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# Unveiling complexity: Statistical physics approaches to ecological communities

Ada Altieri, Silvia de Monte

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Ada Altieri, Silvia de Monte. Unveiling complexity: Statistical physics approaches to ecological communities. EPL - Europhysics Letters, 2025, 150 (5), pp.51002. 10.1209/0295-5075/addae2. hal-05377173

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PERSPECTIVE • OPEN ACCESS

# Unveiling complexity: Statistical physics approaches to ecological communities

To cite this article: Ada Altieri and Silvia De Monte 2025 EPL 150 51002

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Perspective

Unveiling complexity: Statistical physics approaches to ecological
communities

ADA ALTIERI<fnref n="1" />(a) and SILVIA DE MONTE<fnref n="2" />,<fnref n="3" />

received 16 December 2024; accepted in final form 20 May 2025
published online 16 June 2025

**Abstract** – Species-rich 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.

open access

perspective

Copyright © 2025 The author(s)

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**Introduction.** — In an era of ongoing global changes in climate and biodiversity, ecology is increasingly concerned with the dynamics of biological communities on the short and long time scales. Robust theoretical frameworks are needed to make realistic predictions and help understand how diverse ecosystems maintain their stability and function.

Communities are ensembles of species of similar ecol-
ogy that coexist within the same habitat and are charac-
terized by their abundance, i.e., the number or fraction
of individuals they are composed of. Many communities
are so complex that it is virtually impossible to describe
the processes that shape them in quantitative terms, be
they competition for shared resources and space, or the
exchange of metabolic byproducts, toxins, and signals.
Microbial communities, in particular, contain an enor-
mous number of species of different phylogenetic origins
and traits, which collectively provide community functions
—such as community biomass, primary productivity, or
contribution to geochemical cycles— where single-species
contributions are often redundant or synergistic [1]. The
possibility of assembling synthetic microbial communities
in the lab is moreover revealing more details of their

ecological dynamics at the species [2–5] and even sub-
species [6] levels, paving the way for a more thorough in-
tegration of models and observations.

Highly diverse microbial communities are ideal for ad-
dressing the relation between species-level ecological pro-
cesses and community-level emergent features. In partic-
ular, *macroecology* [7,8] studies general laws that describe
community patterns, such as the distribution of species
abundances, and the processes that underlie their emer-
gence and function. Statistical patterns, which are inde-
pendent of the exact species composition of the commu-
nity, can be seen as emergent macroscopic features of a
system with many degrees of freedom [9].

In physics and complex systems science, the averaging
of a large number of (equivalent) microscopic variables
into macroscopic observables is associated with the
existence of *typical* behaviors and *universality classes*. A
common approach to explain such behaviors is to reduce
models to their essential components. Effective models
that retain only a few, fundamental features are then used
to reproduce general properties of the real community,
*e.g.*, of the systems dynamics or some macroscopic ob-
servable. In this spirit, May [10] introduced a simplified
framework for determining when complex systems are

<footnote n="1">¹ Laboratoire Matière et Systèmes Complexes (MSC), Université Paris Cité, CNRS - 75013 Paris, France</footnote>

<footnote n="2">² Institut de Biologie de l'ENS, Département de Biologie, École Normale Supérieure, CNRS, INSERM
Université Paris Science & Lettres - Paris 75005, France</footnote>

<footnote n="3">³ Max Planck Institute for Evolutionary Biology, Department of Theoretical Biology - Plön 24306, Germany</footnote>

<footnote>(a)E-mail: ada.altieri@u-paris.fr (corresponding author)</footnote>
expected to be stable. May's approach posits the existence of an ecological equilibrium state and represents the effects of all remaining unknown variability with a random stability matrix. To address the overwhelming complexity of large ecological communities and study how stability changed with the ecosystem parameters (e.g., the number of species S and their average connectivity C), May considered the associated Jacobian matrix $\hat{J}$ of size $S \times S$, where $J_{ij}$ is zero when species are not connected, and otherwise quantifies the influence of species j over i near a fixed point with positive or zero abundances. Diagonal elements accounting for self-interaction were set to -1, whereas off-diagonal ones were randomly drawn from a Gaussian distribution with zero mean and variance $\sigma^2$ —referred to as the *heterogeneity parameter*. In the large S limit, the statistical physics of random matrices enabled May to derive the associated eigenvalue distribution and demonstrate that stability is lost when $\sigma\sqrt{SC}$ increases beyond 1, hence for diverse ecosystems with a large number of species and high connectivity C.

