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In this paper we derive sufficient conditions for the permanence and ergodicity of a stochastic predator–prey model with a Beddington–DeAngelis functional response. The conditions obtained are in fact very close to the necessary conditions. Both nondegenerate and degenerate diffusions are considered. One of the distinctive features of our results is that they enable the characterization of the support of a unique invariant probability measure. It proves the convergence in total variation norm of the transition probability to the invariant measure. Comparisons to the existing literature and matters related to other stochastic predator–prey models are also given.

In this paper, we provide efficient estimators and honest confidence bands for a variety of treatment effects including local average (LATE) and local quantile treatment effects (LQTE) in data-rich environments. We can handle very many control variables, endogenous receipt of treatment, heterogeneous treatment effects, and function-valued outcomes. Our framework covers the special case of exogenous receipt of treatment, either conditional on controls or unconditionally as in randomized control trials. In the latter case, our approach produces efficient estimators and honest bands for (functional) average treatment effects (ATE) and quantile treatment effects (QTE). To make informative inference possible, we assume that key reduced-form predictive relationships are approximately sparse. This assumption allows the use of regularization and selection methods to estimate those relations, and we provide methods for postregularization and post-selection inference that are uniformly valid (honest) across a wide range of models. We show that a key ingredient enabling honest inference is the use of orthogonal or doubly robust moment conditions in estimating certain reducedform functional parameters. We illustrate the use of the proposed methods with an application to estimating the effect of 401(k) eligibility and participation on accumulated assets. The results on program evaluation are obtained as a consequence of more general results on honest inference in a general moment-condition framework, which arises from structural equation models in econometrics. Here, too, the crucial ingredient is the use of orthogonal moment conditions, which can be constructed from the initial moment conditions. We provide results on honest inference for (function-valued) parameters within this general framework where any high-quality, machine learning methods (e.g., boosted trees, deep neural networks, random forest, and their aggregated and hybrid versions) can be used to learn the nonparametric/high-dimensional components of the model. These include a number of supporting auxiliary results that are of major independent interest: namely, we (1) prove uniform validity of a multiplier bootstrap, (2) offer a uniformly valid functional delta method, and (3) provide results for sparsitybased estimation of regression functions for function-valued outcomes.

Knowledge of feeding rates is the basis to understand interaction strength and subsequently the stability of ecosystems and biodiversity. Feeding rates, as all biological rates, depend on consumer and resource body masses and environmental temperature. Despite five decades of research on functional responses as quantitative models of feeding rates, a unifying framework of how they scale with body masses and temperature is still lacking. This is perplexing, considering that the strength of functional responses (i.e. interaction strengths) is crucially important for the stability of simple consumer–resource systems and the persistence, sustainability and biodiversity of complex communities. Here, we present the largest currently available database on functional response parameters and their scaling with body mass and temperature. Moreover, these data are integrated across ecosystems and metabolic types of species. Surprisingly, we found general temperature dependencies that differed from the Arrhenius terms predicted by metabolic models. Additionally, the body-mass-scaling relationships were more complex than expected and differed across ecosystems and metabolic types. At local scales (taxonomically narrow groups of consumer–resource pairs), we found hump-shaped deviations from the temperature and body-mass-scaling relationships. Despite the complexity of our results, these body-mass- and temperature-scaling models remain useful as a mechanistic basis for predicting the consequences of warming for interaction strengths, population dynamics and network stability across communities differing in their size structure.

1. In community and population ecology, there is a chronic gap between the classic Eltonian ecology describing patterns in abundance and body mass across species and ecosystems and the more process oriented foraging ecology addressing interactions and quantitative population dynamics. However, this dichotomy is arbitrary, because body mass also determines most species traits affecting foraging interactions and population dynamics. 2. In this review, allometric (body-mass dependent) scaling of handling times and attack rates are documented, whereas body-mass effects on Hill exponents (varying between hyperbolic type II and sigmoid type III functional responses) and predator interference coefficients are lacking. This review describes how these allometric relationships define a biological plausible parameter space for population dynamic models. 3. Consistent with the classic Eltonian description, species co-existence in consumer-resource models and tri-trophic food chains is restricted to intermediate consumer-resource body-mass ratios. Allometric population dynamic models allow understanding the processes of energy limitation and unstable dynamics leading to this restriction. Complex food webs are stabilized by high predator-prey body-mass ratios, which are consistent with those found in natural ecosystems. These high body-mass ratios yield positive diversity-stability and complexity-stability relationships thus supporting the classic picture of ecosystem stability. 4. Allometric-trophic network models, based on body mass and trophic information from ecosystems, bridge the gap between Eltonian community patterns and process-oriented foraging ecology and provide a new means to describe the dynamics and functioning of natural ecosystems.

