Beware z-scores

Song, Rohr, and Saavedra (2017) have proposed a methodology to compare network properties across systems with different sizes and constraints, in response to the fact that z‐scores cannot be used for such purposes. Simmons, Hoeppke, and Sutherland (2019) have shown that part of the methodology can be improved. Here, we show that all previous results hold and are strengthened by the new methodology

A guideline to study the feasibility domain of multi-trophic and changing ecological communities

The feasibility domain of an ecological community can be described by the set of environmental abiotic and biotic conditions under which all co-occurring and interacting species in a given site and time can have positive abundances. Mathematically, the feasibility domain corresponds to the parameter space compatible with positive (feasible) solutions at equilibrium for all the state variables in a system under a given model of population dynamics. Under specific dynamics, the existence of a feasible equilibrium is a necessary condition for species persistence regardless of whether the feasible equilibrium is dynamically stable or not. Thus, the size of the feasibility domain can also be used as an indicator of the tolerance of a community to random environmental variations. This has motivated a rich research agenda to estimate the feasibility domain of ecological communities. However, these methodologies typically assume that species interactions are static, or that input and output energy flows on each trophic level are unconstrained. Yet, this is different to how communities behave in nature. Here, we present a step-by-step quantitative guideline providing illustrative examples, computational code, and mathematical proofs to study systematically the feasibility domain of ecological communities under changes of interspecific interactions and subject to different constraints on the trophic energy flows. This guideline covers multi-trophic communities that can be formed by any type of interspecific interactions. Importantly, we show that the relative size of the feasibility domain can significantly change as a function of the biological information taken into consideration. We believe that the availability of these methods can allow us to increase our understanding about the limits at which ecological communities may no longer tolerate further environmental perturbations, and can facilitate a stronger integration of theoretical and empirical research

Data from: Will a small randomly-assembled community be feasible and stable?

How likely is that few species can randomly assemble into a feasible and stable community? Some studies have answered that as long as the community is feasible, it will nearly always be stable. In contrast, other studies have answered that the likelihood is almost null. Here, we show that the origin of this debate has been the underestimation of the association of the parameter space of intrinsic growth rates with the feasibility and stability properties of small randomly-assembled communities. In particular, we demonstrate that not all parameterizations and sampling distributions of intrinsic growth rates lead to the same probabilities of stability and feasibility, which could mistakenly lead to under or overestimate the stability properties of feasible communities. Additionally, we find that stability imposes a filtering of species abundances towards more even distributions in small feasible randomly-assembled communities. This indicates that the stability of feasible communities is inherently linked to the starting distribution of species abundances, a characteristic that many times has been ignored, but should be incorporated in manageable lab and field experiments. Overall, these findings show that a more systematic exploration of the feasible parameter space is necessary to derive general conclusions about the stability properties of ecological communitie

Telling ecological networks apart by their structure: An environment-dependent approach.

The network architecture of an ecological community describes the structure of species interactions established in a given place and time. It has been suggested that this architecture presents unique features for each type of ecological interaction: e.g., nested and modular architectures would correspond to mutualistic and antagonistic interactions, respectively. Recently, Michalska-Smith and Allesina (2019) proposed a computational challenge to test whether it is indeed possible to differentiate ecological interactions based on network architecture. Contrary to the expectation, they found that this differentiation is practically impossible, moving the question to why it is not possible to differentiate ecological interactions based on their network architecture alone. Here, we show that this differentiation becomes possible by adding the local environmental information where the networks were sampled. We show that this can be explained by the fact that environmental conditions are a confounder of ecological interactions and network architecture. That is, the lack of association between network architecture and type of ecological interactions changes by conditioning on the local environmental conditions. Additionally, we find that environmental conditions are linked to the stability of ecological networks, but the direction of this effect depends on the type of interaction network. This suggests that the association between ecological interactions and network architectures exists, but cannot be fully understood without attention to the environmental conditions acting upon them

Will a small randomly assembled community be feasible and stable?

