Equilibrium in a large Lotka–Volterra system with pairwise correlated interactions

Clenet, Maxime; Ferchichi, E; Najim, Jamal

doi:10.1016/j.spa.2022.08.004

Cited by 9 publications

(8 citation statements)

References 20 publications

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“…These results are established in [62] and [57] for the i.i.d. case and the elliptic case.In the case where bold-italicr≠bold1 is still positive componentwise, there is not a sharp threshold at αN∗=2log⁡(N) but rather a transition buffer false[αtrueprefixmin,N∗,αtrueprefixmax,N∗false] from non-feasibility (αN<αtrueprefixmin,N∗) to feasibility (αN>αtrueprefixmax,N∗).…”

Section: Equilibrium Coexistence and Stabilitysupporting

confidence: 55%

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Complex systems in ecology: a guided tour with large Lotka–Volterra models and random matrices

Akjouj,

Barbier,

Clenet

et al. 2024

Proc. R. Soc. A.

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Ecosystems represent archetypal complex dynamical systems, often modelled by coupled differential equations of the form d x i d t = x i ϕ i ( x 1 , … , x N ) , where N represents the number of species and x i , the abundance of species i . Among these families of coupled differential equations, Lotka–Volterra (LV) equations, corresponding to ϕ i ( x 1 , … , x N ) = r i − x i + ( Γ x ) i , play a privileged role, as the LV model represents an acceptable trade-off between complexity and tractability. Here, r i is the intrinsic growth of species i and Γ stands for the interaction matrix: Γ i j represents the effect of species j over species i . For large N , estimating matrix Γ is often an overwhelming task and an alternative is to draw Γ at random, parameterizing its statistical distribution by a limited number of model features. Dealing with large random matrices, we naturally rely on random matrix theory (RMT). The aim of this review article is to present an overview of the work at the junction of theoretical ecology and large RMT. It is intended to an interdisciplinary audience spanning theoretical ecology, complex systems, statistical physics and mathematical biology.

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Section: Equilibrium Coexistence and Stabilitysupporting

confidence: 55%

“…-These results are established in [62] and [57] for the i. Assume that Γ is given by model (2.4)-(ii).…”

Section: (Iii) Remarksmentioning

confidence: 86%

Complex systems in ecology: a guided tour with large Lotka–Volterra models and random matrices

Akjouj,

Barbier,

Clenet

et al. 2024

Proc. R. Soc. A.

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“…Older results in the same vein can be found in, e.g., [OD92,Tok04]. Heuristic evaluations of the asymptotic behavior of µ u‹ were proposed in [CEN22,CNM22], most generally in the elliptical non-centered case.…”

Section: Introductionmentioning

confidence: 82%

Approximate Message Passing for sparse matrices with application to the equilibria of large ecological Lotka-Volterra systems

Hachem¹

2023

Preprint

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This paper is divided into two parts. The first part is devoted to the study of a class of Approximate Message Passing (AMP) algorithms which are widely used in the fields of statistical physics, machine learning, or communication theory. The AMP algorithms studied in this part are those where the measurement matrix has independent elements, up to the symmetry constraint when this matrix is symmetric, with a variance profile that can be sparse. The AMP problem is solved by adapting the approach of Bayati, Lelarge, and Montanari (2015) to this matrix model. The Lotka-Volterra (LV) model is the standard model for studying the dynamical behavior of large dimensional ecological food chains. The second part of this paper is focused on the study of the statistical distribution of the globally stable equilibrium vector of a LV system in the situation where the random symmetric interaction matrix among the living species is sparse, and in the regime of large dimensions. This equilibirium vector is the solution of a Linear Complementarity Problem, which distribution is shown to be characterized through the AMP approach developed in the first part. In the large dimensional regime, this distribution is close to a mixture of a large number of truncated Gaussians.

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“…the equilibrium with all n species), to study the set of persistent species that are actually assembled through the dynamics of Eq. 2, possibly following the extinction of some species from the pool 41,[44][45][46] .…”

Section: Methodsmentioning

confidence: 99%

“…Rather than asking whether a particular set of species can coexist, we examine metacommunities that emerge through the model dynamics from an initial random pool [41][42][43] . This probabilistic approach is a powerful way to identify characteristic outcomes in complex models [44][45][46] . We investigate not only the expected number of coexisting species from a pool of n species, but also the distribution of this quantity, as well as typical features of self-assembled metacommunities.…”

Section: Introductionmentioning

confidence: 99%

Coexistence of many species under a random competition-colonization trade-off

Miller

Clenet

Libera

et al. 2023

Preprint

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The competition-colonization trade-off is a well-studied coexistence mechanism for metacommunities. In this setting, it is believed that coexistence of all species requires their traits to satisfy restrictive conditions limiting their similarity. To investigate whether diverse metacommunities can assemble in a competition-colonization trade-off model, we study their assembly from a probabilistic perspective. From a pool of species with parameters (traits) sampled at random, we compute the probability that any number of species coexist and characterize the set of species that emerges through assembly. Remarkably, almost exactly half of the species in a large pool typically coexist, with no saturation as the size of the pool grows, and with little dependence on the underlying distribution of traits. Through a mix of analytical results and simulations, we show that this unlimited niche packing emerges as assembly actively moves communities toward overdispersed configurations in niche space. Our findings also apply to a realistic assembly scenario where species invade one-at-a-time from a fixed regional pool. When diversity arises de novo in the metacommunity, richness still grows without bound, but more slowly. Together, our results suggest that the competition-colonization trade-off can support the robust emergence of diverse communities, even when coexistence of the full species pool is exceedingly unlikely.

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Equilibrium in a large Lotka–Volterra system with pairwise correlated interactions

Cited by 9 publications

References 20 publications

Complex systems in ecology: a guided tour with large Lotka–Volterra models and random matrices

Complex systems in ecology: a guided tour with large Lotka–Volterra models and random matrices

Approximate Message Passing for sparse matrices with application to the equilibria of large ecological Lotka-Volterra systems

Coexistence of many species under a random competition-colonization trade-off

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