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Climate-mediated changes in the spatiotemporal distribution of thermal stress can destabilize animal populations and promote extinction risk. Using quantile, spectral, and wavelet analyses of temperature projections from the latest generation of earth system models, we show that significant regional differences are expected to arise in the way that temperatures will increase over time. When integrated into empirically-parameterized mathematical models that simulate the dynamical and cumulative effects of thermal stress on the performance of 38 global ectotherm species, the projected spatiotemporal changes in temperature fluctuations are expected to give rise to complex regional changes in population abundance and stability over the course of the 21st century. However, despite their idiosyncratic effects on stability, projected temperatures universally increase extinction risk. These results show that population changes under future climate conditions may be more extensive and complex than the current literature suggests based on the statistical relationship between biological performance and average temperature.

We demonstrate that the issues described in the Wagg et al. (2019) Comment on our paper (Pillai and Gouhier, 2019) are all due to misunderstandings about the implications of pairwise effects, the nature of the null baseline in both our framework and in the Loreau-Hector (LH) partitioning scheme (i.e., the midpoint of the monocultures), and the impact of nonlinearity on the LH partitioning results. Specifically, we show that (i) pairwise effects can be computed over any time horizon and thus do not imply stable coexistence, (ii) the midpoint of the monocultures corresponds to a neutral community so coexistence was always part of the LH baseline, and (iii) contrary to what Wagg et al. suggested, generalized diversity-interaction models do not account for (and may in fact exacerbate) the problem of nonlinearity in monocultures, which inflates the LH net biodiversity effect and generates incorrect estimates of selection and complementarity. Hence, all of our original claims about the triviality inherent in biodiversity-ecosystem functioning research and the issues with the LH partitioning scheme hold.

The Loreau and Hector (2019) Comment on our paper (Pillai and Gouhier, 2019) failed to address the two core elements of our critique, both the circularity of the BEF research program, in general, and the mathematical flaws of the Loreau-Hector partitioning scheme, in particular. Loreau and Hector avoided dealing with the first part of our critique by arguing against a non-existent claim that all biodiversity effects could be reduced to coexistence, while the mathematical flaws in the Loreau-Hector partitioning method that we described in the second part of our critique were ignored altogether. Here, we address these misconceptions and demonstrate that all of the claims that were made in our original paper hold. We conclude that (i) BEF studies need to adopt baselines that account for coexistence in order to avoid overestimating the effects of biodiversity and (ii) the LH partitioning method should not be used unless the linearity of the abundance-ecosystem functioning relationship in monocultures can be verified for all species.

We demonstrate that the conclusions drawn by Bernhard et al. (2018) regarding the ability of nonlinear averaging to accurately predict organismal performance under fluctuating temperatures are flawed because of a series of experimental and statistical issues that include the presence of a hidden treatment effect, the use of a single low frequency temperature fluctuation that could easily be tracked by the fast growing organism, and the decision to quantify performance via population growth rate, a metric that can mask significant variation in population size.

We demonstrate that the conclusions drawn by Lefcheck et al. (2019) regarding the positive effects of fish diversity on coral reef ecosystem functioning across scales are flawed because of a series of conceptual and statistical issues that include spurious correlations, the conflation of population size and species diversity effects and a failure to recognize that observing a biodiversity effect at multiple sites is not equivalent to observing it at multiple scales.

In biodiversity and ecosystem functioning (BEF) research, the Loreau-Hector (LH) statistical scheme is widely-used to partition the effect of biodiversity on ecosystem properties into a "complementarity effect" and a "selection effect". This selection effect was originally considered analogous to the selection term in the Price equation from evolutionary biology. However, a key paper published over thirteen years ago challenged this interpretation by devising a new tripartite partitioning scheme that purportedly quantified the role of selection in biodiversity experiments more accurately. This tripartite method, as well as its recent spatiotemporal extension, were both developed as an attempt to apply the Price equation in a BEF context. Here, we demonstrate that the derivation of this tripartite method, as well as its spatiotemporal extension, involve a set of incoherent and nonsensical mathematical arguments driven largely by na\"ive visual analogies with the original Price equation, that result in neither partitioning scheme quantifying any real property in the natural world. Furthermore, we show that Loreau and Hector's original selection effect always represented a true analog of the original Price selection term, making the tripartite partitioning scheme a nonsensical solution to a non-existent problem [...]

Resolving the relationship between biodiversity and ecosystem functioning has been one of the central goals of modern ecology. Early debates about the relationship were finally resolved with the advent of a statistical partitioning scheme that decomposed the biodiversity effect into a "selection" effect and a "complementarity" effect. We prove that both the biodiversity effect and its statistical decomposition into selection and complementarity are fundamentally flawed because these methods use a na\"ive null expectation based on neutrality, likely leading to an overestimate of the net biodiversity effect, and they fail to account for the nonlinear abundance-ecosystem functioning relationships observed in nature. Furthermore, under such nonlinearity no statistical scheme can be devised to partition the biodiversity effects. We also present an alternative metric providing a more reasonable estimate of biodiversity effect. Our results suggest that all studies conducted since the early 1990s likely overestimated the positive effects of biodiversity on ecosystem functioning., Comment: Keywords: biodiversity; ecosystem functioning; complementarity effect; selection effect; Jensen's inequality; Price equation; species coexistence