The rapid advancement of environmental sequencing technologies, such as metagenomics, has significantly enhanced our ability to study microbial communities. The eubiotic composition of these communities is crucial for maintaining ecological functions and host health. Species diversity is only one facet of a healthy community’s organization; together with abundance distributions and interaction structures, it shapes reproducible macroecological states, that is, joint statistical fingerprints that summarize whole-community behavior. Despite recent developments, a theoretical framework connecting empirical data with ecosystem modeling is still in its infancy, particularly in the context of disordered systems. Here, we present a novel framework that couples statistical physics tools for disordered systems with metagenomic data, explicitly linking diversity, interactions, and stability to define and compare these macroecological states. By employing the generalized Lotka–Volterra model with random interactions, we reveal two different emergent patterns of species interaction networks and species abundance distributions for healthy and diseased microbiomes. On the one hand, healthy microbiomes have similar community structures across individuals, characterized by strong species interactions and abundance diversity consistent with neutral stochastic fluctuations. On the other hand, diseased microbiomes show greater variability driven by deterministic factors, thus resulting in less ecologically stable and more divergent communities. Our findings suggest the potential of disordered system theory to characterize microbiomes and to capture the role of ecological interactions on stability and functioning.

The self-organization of microbial ecosystems involves a large variety of mechanisms, ranging from biochemical signaling to population dynamics. Among these, the role of motility regulation has been little studied, despite the importance of active migration processes. Here we show how weak, random motility regulation suffices to induce complex forms of organization in bacterial mixtures comprising a large number of coexisting strains. First, we simulate microscopic models of run-and-tumble bacteria whose self-propulsion speeds are weakly regulated by the local density of each strain, mimicking the impact of weak, random metabolic interactions. Our simulations reveal that, as the heterogeneity of the interaction network increases, the system undergoes a phase transition leading to the emergence of distinct, spatially segregated communities. To account for these results and assess their robustness, we use random-matrix theory to analyze the hydrodynamic description of the bacterial mixture, obtaining a quantitative agreement with our microscopic simulations. Our results hold for a variety of motility-regulation mechanisms and highlight the need to characterize the role of motility regulation in experimentally relevant situations.

Large ecosystems balance competition and cooperation, yet standard generalized Lotka--Volterra models make mutualism destabilizing by amplifying disorder and driving unbounded growth. We show that Monod-like saturation resolves this paradox: dynamical mean-field theory and random-matrix analysis reveal a broader stable phase and enhanced survival. Network architecture provides a second control mechanism, but nestedness offers no intrinsic stability advantage. Instead, it is a byproduct of degree distributions with high connectivity necessary for stability.

The study of ecological systems is gaining momentum in modern scientific research, driven by an abundance of empirical data and advancements in bioengineering techniques. However, a full understanding of their dynamical and thermodynamical properties, also in light of the ongoing biodiversity crisis, remains a formidable endeavor. From a theoretical standpoint, modeling the interactions within these complex systems – such as bacteria in microbial communities, plant-pollinator networks in forests, or starling murmurations – presents a significant challenge. Given the intrinsically high dimensionality of these datasets, powerful approaches rely on random matrix theory and methods from disordered systems. In these lectures, we will explore two cornerstone models in theoretical ecology: the MacArthur/Resource-Consumer model, and the Generalized Lotka-Volterra model, with a special focus on systems composed of a large number of interacting species. In the second part, we will highlight timely research directions, particularly to bridge the gap with empirical observations and detect macroecological patterns.

ecosystems, notably microbial communities, display regular patterns in diversity, abundance distribution, and function, independently of their specific composition. The effort to understand how such regularities are achieved and maintained has driven the development of theoretical approaches inspired by complex systems and statistical physics. We introduce three classical frameworks for modelling biological communities -neutral models, the Generalized Lotka-Volterra equations, and a Consumer-Resource model-and discuss their hypotheses and empirical support. We further discuss a few theoretical challenges for future research, emphasizing the pivotal role of interfacing theory with ecological data.

