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Abstract Spatial environmental heterogeneity maintains species diversity in ecological communities, but it is difficult to model and its effects on individuals are often ignored in empirical studies of species coexistence. This is problematic because many studies estimate plant competition at the neighbourhood level, and modelling individual demographic responses without neighbourhood‐level environmental data could lead to biased results. We aimed to understand how and when ignoring spatial heterogeneity produces biased estimates of the strength of intraspecific and interspecific competition and, therefore, incorrect predictions about species coexistence.First, we used a spatially explicit individual‐based model (IBM), with neighbourhood competition depending on neighbour size and distance, to simulate two plant populations that coexist via differing responses to spatial heterogeneity. Next, we mimicked the action of ecologists collecting data in observational studies or short‐term experimental studies and sampled from the simulated populations. Then, we fitted demographic models for each species including parameters for the intensity of intra‐ and interspecific competitionwithoutinformation about spatial heterogeneity. Finally, we simulated invasions of species at low density using the IBM and estimated parameters to compute invasion growth rates (IGR).Demographic modelling with data from simulated observational studies resulted in underestimates of intraspecific competition and overestimates of interspecific competition. When we simulated invasions with the IBM and those estimated competition coefficients, the IGRs were negative or around zero and incorrectly predicted competitive exclusion. In simulated experimental studies, the estimates of competition were more accurate, but the model still incorrectly predicted competitive exclusion.Synthesis: Our study demonstrates how biases can arise in neighbourhood competition studies and highlights the importance of explicitly incorporating spatial heterogeneity into empirical coexistence studies. Failure to account for environmental heterogeneity at the level of individual plants may lead to mistaken conclusions about coexistence outcomes. 

Abstract Many animals show avoidance behavior in response to disease. For instance, in some species of frogs, individuals that survive infection of the fungal disease chytridiomycosis may learn to avoid areas where the pathogen is present. As chytridiomycosis has caused substantial declines in many amphibian populations worldwide, it is a highly relevant example for studying these behavioral dynamics. Here we develop compartmental ODE models to study the epidemiological consequences of avoidance behavior of animals in response to waterborne infectious diseases. Individuals with avoidance behavior are less likely to become infected, but avoidance may also entail increased risk of mortality. We compare the outbreak dynamics with avoidance behavior that is innate (present from birth) or learned (gained after surviving infection). We also consider how management to induce learned avoidance might affect the resulting dynamics. Using methods from dynamical systems theory, we calculate the basic reproduction number$$R_0$$for each model, analyze equilibrium stability of the systems, and perform a detailed bifurcation analysis. We show that disease persistence when$$R_0< 1$$is possible with learned avoidance, but not with innate avoidance. Our results imply that management to induce behavioral avoidance can actually cause such a scenario, but it is also less likely to occur for high-mortality diseases (e.g., chytridiomycosis). Furthermore, the learned avoidance model demonstrates a variety of codimension-1 and -2 bifurcations not found in the innate avoidance model. Simulations with parameters based on chytridiomycosis are used to demonstrate these features and compare the outcomes with innate, learned, and no avoidance behavior. 

Abstract Integral projection models (IPMs) are widely used for studying continuously size‐structured populations. IPMs require a growth sub‐model that describes the probability of future size conditional on current size and any covariates. Most IPM studies assume that this distribution is Gaussian, despite calls for non‐Gaussian models that accommodate skewness and excess kurtosis. We provide a general workflow for accommodating non‐Gaussian growth patterns while retaining important covariates and random effects. Our approach emphasizes visual diagnostics from pilot Gaussian models and quantile‐based metrics of skewness and kurtosis that guide selection of a non‐Gaussian alternative, if necessary. Across six case studies, skewness and excess kurtosis were common features of growth data, and non‐Gaussian models consistently generated simulated data that were more consistent with real data than pilot Gaussian models. However, effects of “improved” growth modeling on IPM results were moderate to weak and differed in direction or magnitude between different outputs from the same model. Using tools not available when IPMs were first developed, it is now possible to fit non‐Gaussian models to growth data without sacrificing ecological complexity. Doing so, as guided by careful interrogation of the data, will result in models that better represent the populations for which they are intended. 