Models of habitat preference are widely used to quantify animal–habitat relationships, to describe and predict differential space use by animals, and to identify habitat that is important to an animal (i.e. that is assumed to influence fitness). Quantifying habitat preference involves the statistical comparison of samples of habitat use and availability. Preference is therefore contingent upon both of these samples. The inferences that can be made from use versus availability designs are influenced by subjectivity in defining what is available to the animal, the problem of quantifying the accessibility of available resources and the framework in which preference is modelled. Here, we describe these issues, document the conditional nature of preference and establish the limits of inferences that can be drawn from these analyses. We argue that preference is not interpretable as reflecting the intrinsic behavioural motivations of the animal, that estimates of preference are not directly comparable among different samples of availability and that preference is not necessarily correlated with the value of habitat to the animal. We also suggest that preference is context-dependent and that functional responses in preference resulting from changing availability are expected. We conclude by describing advances in analytical methods that begin to resolve these issues.

Invasion ecology urgently requires predictive methodologies that can forecast the ecological impacts of existing, emerging and potential invasive species. We argue that many ecologically damaging invaders are characterised by their more efficient use of resources. Consequently, comparison of the classical ‘functional response’ (relationship between resource use and availability) between invasive and trophically analogous native species may allow prediction of invader ecological impact. We review the utility of species trait comparisons and the history and context of the use of functional responses in invasion ecology, then present our framework for the use of comparative functional responses. We show that functional response analyses, by describing the resource use of species over a range of resource availabilities, avoids many pitfalls of ‘snapshot’ assessments of resource use. Our framework demonstrates how comparisons of invader and native functional responses, within and between Type II and III functional responses, allow testing of the likely population-level outcomes of invasions for affected species. Furthermore, we describe how recent studies support the predictive capacity of this method; for example, the invasive ‘bloody red shrimp’ Hemimysis anomala shows higher Type II functional responses than native mysids and this corroborates, and could have predicted, actual invader impacts in the field. The comparative functional response method can also be used to examine differences in the impact of two or more invaders, two or more populations of the same invader, and the abiotic (e.g. temperature) and biotic (e.g. parasitism) context-dependencies of invader impacts. Our framework may also address the previous lack of rigour in testing major hypotheses in invasion ecology, such as the ‘enemy release’ and ‘biotic resistance’ hypotheses, as our approach explicitly considers demographic consequences for impacted resources, such as native and invasive prey species. We also identify potential challenges in the application of comparative functional responses in invasion ecology. These include incorporation of numerical responses, multiple predator effects and trait-mediated indirect interactions, replacement versus non-replacement study designs and the inclusion of functional responses in risk assessment frameworks. In future, the generation of sufficient case studies for a meta-analysis could test the overall hypothesis that comparative functional responses can indeed predict invasive species impacts.

The curvature of generalized Holling type functional response curves is controlled by a shape parameter b yielding hyperbolic type II (b = 1) to increasingly sigmoid type III (b > 1) responses. Empirical estimates of b vary considerably across taxa. Larger consumer–resource body mass ratios have been suggested to generate more pronounced type III responses and therefore to promote dynamic stability. The dependence of consumer–resource stability on b has, however, not been systematically explored, and the accurate empirical determination of b is challenging. Specifically, the shape of the functional response of the pelagic grazer Daphnia feeding on phytoplankton, and its consequences for stability, remain controversial. We derive a novel analytical condition relating b to local stability of consumer–resource interactions and use it to predict stability of empirically parameterized models of Daphnia and phytoplankton under enrichment. Functional response parameters were experimentally derived for two species of Daphnia feeding separately on single cultures of two different phytoplankton species. All experimentally studied Daphnia–algae systems exhibited type III responses. Parameterized type III responses are predicted to stabilize the modeled Daphnia–phytoplankton dynamics in some species pairs but not in others. Remarkably, stability predictions differ depending on whether functional response parameters are derived from clearance vs. ingestion rates. Accurate parameter estimation may therefore require fitting to both rates. In addition, our estimates of b for filter‐feeding Daphnia are much smaller than predicted for actively hunting predators at similar consumer–resource body mass ratios. This suggests that the relationship between functional response shape and body mass ratios may vary with predation mode.

