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Research into the processes governing species richness has often assumed that the environment is fixed, whereas realistic environments are often characterised by random fluctuations over time. This temporal environmental stochasticity (TES) changes the demographic rates of species populations, with cascading effects on community dynamics and species richness. Theoretical and applied studies have used process-based mathematical models to determine how TES affects species richness, but under a variety of frameworks. Here, we critically review such studies to synthesise their findings and draw general conclusions. We first provide a broad mathematical framework encompassing the different ways in which TES has been modelled. We then review studies that have analysed models with TES under the assumption of negligible interspecific interactions, such that a community is conceptualised as the sum of independent species populations. These analyses have highlighted how TES can reduce species richness by increasing the frequency at which a species becomes rare and therefore prone to extinction. Next, we review studies that have relaxed the assumption of negligible interspecific interactions. To simplify the corresponding models and make them analytically tractable, such studies have used mean-field theory to derive fixed parameters representing the typical strength of interspecific interactions under TES. The resulting analyses have highlighted community-level effects that determine how TES affects species richness, for species that compete for a common limiting resource. With short temporal correlations of environmental conditions, a non-linear averaging effect of interspecific competition strength over time gives an increase in species richness. In contrast, with long temporal correlations of environmental conditions, strong selection favouring the fittest species between changes in environmental conditions results in a decrease in species richness. We compare such results with those from invasion analysis, which examines invasion growth rates (IGRs) instead of species richness directly. Qualitative differences sometimes arise because the IGR is the expected growth rate of a species when it is rare, which does not capture the variation around this mean or the probability of the species becoming rare. Our review elucidates key processes that have been found to mediate the negative and positive effects of TES on species richness, and by doing so highlights key areas for future research., Competing Interests: Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2023 Elsevier Inc. All rights reserved.)

In a critique of our recent review on measuring habitat complexity in ecology, Madin et al. (2023) advocate the use of fractal dimension in ecology and defend their geometric constraint theory of habitat complexity. We explain the flaws in their arguments and highlight points where they misinterpreted our statements., (© 2023 The Authors. Ecology Letters published by John Wiley & Sons Ltd.)

Ecology lacks a holistic approach that can model phenomena across temporal and spatial scales, largely because of the challenges in modelling systems with a large number of interacting constituents. This hampers our understanding of complex ecosystems and the impact that human interventions (e.g., deforestation, wildlife harvesting and climate change) have on them. Here we use density functional theory, a computational method for many-body problems in physics, to develop a computational framework for ecosystem modelling. Our methods accurately fit experimental and synthetic data of interacting multi-species communities across spatial scales and can project to unseen data. As the key concept we establish and validate a cost function that encodes the trade-offs between the various ecosystem components. We show how this single general modelling framework delivers predictions on par with established, but specialised, approaches for systems from predatory microbes to territorial flies to tropical tree communities. Our density functional framework thus provides a promising avenue for advancing our understanding of ecological systems., (© 2023. The Author(s).)

Habitat complexity has been considered a key driver of biodiversity and other ecological phenomena for nearly a century. However, there is still no consensus over the definition of complexity or how to measure it. Up-to-date and clear guidance on measuring complexity is urgently needed, particularly given the rise of remote sensing and advent of technologies that allow environments to be scanned at unprecedented spatial extents and resolutions. Here we review how complexity is measured in ecology. We provide a framework for metrics of habitat complexity, and for the related concept of spatial heterogeneity. We focus on the two most commonly used complexity metrics in ecology: fractal dimension and rugosity. We discuss the pros and cons of these metrics using practical examples from our own empirical data and from simulations. Fractal dimension is particularly widely used, and we provide a critical examination of it drawing on research from other scientific fields. We also discuss informational metrics of complexity and their potential benefits. We chart a path forward for research on measuring habitat complexity by presenting, as a guide, sets of essential and desirable criteria that a metric of complexity should possess. Lastly, we discuss the applied significance of our review., (© 2022 The Authors. Ecology Letters published by John Wiley & Sons Ltd.)