Abstract Model distillation has been a popular method for producing interpretable machine learning. It uses an interpretable “student” model to mimic the predictions made by the black box “teacher” model. However, when the student model is sensitive to the variability of the data sets used for training even when keeping the teacher fixed, the corresponded interpretation is not reliable. Existing strategies stabilize model distillation by checking whether a large enough sample of pseudo-data is generated to reliably reproduce student models, but methods to do so have so far been developed separately for each specific class of student model. In this paper, we develop a generic approach for stable model distillation based on central limit theorem for the estimated fidelity of the student to the teacher. We start with a collection of candidate student models and search for candidates that reasonably agree with the teacher. Then we construct a multiple testing framework to select a sample size such that the consistent student model would be selected under different pseudo samples. We demonstrate the application of our proposed approach on three commonly used intelligible models: decision trees, falling rule lists and symbolic regression. Finally, we conduct simulation experiments on Mammographic Mass and Breast Cancer datasets and illustrate the testing procedure throughout a theoretical analysis with Markov process. The code is publicly available athttps://github.com/yunzhe-zhou/GenericDistillation. 

Abstract Environmental factors and individual attributes, and their interactions, impact survival, growth and reproduction of an individual throughout its life. In the clonal rotiferBrachionus, low food conditions delay reproduction and extend lifespan. This species also exhibits maternal effect senescence; the offspring of older mothers have lower survival and reproductive output. In this paper, we explored the population consequences of the individual‐level interaction of maternal age and low food availability.We built matrix population models for both ad libitum and low food treatments, in which individuals are classified both by their age and maternal age. Low food conditions reduced population growth rate () and shifted the population structure to older maternal ages, but did not detectably impact individual lifetime reproductive output.We analysed hypothetical scenarios in which reduced fertility or survival led to approximately stationary populations that maintained the shape of the difference in demographic rates between the ad libitum and low food treatments. When fertility was reduced, the populations were more evenly distributed across ages and maternal ages, while the lower‐survival models showed an increased concentration of individuals in the youngest ages and maternal ages.Using life table response experiment analyses, we compared populations grown under ad libitum and low food conditions in scenarios representing laboratory conditions, reduced fertility and reduced survival. In the laboratory scenario, the reduction in population growth rate under low food conditions is primarily due to decreased fertility in early life. In the lower‐fertility scenario, contributions from differences in fertility and survival are more similar, and show trade‐offs across both ages and maternal ages. In the lower‐survival scenario, the contributions from decreased fertility in early life again dominate the difference in .These results demonstrate that processes that potentially benefit individuals (e.g. lifespan extension) may actually reduce fitness and population growth because of links with other demographic changes (e.g. delayed reproduction). Because the interactions of maternal age and low food availability depend on the population structure, the fitness consequences of an environmental change can only be fully understood through analysis that takes into account the entire life cycle. 

Abstract Varying coefficient models are a flexible extension of generic parametric models whose coefficients are functions of a set of effect-modifying covariates instead of fitted constants. They are capable of achieving higher model complexity while preserving the structure of the underlying parametric models, hence generating interpretable predictions. In this paper we study the use of gradient boosted decision trees as those coefficient-deciding functions in varying coefficient models with linearly structured outputs. In contrast to the traditional choices of splines or kernel smoothers, boosted trees are more flexible since they require no structural assumptions in the effect modifier space. We introduce our proposed method from the perspective of a localized version of gradient descent, prove its theoretical consistency under mild assumptions commonly adapted by decision tree research, and empirically demonstrate that the proposed tree boosted varying coefficient models achieve high performance qualified by their training speed, prediction accuracy and intelligibility as compared to several benchmark algorithms. 

Abstract Matrix population models are frequently built and used by ecologists to analyse demography and elucidate the processes driving population growth or decline. Life Table Response Experiments (LTREs) are comparative analyses that decompose the realized difference or variance in population growth rate () into contributions from the differences or variances in the vital rates (i.e. the matrix elements). Since their introduction, LTREs have been based on approximations and have not included biologically relevant interaction terms.We used the functional analysis of variance framework to derive an exact LTRE method, which calculates the exact response of to the difference or variance in a given vital rate, for all interactions among vital rates—including higher‐order interactions neglected by the classical methods. We used the publicly available COMADRE and COMPADRE databases to perform a meta‐analysis comparing the results of exact and classical LTRE methods. We analysed 186 and 1487 LTREs for animal and plant matrix population models, respectively.We found that the classical methods often had small errors, but that very high errors were possible. Overall error was related to the difference or variance in the matrices being analysed, consistent with the Taylor series basis of the classical method. Neglected interaction terms accounted for most of the errors in fixed design LTRE, highlighting the importance of two‐way interaction terms. For random design LTRE, errors in the contribution terms present in both classical and exact methods were comparable to errors due to neglected interaction terms. In most examples we analysed, evaluating exact contributions up to three‐way interaction terms was sufficient for interpreting 90% or more of the difference or variance in .Relative error, previously used to evaluate the accuracy of classical LTREs, is not a reliable metric of how closely the classical and exact methods agree. Error compensation between estimated contribution terms and neglected contribution terms can lead to low relative error despite faulty biological interpretation. Trade‐offs or negative covariances among matrix elements can lead to high relative error despite accurate biological interpretation. Exact LTRE provides reliable and accurate biological interpretation, and the R packageexactLTREmakes the exact method accessible to ecologists. 