Disordered systems generically exhibit aging and a glass transition. Previous studies have long suggested that nonreciprocity tends to destroy glassiness. Here, we show that this is not always the case using a bipartite spherical Sherrington-Kirkpatrick model that describes the antagonistic coupling between two identical complex agents modeled as macroscopic spin glasses. Our dynamical mean-field theory calculations reveal an exceptional-point-mediated transition from a static disorder phase to an oscillating amorphous phase as well as nonreciprocal aging with slow dynamics and oscillations.

The rapid advancement of environmental sequencing technologies, such as metagenomics, has significantly enhanced our ability to study microbial communities. The eubiotic composition of these communities is crucial for maintaining ecological functions and host health. Species diversity is only one facet of a healthy community organization; together with abundance distributions and interaction structures, it shapes reproducible macroecological states, i.e., joint statistical fingerprints that summarize whole-community behavior. Despite recent developments, a theoretical framework connecting empirical data with ecosystem modeling is still in its infancy, particularly in the context of disordered systems. Here, we present a novel framework that couples statistical physics tools for disordered systems with metagenomic data, explicitly linking diversity, interactions, and stability to define and compare these macroecological states. By employing the generalized Lotka-Volterra model with random interactions, we reveal two different emergent patterns of species interaction networks and species abundance distributions for healthy and diseased microbiomes. On the one hand, healthy microbiomes have similar community structures across individuals, characterized by strong species interactions and abundance diversity consistent with neutral stochastic fluctuations. On the other hand, diseased microbiomes show greater variability driven by deterministic factors, thus resulting in less ecologically stable and more divergent communities. Our findings suggest the potential of disordered system theory to characterize microbiomes and to capture the role of ecological interactions on stability and functioning.

Disordered systems can exhibit a dramatic slowdown of their dynamics called aging. Contrary to the established understanding that this phenomenon is destroyed by nonreciprocal interactions, we here show that the outcome crucially depends on the structure of the system. Unlike previous studies, which focused on random nonsymmetric interactions between simple microscopic components, we investigate a scenario where nonreciprocally coupled agents are macroscopic entities with complex internal dynamics, modeled as two identical spin glasses. This framework could be relevant for many biological systems, in which nonreciprocal interactions can arise at a coarse-grained level. Our dynamical mean-field theory calculations reveal a finite temperature transition from a static disordered phase to a non-time-translationally-invariant regime. Below this transition, mediated by a spectral singularity known as exceptional points, we find macroscopic oscillations superimposed on aging behavior. Asymptotically, the system rotates in the plane spanned by the two lowest energy modes of the uncoupled system. We contrast these results to the case of random nonreciprocity, where aging is suppressed at any finite temperature, and propose that the two cases correspond to two broader classes of systems, with "microscopic" versus "macroscopic" nonreciprocity, with aging surviving only in the second case.

How diversity is maintained in natural ecosystems is a long-standing question in Theoretical Ecology. By studying a system that combines ecological dynamics, heterogeneous interactions and spatial structure, we uncover a new mechanism for the survival of diversity-rich ecosystems in the presence of demographic fluctuations. For a single species, one finds a continuous phase transition between an extinction and a survival state, that falls into the universality class of Directed Percolation. Here we show that the case of many species with heterogeneous interactions is different and richer. By merging theory and simulations, we demonstrate that with sufficiently strong demographic noise, the system exhibits behavior akin to the single-species case, undergoing a continuous transition. Conversely, at low demographic noise, we observe unique features indicative of the ecosystem's complexity. The combined effects of the heterogeneity in the interaction network and migration enable the community to thrive, even in situations where demographic noise would lead to the extinction of isolated species. The emergence of mutualism induces the development of global bistability, accompanied by sudden tipping points. We present a way to predict the catastrophic shift from high diversity to extinction by probing responses to perturbations as an early warning signal.

We review the main methods used to study spin glasses. In the first part, we focus on methods for fully connected models and systems defined on a tree, such as the replica method, the Thouless-Anderson-Palmer formalism, the cavity method, and the dynamical mean-field theory. In the second part, we deal with the description of low-dimensional systems, mostly in three spatial dimensions, which are mostly studied through numerical simulations. We conclude by mentioning some of the main open problems in the field.

