We investigate the Multiple Equilibria phase of generalized Lotka-Volterra dynamics with random, non-reciprocal interactions. We compute the topological complexity of equilibria, which quantifies how rapidly the number of equilibria of the dynamical equations grows with the total number of species. We perform the calculation for arbitrary degree of non-reciprocity in the interactions, distinguishing between configurations that are dynamically stable to invasions by species absent from the equilibrium, and those that are not. We characterize the properties of typical (i.e., most numerous) equilibria at a given diversity, including their average abundance, mutual similarity, and internal stability. This analysis reveals the existence of two distinct ME phases, which differ in how internally stable equilibria behave under invasions by absent species. We discuss the implications of this finding for the system's dynamical behavior.
Nonreciprocal interactions drive a broad class of outof-equilibrium dynamics and are nowadays at the core of an established line of research. Systems with a large number of heterogeneous components interacting asymmetrically are a particularly interesting realization of nonreciprocity, since their high dimensionality provides a natural framework for analytical approaches. This setting is well justified for modeling biological neural networks [1][2][3][4] and ecosystems composed of many coexisting species such as the microbiota, tropical rainforests, or plankton communities [5][6][7]. The models often incorporate randomness to represent interactions among neurons or species, with different levels of structural organization. The fully unstructured case in which couplings are drawn independently at random was, for instance, famously exploited by R. May in his seminal analysis of the diversity-stability problem in ecology [8].
In ecosystems modeling, the non-reciprocal interaction terms enter the dynamical equations governing the time evolution of species abundances. These (random) interactions compete with single-species terms that encode the intrinsic growth or suppression of the abundances in the absence of other species, as determined by environmental resources and intra-species dynamics [9]. This competition naturally leads to distinct dynamical regimes: at weak randomness (henceforth, variability), the abundances relax to time-independent values, while at stronger variability they persistently fluctuate in time. In presence of non-reciprocity, these fluctuations display signatures of chaotic behavior [10][11][12][13][14][15][16]. While the first type of behavior is simple to characterize analytically, the second poses a quite significant theoretical challenge.
We focus here on a prototypical model exhibiting such a transition, the generalized Lotka-Volterra equations with random interactions (rGLV). The existence of a dynamical transition in this (and equivalent) models is known since the early studies [17][18][19]. In the weak variability phase, the dynamical equations admit a unique * thomas.louis-sarrola@universite-paris-saclay.fr fixed point (or equilibrium) configuration that is stable with respect to perturbations of the species abundances, both for the species coexisting at the fixed point and also for those that are absent, and can potentially invade. When the variability reaches a critical value, this equilibrium loses its stability, and the system correspondingly displays a qualitatively different dynamics. It is expected that this complex dynamical phase appears concomitantly with the emergence of a multitude of fixed points of the dynamical equations [20]. In the limiting case of reciprocal interactions, this expectation is reinforced by the similarity between the rGLV model and spin-glass models [19,21,22]; in this limit indeed the rGLV equations describe a dynamical descent into an energy landscape (conservative dynamics), and the stable fixed points are local minima of such landscape. As in other high-dimensional optimization problems with randomness [23][24][25], one expects a multitude of marginally stable local minima [26] and a dynamical descent that exhibits slowing down and aging due to marginality [27]. The existence of exponentially many fixed points has also been shown in the opposite limit of strong nonreciprocity [28,29], and dynamical studies in this setting [30,31] hint that certain fixed points may exert an influence on the (chaotic) dynamics in this case, too. More broadly, investigating to what extent the out-ofequilibrium dynamics of non-reciprocal systems can be understood through the fixed points of the dynamics is an open challenge that is attracting increasing interest [32][33][34][35]. In this work, we provide a statistical characterization of the fixed points of the gLVE for arbitrary degrees of non-reciprocity, thereby establishing the basis for assessing their impact on dynamics.
Fixed points and stability. We consider the rGLV equations
where N i is the properly scaled abundance of species i = 1, • • • , S, F i ( ⃗ N ) is the effective growth rate associated to FIG. 1: Phase diagram in the variability-reciprocity space (σ, γ), highlighting the existence of three distinct phases. The black dotted line marks the boundary between the Unique Fixed Point (UFP) phase (gray), where a unique uninvadable and internally stable equilibrium equilibrium exists, and the Multiple Equilibria (ME) phase. The latter is split into a Fragile (orange) phase, where all internally stable equilibria are unstable to invasions, and a Robust (blue) phase, where uninvadable internally stable equilibria exist in exponential number. The red crosses identify the critical line γ F R , the red full line is the hyperbolic fit γ F R ≈ 0.318/σ + 0.549. species i, and κ i denote the species carrying capacities. The interaction couplings between species
a ij a kl = δ ik δ jl + γ δ il δ jk (2) are
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