators on large herbivore abundance and population dynamics in species‐rich African savanna ecosystems compared with less species‐diverse northern temperate or subarctic ecosystems. Demographically, mortality inflicted by predation depends on the relative size and life history stage of the prey. Because all animals eventually die and are consumed by various carnivores, the additive component of the mortality inflicted is somewhat less than the predation rate. Prey vulnerability varies annually and seasonally, and between day and night. Spatial variation in the risk of predation depends on vegetation cover as well as on the availability of food resources. During times of food shortage, herbivores become prompted to occupy more risky habitats retaining more food. Predator concentrations dependent on the abundance of primary prey species may restrict the occurrence of other potential prey species less resistant to predation. The presence of multiple herbivore species of similar size in African savannas allows the top predator, the lion, to shift its prey selection flexibly dependent on changing prey vulnerability. Hence top–down and bottom–up influences on herbivore populations are intrinsically entangled. Models coupling the population dynamics of predators and prey need to accommodate the changing influences of prey demography, temporal variation in environmental conditions, and spatial variation in the relative vulnerability of alternative prey species to predation. Synthesis While re‐established predators have had major impacts on prey populations in northern temperate regions, multiple large herbivore species typically coexist along with diverse carnivores in African savanna ecosystems. In order to explain these contrasting outcomes, certain functional heterogeneities must be recognised, including relative vulnerability of alternative prey, temporal variation in the risk of predation, demographic differences in susceptibility to predation, and spatial contrasts in exposure to predation. Food shortfalls prompt herbivores to exploit more risky habitats, meaning that top–down and bottom–up influences on prey populations are intrinsically entangled. Models coupling the interactive dynamics of predator and prey populations need to incorporate these varying influences on relative prey vulnerability.     

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.     

Do the antipredator strategies of shared prey mediate intraguild predation and mesopredator suppression?

Understanding the conditions that facilitate top predator effects upon mesopredators and prey is critical for predicting where these effects will be significant. Intraguild predation (IGP) and the ecology of fear are hypotheses used to describe the effects of top predators upon mesopredators and prey species, but make different assumptions about organismal space use. The IGP hypothesis predicts that mesopredator resource acquisition and risk are positively correlated, creating a fitness deficit. But if shared prey also avoid a top predator, then mesopredators may not have to choose between risk and reward. Prey life history may be a critical predictor of how shared prey respond to predation and may mediate mesopredator suppression. We used hierarchical models of species distribution and abundance to test expectations of IGP using two separate triangular relationships between a large carnivore, smaller intraguild carnivore, and shared mammalian prey with different life histories. Following IGP, we expected that a larger carnivore would suppress a smaller carnivore if the shared prey species did not spatially avoid the large carnivore at broad scales. If prey were fearful over broad scales, we expected less evidence of mesopredator suppression. We tested these theoretical hypotheses using remote camera detections across a large spatial extent. Lagomorphs did not appear to avoid coyotes, and fox detection probability was lower as coyote abundance increased. In contrast, white‐tailed deer appeared to avoid areas of increased wolf use, and coyote detection probability was not reduced at sites where wolves occurred. These findings suggest that mesopredator suppression by larger carnivores may depend upon the behavior of shared prey, specifically the spatial scale at which they perceive risk. We further discuss how extrinsic environmental factors may contribute to mesopredator suppression.     

Landscapes of fear or competition? Predation did not alter habitat choice by Arctic rodents

In systems where predation plays a key role in the dynamics of prey populations, such as in Arctic rodents, it is reasonable to assume that differential patterns of habitat use by prey species represent adaptive responses to spatial variation in predation. However, habitat selection by collared (Dicrostonyx groenlandicus) and brown (Lemmus trimucronatus) lemmings depends on intra- and inter-specific densities, and there has been little agreement on the respective influences of food abundance, predators, and competition for habitat on lemming dynamics. Thus, we investigated whether predation affected selection of sedge-meadow versus upland tundra by collared lemmings in the central Canadian Arctic. We first controlled for the effects of competition on lemming habitat selection. We then searched for an additional signal of predation by comparing habitat selection patterns between 12 control plots and one large grid where lemmings were protected from predators by fencing in 1996 and 1997, but not during 5 subsequent years when we monitored habitat use in the grid as well as in the control plots. Dicrostonyx used upland preferentially over meadows and was more numerous in 1996 and 2011 than in other sample years. Lemmus was also more abundant in 1996 than in subsequent years, but its abundance was too low in the exclosure to assess whether exclusion of predators influenced its habitat selection. Contrary to the effects of competition, predation had a negligible impact on the spatial dynamics of Dicrostonyx, at least during summer. These results suggest that any differences in predation risk between the two habitats have little direct influence on the temporal dynamics of Dicrostonyx even if induced through predator–prey cycles.     

Scaling up population dynamic processes in a ladybird–aphid system

Here, we study how scaling up to the metapopulation level affects predictions of a population dynamics model motivated by an aphidophagous predator–aphid system. The model incorporates optimization of egg distribution in predatory females, cannibalism among their offspring, and self-regulation of the prey population. These factors determine the within-year dynamics of the system and translate the numbers of prey and predator individuals at the beginning of the season into their numbers at the end of the season at the level of one patch—one suitable host plant or a group of these. At the end of each season, all populations of prey and all populations of predators are mixed (this simulates aphid host-alternation and ladybird migration to hibernation sites), and then redistributed at the beginning of the next season. Prey individuals are distributed at random among the patches as a “prey rain”, while adult predators that survived from the previous season optimize the distribution of their offspring, in that they prefer patches with sufficient amount of prey and absence of other predators. This redistribution followed by within-season dynamics is then iterated over many seasons. We look at whether small-scale trends in population dynamics predicted by this model are consistent with large-scale outcomes. Specifically, we show that even on the metapopulation scale, the impact of predators on prey metapopulation is relatively low. We further show how the dates of predator arrival to and departure from the system affect the qualitative behaviour of the model predictions.