Species distribution models (SDMs) are frequently data‐limited. In aquatic habitats, emerging environmental DNA (eDNA) sampling methods can be quicker and more cost‐efficient than traditional count and capture surveys, but their utility for fitting SDMs is complicated by dilution, transport, and loss processes that modulate DNA concentrations and mix eDNA from different locations. Past models for estimating organism densities from measured species‐specific eDNA concentrations have accounted for how these processes affect expected concentrations. We built off this previous work to construct a linear hierarchical model that also accounts for how they give rise to spatially correlated concentration errors. We applied our model to 60 simulated stream networks and three types of species niches in order to answer two questions: 1) what is the D‐optimal sampling design, i.e. where should eDNA samples be positioned to most precisely estimate species–environment relationships? and 2) How does parameter estimation accuracy depend on the stream network's topological and hydrologic properties? We found that correcting for eDNA dynamics was necessary to obtain consistent parameter estimates, and that relative to a heuristic benchmark design, optimizing sampling locations improved design efficiency by an average of 41.5%. Samples in the D‐optimal design tended to be positioned near downstream ends of stream reaches high in the watershed, where eDNA concentration was high and mostly from homogeneous source areas, and they collectively spanned the full ranges of covariates. When measurement error was large, it was often optimal to collect replicate samples from high‐information reaches. eDNA‐based estimates of species–environment regression parameters were most precise in stream networks that had many reaches, large geographic size, slow flows, and/or high eDNA loss rates. Our study demonstrates the importance and viability of accounting for eDNA dilution, transport, and loss in order to optimize sampling designs and improve the accuracy of eDNA‐based species distribution models. 

Species distribution models (SDMs) are frequently data-limited. In aquatic habitats, emerging environmental DNA (eDNA) sampling methods can be quicker and more cost-efficient than traditional count and capture surveys, but their utility for fitting SDMs is complicated by dilution, transport, and loss processes that modulate DNA concentrations and mix eDNA from different locations. Past models for estimating organism densities from measured species-specific eDNA concentrations have accounted for how these processes affect expected concentrations. We built off this previous work to construct a linear hierarchical model that also accounts for how they give rise to spatially correlated concentration errors. We applied our model to 60 simulated stream networks and three types of species niches in order to answer two questions: 1) what is the D-optimal sampling design, i.e. where should eDNA samples be positioned to most precisely estimate species–environment relationships? and 2) How does parameter estimation accuracy depend on the stream network’s topological and hydrologic properties? We found that correcting for eDNA dynamics was necessary to obtain consistent parameter estimates, and that relative to a heuristic benchmark design, optimizing sampling locations improved design efficiency by an average of 41.5%. Samples in the D-optimal design tended to be positioned near downstream ends of stream reaches high in the watershed, where eDNA concentration was high and mostly from homogeneous source areas, and they collectively spanned the full ranges of covariates. When measurement error was large, it was often optimal to collect replicate samples from high-information reaches. eDNA-based estimates of species–environment regression parameters were most precise in stream networks that had many reaches, large geographic size, slow flows, and/or high eDNA loss rates. Our study demonstrates the importance and viability of accounting for eDNA dilution, transport, and loss in order to optimize sampling designs and improve the accuracy of eDNA-based species distribution models. 

Abstract: One strand of modern coexistence theory (MCT) partitions invader growth rates (IGR) to quantify how different mechanisms contribute to species coexistence, highlighting fluctuation‐dependent mechanisms. A general conclusion from the classical analytic MCT theory is that coexistence mechanisms relying on temporal variation (such as the temporal storage effect) are generally less effective at promoting coexistence than mechanisms relying on spatial or spatiotemporal variation (primarily growth‐density covariance). However, the analytic theory assumes continuous population density, and IGRs are calculated for infinitesimally rare invaders that have infinite time to find their preferred habitat and regrow, without ever experiencing intraspecific competition. Here we ask if the disparity between spatial and temporal mechanisms persists when individuals are, instead, discrete and occupy finite amounts of space. We present a simulation‐based approach to quantifying IGRs in this situation, building on our previous approach for spatially non‐varying habitats. As expected, we found that spatial mechanisms are weakened; unexpectedly, the contribution to IGR from growth‐density covariance could even become negative, opposing coexistence. We also found shifts in which demographic parameters had the largest effect on the strength of spatial coexistence mechanisms. Our substantive conclusions are statements about one model, across parameter ranges that we subjectively considered realistic. Using the methods developed here, effects of individual discreteness should be explored theoretically across a broader range of conditions, and in models parameterized from empirical data on real communities.