How likely is it that few species can randomly assemble into a feasible and stable community? Some studies have answered that as long as the community is feasible, it will nearly always be stable. In contrast, other studies have answered that the likelihood is almost null. Here, we show that the origin of this debate has been the underestimation of the association of the parameter space of intrinsic growth rates with the feasibility and stability properties of small randomly‐assembled communities. In particular, we demonstrate that not all parameterizations and sampling distributions of intrinsic growth rates lead to the same probabilities of stability and feasibility, which could mistakenly lead to under‐ or overestimate the stability properties of feasible communities. Additionally, we find that stability imposes a filtering of species abundances “towards” more even distributions in small feasible randomly‐assembled communities. This indicates that the stability of feasible communities is inherently linked to the starting distribution of species abundances, a characteristic that many times has been ignored, but should be incorporated in manageable lab and field experiments. Overall, the return to this debate is a central reminder that a more systematic exploration of the feasible parameter space is necessary to derive general conclusions about the stability properties of ecological communities. Keywords: feasibility; intrinsic growth rates; random matrices; small communities; species abundances; stabilit

Bridging parametric and nonparametric measures of species interactions unveils new insights of non‐equilibrium dynamics

A central theme in ecological research is to understand how species interactions contribute to community dynamics. Species interactions are the basis of parametric (model-driven) and nonparametric (model-free) approaches in theoretical and empirical work. However, despite their different interpretations across these approaches, these measures have occasionally been used interchangeably, limiting our opportunity to use their differences to gain new insights about ecological systems. Here, we revisit two of the most used measures across these approaches: species interactions measured as constant direct effects (typically used in parametric approaches) and local aggregated effects (typically used in nonparametric approaches). We show two fundamental properties of species interactions that cannot be revealed without bridging these definitions. First, we show that the local aggregated intraspecific effect summarizes all potential pathways through which one species impacts itself, which are likely to be negative even without any constant direct self-regulation mechanism. This property has implications for the long-held debate on how communities can be stabilized when little evidence of self-regulation has been found among higher-trophic species. Second, we show that a local aggregated interspecific effect between two species is correlated with the constant direct interspecific effect if and only if the population dynamics do not have any higher-order direct effects. This other property provides a rigorous methodology to detect direct higher-order effects in the field and experimental data. Overall, our findings illustrate a practical route to gain further insights about non-equilibrium ecological dynamics and species interactionsNSF (Grant DEB- 2024349

R-code

The file contains: The R-code: (1) To generate a small random interaction matrix. (2) To generare the parameterization of intrinsic growth rates corresponding to the random interaction matrix. This vector of intrinsic growth rates can be generated either randomly with no other constraints, all elements with the same value, randomly inside the feasibility domain, or the vector that corresponds to the geometric centroid of the feasibility domain. (3) To check whether the community with the given random interaction matrix and chosen parameterization is feasible and stable

Statistical analysis for year 1996;Statistical analysis for year 1997 from Structural stability as a consistent predictor of phenological events

The timing of the first and last seasonal appearance of a species in a community typically follows a pattern that is governed by temporal factors. While it has been shown that changes in the environment are linked to phenological changes, the direction of this link appears elusive and context-dependent. Thus, finding consistent predictors of phenological events is of central importance for a better assessment of expected changes in the temporal dynamics of ecological communities. Here we introduce a measure of structural stability derived from species interaction networks as an estimator of the expected range of environmental conditions compatible with the existence of a community. We test this measure as a predictor of changes in species richness recorded on a daily basis in a high-arctic plant–pollinator community during two spring seasons. We find that our measure of structural stability is the only consistent predictor of changes in species richness among different ecological and environmental variables. Our findings suggest that measures based on the notion of structural stability can synthesize the expected variation of environmental conditions tolerated by a community, and explain more consistently the phenological changes observed in ecological communities

code

"main text.nb" (written in Mathematica) reproduces all five figures in the main text. "Supplement 7.R" (written in R) reproduces Supplement 7