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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…

Ecosystems with a large number of species are often modelled as Lotka-Volterra dynamical systems built around a large random interaction matrix. Under some known conditions, a global equilibrium exists and is unique. In this article, we…

Lotka-Volterra (LV) equations play a key role in the mathematical modeling of various ecological, biological and chemical systems. When the number of species (or, depending on the viewpoint, chemical components) becomes large, basic but…

Random matrix theory successfully connects the structure of interactions of large ecological communities to their ability to respond to perturbations. One of the most debated aspects of this approach is the missing role of population…

The Lotka-Volterra system is a set of ordinary differential equations describing growth of interacting ecological species. This model has gained renewed interest in the context of random interaction networks. One of the debated questions is…

We compute the typical number of equilibria of the Generalized Lotka-Volterra equations describing species-rich ecosystems with random, non-reciprocal interactions using the replicated Kac-Rice method. We characterize the…

Ecological interaction networks are rarely homogeneous: species naturally form communities with distinct interaction structures, resulting in block-structured variance and correlation profiles in the interaction matrix. We study the…

We consider the problem of rational decision making in the presence of nonlinear constraints. By using tools borrowed from spin glass and random matrix theory, we focus on the portfolio optimisation problem. We show that the number of…

We discuss the portfolio optimization problem with the obligatory deposits constraint. Recently it has been shown that as a consequence of this nonlinear constraint, the solution consists of an exponentially large number of optimal…

Empirical observations show that ecological communities can have a huge number of coexisting species, also with few or limited number of resources. These ecosystems are characterized by multiple type of interactions, in particular…

Empirical observations show that ecological communities can have a huge number of coexisting species, also with few or limited number of resources. These ecosystems are characterized by multiple type of interactions, in particular…

Does an ecological community allow stable coexistence? Identifying the general principles that determine the answer to this question is a central problem of theoretical ecology. Random matrix theory approaches have uncovered the general…

We study the equilibrium phases of a generalized Lotka-Volterra model characterized by a species interaction matrix which is random, sparse and symmetric. Dynamical fluctuations are modeled by a demographic noise with amplitude proportional…

In a diverse population, where many species are present, competitors can fight for surviving at individual and collective levels. In particular, species, which would beat each other individually, may form a specific alliance that ensures…

We study the equilibria of a large Lokta-Volterra system of coupled differential equations in the case where the interaction coefficients form a large random matrix. In the case where this random matrix follows an elliptic model , we study…

The random Lotka-Volterra model is widely used to describe the dynamical and thermodynamic features of ecological communities. In this work, we consider random symmetric interactions between species and analyze the strongly competitive…

In complex ecological communities, species may self-organize into clusters or clumps where highly similar species can coexist. The emergence of such species clusters can be captured by the interplay between neutral and niche theories. Based…

Explaining the emergence of self-organized biodiversity and species abundance distribution patterns remians a fundamental challenge in ecology. While classical frameworks, such as neutral theory and models based on pairwise species…

Understanding the behaviors of ecological systems is challenging given their multi-faceted complexity. To proceed, theoretical models such as Lotka-Volterra dynamics with random interactions have been investigated by the dynamical…

Classical approaches to ecological stability rely on fully connected interaction models, yet real ecosystems are sparse and structured--a feature that qualitatively reshapes their collective dynamics. Here, we establish a thermodynamically…