The worldwide loss of species diversity brings urgency to understanding how diverse ecosystems maintain stability. Whereas early ecological ideas and classic observations suggested that stability increases with diversity, ecological theory makes the opposite prediction, leading to the long-standing “diversity-stability debate.” Here, we show that this puzzle can be resolved if growth scales as a sublinear power law with biomass (exponent <1), exhibiting a form of population self-regulation analogous to models of individual ontogeny. We show that competitive interactions among populations with sublinear growth do not lead to exclusion, as occurs with logistic growth, but instead promote stability at higher diversity. Our model realigns theory with classic observations and predicts large-scale macroecological patterns. However, it makes an unsettling prediction: Biodiversity loss may accelerate the destabilization of ecosystems.

We study how migration and interactions can rescue species-rich, spatially structured ecosystems from extinction due to demographic noise. For a single species, one finds an extinction and a survival state depending on the strength of the demographic noise with respect to migration. The transition between these two regimes is a second-order out-of-equilibrium phase transition, which falls into the universality class of Directed Percolation. Here we show that the case of many species with heterogeneous interactions is different and richer. When the demographic fluctuations are strong enough, the transition is continuous and analogous to the single-species case. Ecological interactions play a secondary role in it. At small demographic noise, instead, we find novel features that are a signature of the complexity of the ecosystem. Thanks to interactions and migration, the metacommunity is able to thrive even when demographic noise would drive single species to extinction. This is accompanied by the development of mutualism between non-extinct species. The transition becomes discontinuous and a global bistability emerges. We present a way to predict the catastrophic shift from high diversity to extinction by probing responses to perturbations as early warning signal. Our analytical framework is grounded in statistical physics and focuses on a model of metacommunities based on Lotka-Volterra equations with random interactions.

How diversity is maintained in natural ecosystems is a long-standing question in Theoretical Ecology. By studying a system that combines ecological dynamics, heterogeneous interactions and spatial structure, we uncover a new mechanism for the survival of diversity-rich ecosystems in the presence of demographic fluctuations. For a single species, one finds a continuous phase transition between an extinction and a survival state, that falls into the universality class of Directed Percolation. Here we show that the case of many species with heterogeneous interactions is different and richer. By merging theory and simulations, we demonstrate that with sufficiently strong demographic noise, the system exhibits behavior akin to the single-species case, undergoing a continuous transition. Conversely, at low demographic noise, we observe unique features indicative of the ecosystem's complexity. The combined effects of the heterogeneity in the interaction network and migration enable the community to thrive, even in situations where demographic noise would lead to the extinction of isolated species. The emergence of mutualism induces the development of global bistability, accompanied by sudden tipping points. We present a way to predict the catastrophic shift from high diversity to extinction by probing responses to perturbations as an early warning signal.

We compute the typical number of equilibria of the generalized Lotka-Volterra equations describing species-rich ecosystems with random, nonreciprocal interactions using the replicated Kac-Rice method. We characterize the multiple-equilibria phase by determining the average abundance and similarity between equilibria as a function of their diversity (i.e., of the number of coexisting species) and of the variability of the interactions. We show that linearly unstable equilibria are dominant, and that the typical number of equilibria differs with respect to the average number.

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 interaction case. We investigate different scalings for the distribution of the interactions with the number of species and try to bridge the gap with previous works. Our results show two different behaviors for the mean abundance at zero and finite temperature, respectively, with a continuous crossover between the two. We confirm and extend previous results obtained for weak interactions: at zero temperature, even in the strong competitive interaction limit, the system is in a multiple-equilibria phase, whereas at finite temperature only a unique stable equilibrium can exist. Finally, we establish the qualitative phase diagrams and compare the species abundance distributions in the two cases.

In this report, I will review some of the most used models in theoretical ecology along with appealing reformulations and recent results in terms of diversity, stability, and functioning of large well-mixed ecological communities.

We analyze the role of the Allee effect - a positive correlation between population density and mean individual fitness - for ecological communities formed by a large number of species. Our study is performed using the generalized Lotka-Volterra model with random interactions between species. We obtain the phase diagram and analyze the nature of the multiple equilibria phase. Remarkable differences emerge with respect to the logistic growth case, thus revealing the major role played by the functional response in determining aggregate behaviors of large ecosystems.