Motivated by recent work studying massive imaging data in the neuroimaging literature, we propose multivariate varying coefficient models (MVCM) for modeling the relation between multiple functional responses and a set of covariates. We develop several statistical inference procedures for MVCM and systematically study their theoretical properties. We first establish the weak convergence of the local linear estimate of coefficient functions, as well as its asymptotic bias and variance, and then we derive asymptotic bias and mean integrated squared error of smoothed individual functions and their uniform convergence rate. We establish the uniform convergence rate of the estimated covariance function of the individual functions and its associated eigenvalue and eigenfunctions. We propose a global test for linear hypotheses of varying coefficient functions, and derive its asymptotic distribution under the null hypothesis. We also propose a simultaneous confidence band for each individual effect curve. We conduct Monte Carlo simulation to examine the finite-sample performance of the proposed procedures. We apply MVCM to investigate the development of white matter diffusivities along the genu tract of the corpus callosum in a clinical study of neurodevelopment.

Invasive alien species (IAS) can cause substantive ecological impacts, and the role of temperature in mediating these impacts may become increasingly significant in a changing climate. Habitat conditions and physiological optima offer predictive information for IAS impacts in novel environments. Here, using meta-analysis and laboratory experiments, we tested the hypothesis that the impacts of IAS in the field are inversely correlated with the difference in their ambient and optimal temperatures. A meta-analysis of 29 studies of consumptive impacts of IAS in inland waters revealed that the impacts of fishes and crustaceans are higher at temperatures that more closely match their thermal growth optima. In particular, the maximum impact potential was constrained by increased differences between ambient and optimal temperatures, as indicated by the steeper slope of a quantile regression on the upper 25th percentile of impact data compared to that of a weighted linear regression on all data with measured variances. We complemented this study with an experimental analysis of the functional response (the relationship between predation rate and prey supply) of two invasive predators (freshwater mysid shrimp, Hemimysis anomala and Mysis diluviana ) across relevant temperature gradients; both of these species have previously been found to exert strong community-level impacts that are corroborated by their functional responses to different prey items. The functional response experiments showed that maximum feeding rates of H. anomala and M. diluviana have distinct peaks near their respective thermal optima. Although variation in impacts may be caused by numerous abiotic or biotic habitat characteristics, both our analyses point to temperature as a key mediator of IAS impact levels in inland waters and suggest that IAS management should prioritize habitats in the invaded range that more closely match the thermal optima of targeted invaders.

Describing the mechanisms that drive variation in species interaction strengths is central to understanding, predicting, and managing community dynamics. Multiple factors have been linked to trophic interaction strength variation, including species densities, species traits, and abiotic factors. Yet most empirical tests of the relative roles of multiple mechanisms that drive variation have been limited to simplified experiments that may diverge from the dynamics of natural food webs. Here, we used a field-based observational approach to quantify the roles of prey density, predator density, predator-prey body-mass ratios, prey identity, and abiotic factors in driving variation in feeding rates of reticulate sculpin (Cottus perplexus). We combined data on over 6,000 predator-prey observations with prey identification time functions to estimate 289 prey-specific feeding rates at nine stream sites in Oregon. Feeding rates on 57 prey types showed an approximately log-normal distribution, with few strong and many weak interactions. Model selection indicated that prey density, followed by prey identity, were the two most important predictors of prey-specific sculpin feeding rates. Feeding rates showed a positive relationship with prey taxon densities that was inconsistent with predator saturation predicted by current functional response models. Feeding rates also exhibited four orders-of-magnitude in variation across prey taxonomic orders, with the lowest feeding rates observed on prey with significant anti-predator defenses. Body-mass ratios were the third most important predictor variable, showing a hump-shaped relationship with the highest feeding rates at intermediate ratios. Sculpin density was negatively correlated with feeding rates, consistent with the presence of intraspecific predator interference. Our results highlight how multiple co-occurring drivers shape trophic interactions in nature and underscore ways in which simplified experiments or reliance on scaling laws alone may lead to biased inferences about the structure and dynamics of species-rich food webs.