Isolating the effects of fragmentation per se (i.e., spatial configuration of habitat patches) on species richness is an ongoing challenge as habitat configuration often covaries with the amount of habitat. Consequently, there is a lack of experimental evidence for configurational effects on species richness in the whole landscape. Here, we developed a novel experimental system for testing the independent and interactive effects of habitat area and configuration on tropical intertidal species richness. Our results confirmed the expectation that average species richness would increase monotonically with habitat area. More intriguingly, we found mixed evidence for a non-monotonic relationship between species richness and fragmentation per se, with the highest richness at intermediate fragmentation configuration, that is, when habitat tiles were placed in a "several-small" configuration. The effect of habitat configuration was not due to passive sampling (since area was controlled for), variation in total individual abundance, or niche specialization of species to different landscape configurations. We postulate that a combination of processes, including local negative density dependence and dispersal limitation, could give rise to the observed pattern. We emphasize the importance of considering configurational effects on biodiversity at broader spatial scales and for more experimental research to delve into the mechanisms driving the patterns seen here., (© 2019 The Authors. Ecology published by Wiley Periodicals, Inc. on behalf of Ecological Society of America.)

Local adaptation to rare habitats is difficult due to gene flow, but can occur if the habitat has higher productivity. Differences in offspring phenotypes have attracted little attention in this context. We model a scenario where the rarer habitat improves offspring's later competitive ability - a carryover effect that operates on top of local adaptation to one or the other habitat type. Assuming localised dispersal, so the offspring tend to settle in similar habitat to the natal type, the superior competitive ability of offspring remaining in the rarer habitat hampers immigration from the majority habitat. This initiates a positive feedback between local adaptation and trait divergence, which can thereafter be reinforced by coevolution with dispersal traits that match ecotype to habitat type. Rarity strengthens selection on dispersal traits and promotes linkage disequilibrium between locally adapted traits and ecotype-habitat matching dispersal. We propose that carryover effects may initiate isolation by ecology., (© 2018 The Authors. Ecology Letters published by CNRS and John Wiley & Sons Ltd.)

LaManna et al (Reports, 30 June 2017, p. 1389) found higher conspecific negative density dependence in tree communities at lower latitudes, yielding a possible mechanistic explanation for the latitudinal diversity gradient. We show that their results are artifacts of a selective data transformation and a forced zero intercept in their fitted model. A corrected analysis shows no latitudinal trend., (Copyright © 2018, American Association for the Advancement of Science.)

Many problems in ecology require the estimation of rates of dispersal of individuals or propagules across physical boundaries. Such problems arise in invasion ecology, forest dynamics, and the neutral theory of biodiversity. In a forest plot, for example, one might ask what proportion of the seed rain originates from outside the plot. A recent study presented analytical approximations that relate the rate of immigration across a boundary to plot geometry and to the parameters of a dispersal kernel in one- and two-dimensional environments. In this study, we provide a more rigorous derivation of these expressions and we derive a more general expression that applies in environments of arbitrary dimension. We discuss potential applications of the one-, two-, and three-dimensional results to ecological problems.

Info-gap decision theory facilitates decision making for problems in which uncertainty is large and probability distributions of uncertain variables are unknown. The info-gap framework allows the decision maker to maximize robustness to failure in the presence of uncertainty, where uncertainty is in the parameters of the model and failure is defined as the model output falling below some minimally acceptable performance threshold. Info-gap theory has found particular application to problems in conservation biology and ecological economics. In this study, we applied info-gap theory to an ecosystem services tradeoff case study in which a decision maker aiming to maximize ecosystem service investment returns must choose between two alternative land uses: native vegetation conservation or the establishment of an exotic timber plantation. The uncertain variables are the carbon price and the water price. With a "no-information" uncertainty model that assumes equal relative uncertainty across both variables, info-gap theory identifies a minimally acceptable reward threshold above which conservation is preferred, but below which plantation establishment is preferred. However, with an uncertainty model that allows the carbon price to be substantially more uncertain than the water price, conservation of native vegetation becomes an economically more robust investment option than establishing alien pine plantations. We explored the sensitivity of the results to the use of alternative uncertainty models, including asymmetric uncertainty in individual variables. We emphasize the general finding that the results of info-gap analyses can be sensitive to the choice of uncertainty model and that, therefore, future applications to ecological problems should be careful to incorporate all available qualitative and quantitative information relating to uncertainties or should at least justify the no-information uncertainty model.

