Chiara Cammarota, Giulio Biroli, Guy Bunin, Institut de Physique Théorique - UMR CNRS 3681 (IPHT), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Physique Statistique de l'ENS (LPS), Fédération de recherche du Département de physique de l'Ecole Normale Supérieure - ENS Paris (FRDPENS), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Massachusetts Institute of Technology (MIT), Department of Physics [Haifa], Technion - Israel Institute of Technology [Haifa], King‘s College London, Grant from the Simons Foundation number 454935 (G.B.), Simons FE, École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Saclay-Commissariat à l'énergie atomique et aux énergies alternatives (CEA), and De Laborderie, Emmanuelle
In this work we study the stability of the equilibria reached by ecosystems formed by a large number of species. The model we focus on are Lotka-Volterra equations with symmetric random interactions. Our theoretical analysis, confirmed by our numerical studies, shows that for strong and heterogeneous interactions the system displays multiple equilibria which are all marginally stable. This property allows us to obtain general identities between diversity and single species responses, which generalize and saturate May's bound. By connecting the model to systems studied in condensed matter physics, we show that the multiple equilibria regime is analogous to a critical spin-glass phase. This relation provides a new perspective as to why many systems in several different fields appear to be poised at the edge of stability and also suggests new experimental ways to probe marginal stability. Many complex systems in Nature organize in states that are poised just at the edge of stability. The growing evidence comes from physics [1], biology [2], ecology [3], neuroscience [4, 5] and economy [6]. One important common trait of all examples is that they are formed by strongly interacting units—species, neurons, agents and particles depending on the situation. The possible explanations of such phenomenon are varied. They include the need for flexibility and adaptiveness to time-varying conditions [2, 7], balance between functionality and stability [7], self-organized criticality [8], self-organized instability [9], and continuous constraints satisfaction [1]. Here we address this problem focusing on generalized Lotka-Volterra (LV) equations. They provide a simple and general setting to study assemblies of interacting degrees of freedom; as such they are used in several fields [10–13]. In particular, they provide a canonical model for ecosystems, with growing connections to systems across biology [10, 11, 14]. The study of stability of equilibria and their properties using LV equations and generalizations has become a very active research subject. Several important results were obtained recently; in particular general techniques to count the number of equilibria and their properties have been developed [15], and criticality and glassiness have been found to be emergent properties of ecosystems [16–18]. Our approach unifies these different perspectives and, by a mapping to condensed matter systems, reveal their generality beyond LV models. Henceforth, in order to describe it, we shall use the terminology employed in theoretical ecology. In the model we consider, an ecological community is assembled from a pool of available species. We focus on the case relevant for the examples cited above, and in many other situations, when the number of species is large. Since the detailed parameters of all interactions are not known in the majority of cases, and in any case not all details are expected to matter [19], we follow the long tradition pioneered by May in ecology [20] and Wigner in physics [21], and sample the interactions randomly. However , we go beyond May's classical work since randomness is here introduced at the level of interactions between all possible species, while the community self-organizes by choosing which species are present. In other words, the number and identity of the species that are present in the community is selected dynamically [22, 23]. Understanding the emergent stability of the equilibria reached dynamically and its dependence on the external parameters is the main purpose of this work. We find, in agreement with [16, 18, 24], that when the interactions are weak or highly uniform, only one equilibrium is present and is determined mainly by self-regulation within each species. For stronger and more heterogeneous interactions, multiple equilibria emerge. Our main result is that when this happens, all possible states of the system are close to be marginally stable for large number of species and this determines the diversity of the ecosystem, see Fig. 1. Marginal stability has several important consequences, in particular it leads to extreme susceptibility to small perturbations. This situation is referred as " critical " in the physics literature [25]. May famously suggested that complexity and interactions limit the stability of ecosystems [20]. Our results provide a complementary perspective: complex ecological communities reduce dynamically their instability through a reduction of the possible number of surviving species, i.e. diversity, and eventually reach a marginally stable state saturating May's bound. Since this phenomenon stems from a dynamical process, it holds for a broad range of system parameters. It is robust against a range of variations in the model, including different functional forms of responses and interactions, as well as noise. Although in many physical cases criticality