Abstract
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 stability 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 suggests new experimental ways to probe marginal stability.
Highlights
In this work we study the stability of the equilibria reached by ecosystems formed by a large number of species
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
By relating the LV model to systems studied in condensed matter physics, the multiple-equilibria regime is shown to be akin to a critical spin-glass phase
Summary
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 [16,17,18] Our approach unifies these different perspectives and, by a mapping to condensed matter systems, reveal their generality beyond LV models. 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.
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