Transitions in ecological systems often occur without apparent warning, and may represent shifts between alternative persistent states. Decreasing ecological resilience (the size of the basin of attraction around a stable state) can signal an impending transition, but this effect is difficult to measure in practice. Recent research has suggested that a decreasing rate of recovery from small perturbations (critical slowing down) is a good indicator of ecological resilience. Here we use analytical techniques to draw general conclusions about the conditions under which critical slowing down provides an early indicator of transitions in two-species predator-prey and competition models. The models exhibit three types of transition: the predator-prey model has a Hopf bifurcation and a transcritical bifurcation, and the competition model has two saddle-node bifurcations (in which case the system exhibits hysteresis) or two transcritical bifurcations, depending on the parameterisation. We find that critical slowing down is an earlier indicator of the Hopf bifurcation in predator-prey models in which prey are regulated by predation rather than by intrinsic density-dependent effects and an earlier indicator of transitions in competition models in which the dynamics of the rare species operate on slower timescales than the dynamics of the common species. These results lead directly to predictions for more complex multi-species systems, which can be tested using simulation models or real ecosystems.

The importance of incorporating landscape dynamics into population viability analysis (PVA) has previously been acknowledged, but the need to repeat the landscape generation process to encapsulate landscape stochasticity in model outputs has largely been overlooked. Reasons for this are that (1) there is presently no means for quantifying the relative effects of landscape stochasticity and population stochasticity on model outputs, and therefore no means for determining how to allocate simulation time optimally between the two; and (2) the process of generating multiple landscapes to incorporate landscape stochasticity is tedious and user-intensive with current PVA software. Here we demonstrate that landscape stochasticity can be an important source of variance in model outputs. We solve the technical problems with incorporating landscape stochasticity by deriving a formula that gives the optimal ratio of population simulations to landscape simulations for a given model, and by providing a computer program that incorporates the formula and automates multiple landscape generation in a dynamic landscape metapopulation (DLMP) model. Using a case study of a bird population, we produce estimates of DLMP model output parameters that are up to four times more precise than those estimated from a single landscape in the same amount of total simulation time. We use the DLMP modeling software RAMAS Landscape to run the landscape and metapopulation models, though our method is general and could be applied to any PVA platform. The results of this study should motivate DLMP modelers to consider landscape stochasticity in their analyses.

Numerous studies have shown reduced performance in plants that are surrounded by neighbours of the same species1,2, a phenomenon known as conspecific negative density dependence (CNDD)3. A long-held ecological hypothesis posits that CNDD is more pronounced in tropical than in temperate forests4,5, which increases community stabilization, species coexistence and the diversity of local tree species6,7. Previous analyses supporting such a latitudinal gradient in CNDD8,9 have suffered from methodological limitations related to the use of static data10-12. Here we present a comprehensive assessment of latitudinal CNDD patterns using dynamic mortality data to estimate species-site-specific CNDD across 23 sites. Averaged across species, we found that stabilizing CNDD was present at all except one site, but that average stabilizing CNDD was not stronger toward the tropics. However, in tropical tree communities, rare and intermediate abundant species experienced stronger stabilizing CNDD than did common species. This pattern was absent in temperate forests, which suggests that CNDD influences species abundances more strongly in tropical forests than it does in temperate ones13. We also found that interspecific variation in CNDD, which might attenuate its stabilizing effect on species diversity14,15, was high but not significantly different across latitudes. Although the consequences of these patterns for latitudinal diversity gradients are difficult to evaluate, we speculate that a more effective regulation of population abundances could translate into greater stabilization of tropical tree communities and thus contribute to the high local diversity of tropical forests.