Jean-François Arnoldi, Michel Loreau, Bart Haegeman. The inherent multidimensionality of temporal variability: How common and rare species shape stability patterns (2019), bioRxiv, 431296, ver. 3 peer-reviewed and recommended by Peer Community in Ecology. 10.1101/431296

Kevin Cazelles and Kevin Shear McCann (2019) Diversity-Stability and the Structure of Perturbations.

In his 1972 paper “Will a Large Complex System Be Stable?” [1], May challenges the idea that large communities are more stable than small ones. This was the beginning of a fundamental debate that still structures an entire research area in ecology: the diversity-stability debate [2]. The most salient strength of May’s work was to use a mathematical argument to refute an idea based on the observations that simple communities are less stable than large ones. Using the formalism of dynamical systems and a major results on the distribution of the eigen values for random matrices, May demonstrated that the addition of random interactions destabilizes ecological communities and thus, rich communities with a higher number of interactions should be less stable. But May also noted that his mathematical argument holds true only if ecological interactions are randomly distributed and thus concluded that this must not be true! This is how the contradiction between mathematics and empirical observations led to new developments in the study of ecological networks.

Since 1972, the theoretical corpus of ecology has advanced, building on the formalism of dynamical systems, ecologists have revealed that ecological interactions are indeed not randomly distributed [3,4], but general rules are still missing and we are far from understanding what determine the exact network topology of a given community. One promising avenue is to understand the relationship between different facets of the concept of stability [5,6]. Indeed, the classical approach to determine whether a system is stable is qualitative: if a system returns to its equilibrium when it is slightly moved away from it, then the system is considered stable. But there are several other aspects that are worth scrutinizing. For instance, when a system returns to its equilibrium, one can characterize the corresponding transient dynamics [7,8], that is asking fundamental questions such as: what is the trajectory of return? How long does it take to return to the equilibrium? Another fundamental question is whether the system remains qualitatively stable when the distributions of interactions strengths change? From a biological standpoint, all of these questions matter as all these aspects of stability may partially explain the actual structure of ecological networks, and hence, frameworks that integrate several facets of stability are much needed.

The study by Arnoldi et al. [9] is a significant step towards such a framework. The strength of their formalism is threefold. First, instead of considering separately the system and its perturbations, they considering the fluctuations of a perturbed ecological systems and thus, perturbations are parts of the ecological system. Second, they use of a broad definition of perturbation that encompasses the types of perturbations (whether the individual respond synchronously or not), their intensity and their direction (how the perturbations are correlated across species). Third, they quantify the instability of the system using variability which integrates the consequences of perturbations over the whole set of species of a community: such a measure is comparable across communities and accounts for the trivial effect of the perturbations on the system dynamics.

Using this framework, the authors show that interactions within a stable community leads to a general relationship between variability and the abundance of individually perturbed species: if individuals of species respond in synchrony to a perturbation, then the more abundant the species perturbed the higher the variability of the system, but the relationship is reverse when individual respond asynchronously. A direct implications of these results for the classical debate is that the diversity-stability relationship is negative for the former type of perturbations (as in May’s seminal paper) but positive for the latter type. Hence, the rigorous work of Arnoldi and colleagues sheds a new light upon the classical debate: the nature of the perturbation regime prevailing within a community affects the slope of the diversity-stability relationships and given the vast diversity of ecological communities, this may very well be one of the reasons why the debate still endures.

From a historical perspective, it is interesting that ecologists have gone from looking at random webs to structured webs and now, in a sense, Arnoldi et al. are unpacking the role of differentially structured perturbations. The work they achieved will doubtlessly be followed by further theoretical investigations. One natural research avenue is to revisit the role of the topology of ecological networks with this framework: how the distribution of interactions and their strength affect the general relationship they unravel? Finally, this study demonstrate that the impact of the abundance of a species on the variability of the system depends on the nature of the perturbation regime and so the distribution of species abundances within a community should be determined by the prevailing perturbation regime which is a prediction that remains to be tested.

**References**

[1] May, Robert M (1972). Will a Large Complex System Be Stable? Nature 238, 413–414. doi: 10.1038/238413a0

[2] McCann, Kevin Shear (2000). The Diversity–Stability Debate. Nature 405, 228–233. doi: 10.1038/35012234

[3] Rooney, Neil, Kevin McCann, Gabriel Gellner, and John C. Moore (2006). Structural Asymmetry and the Stability of Diverse Food Webs. Nature 442, 265–269. doi: 10.1038/nature04887

[4] Jacquet, Claire, Charlotte Moritz, Lyne Morissette, Pierre Legagneux, François Massol, Philippe Archambault, and Dominique Gravel (2016). No Complexity–Stability Relationship in Empirical Ecosystems. Nature Communications 7, 12573. doi: 10.1038/ncomms12573

[5] Donohue, Ian, Helmut Hillebrand, José M. Montoya, Owen L. Petchey, Stuart L. Pimm, Mike S. Fowler, Kevin Healy, et al. (2016). Navigating the Complexity of Ecological Stability. Ecology Letters 19, 1172–1185. doi: 10.1111/ele.12648

[6] Arnoldi, Jean-François, and Bart Haegeman (2016). Unifying Dynamical and Structural Stability of Equilibria. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 472, 20150874. doi: 10.1098/rspa.2015.0874

[7] Caswell, Hal, and Michael G. Neubert (2005). Reactivity and Transient Dynamics of Discrete-Time Ecological Systems. Journal of Difference Equations and Applications 11, 295–310. doi: 10.1080/10236190412331335382

[8] Arnoldi, J-F., M. Loreau, and B. Haegeman (2016). Resilience, Reactivity and Variability: A Mathematical Comparison of Ecological Stability Measures. Journal of Theoretical Biology 389, 47–59. doi: 10.1016/j.jtbi.2015.10.012

[9] Arnoldi, Jean-Francois, Michel Loreau, and Bart Haegeman. (2019). The Inherent Multidimensionality of Temporal Variability: How Common and Rare Species Shape Stability Patterns.” BioRxiv, 431296, ver. 3 peer-reviewed and recommended by PCI Ecology. doi: 10.1101/431296