The conclusion that large, heterogeneous ecosystems are expected to be unstable gave rise to a long-standing *diversity-stability debate*, where the generality of the stabilizing effect of heterogeneity was questioned [11]. Alternative hypotheses on species interactions were proposed, notably in trophic chains [9] or in communities with sublinear self-regulation mechanisms [12], to reflect the variable stability of natural ecosystems [13]. Recent experimental results on synthetic microbial communities indicate that May's expectation of diversity-induced destabilization seems nonetheless to be met when complexity is gradually altered [4]. These observations prompt further characterization of when and how an increase of heterogeneity and number of species is stabilizing.

In this review, we aim to illustrate how heuristic models for community ecology are solved using statistical physics methods and how their predictions compare with empirical observations. First, we will introduce three classical approaches to community ecology: neutral models, the Generalized Lotka-Volterra model, and the MacArthur model. We will briefly outline their underlying hypotheses and statistical features. We will then discuss their relevance for explaining patterns of species-rich microbial communities. Eventually, we will discuss a non-exhaustive set of open theoretical problems and current efforts to solve them.

Our purpose is not to provide a comprehensive summary of all recent contributions of statistical physics to the field of community ecology, but rather to illustrate how these two fields interface. More detailed explanations of theoretical methods can be found in refs. [14,15].

**Models for community ecology.** — Reflecting heuristic assumptions about the origins of abundance variations, numerous models have been proposed to explain how community patterns emerge from the dynamics of the abundance of their constituent populations. These formal approaches entail assumptions on the rates at which

individuals within species reproduce and die, and how such rates are affected by other factors such as the abundance of other species (inter-specific interactions) or of the species itself (intra-specific interactions), environmental fluctuations, immigration or speciation (the appearance of a new species in a local community). Here, we present three classes of models that differ in such hypotheses (see refs. [16] and [17] for the presentation of a broader range of models).

*Neutral models* — exemplified by Hubbell's unified neutral theory of biodiversity and biogeography [18] — assume that all individuals in a community are equivalent, irrespective of the species they belong to. They thus have identical *per capita* growth and death probabilities and the same migration and speciation rates. Even in the absence of interactions, the stochastic nature of these processes generates variation in species abundance.

Neutral approaches entail an intrinsic symmetry between species, such that every species follows the same *typical* dynamics. The stationary distributions of the associated Master and Fokker-Planck equations can then be obtained exactly when demographic rates are sufficiently simple functions of abundances [14,19].

Proposed as a null model for species-rich ecosystems, and subsequently extended to include some degree of variability among species, neutral models capture several macroecological patterns and have been extensively used for studying biodiversity [20,21]. The assumption of equivalence between species, however, has been criticized for being rather a convenient mathematical ansatz than a biologically realistic representation. Indeed, species generally differ in their degree of adaptation to a given environment. Their demographic rates are thus not identical, and are moreover affected by individuals of their own and other species.

Neutral models are, therefore, often opposed to those that explicitly account for differences between species, termed collectively *niche models*. A prominent one, introduced at the beginning of the 20th century by Alfred Lotka and Vito Volterra, postulates that niches are fundamentally determined by interactions with individuals of the same and of other species. The demographic rates of a given species thus depend on intra- and inter-specific interactions. The deterministic dynamic of species abundance is then described by ordinary differential equations accounting for the simplest possible interactions, those involving, at most, pairs of species. When the number of species S is large, the equations describing a spatially homogeneous community are usually referred to as *Generalized Lotka-Volterra (GLV) model* and posit that the increment of the abundance $N_i(t)$ of species i at time t obeys

$$ \frac{dN_i}{dt} = N_i \left[ K_i - N_i - \sum_{j,(j \neq i)} \alpha_{ij} N_j \right] + \omega \eta(N_i(t)) + \lambda_i. \quad (1) $$

Here, the parameters $K_i$ denote the carrying capacity of species i, i.e., its stationary abundance in isolation. The
coefficients $\alpha_{ij}$ quantify the effect of species $j$ on the net growth rate of species $i$ (by this convention, positive values correspond to detrimental effects, like competition, and negative values to facilitation). The term $\eta(N_i(t))$ encodes stochastic fluctuations of intensity $\omega$. It is typically chosen to heuristically represent, in a continuous description, the discreteness of stochastic death and birth events. Such demographic noise scales with the square root of the population and —in the case of a symmetric interaction matrix— satisfies detailed balance. Conversely, environmental noise, which is often assumed to affect per capita demographic rates independently of $N_i$, violates this principle, thereby precluding the derivation of a closed-form Hamiltonian. Considering that $N_i = 0$ is an absorbing state, demographic fluctuations may cause species to go extinct over a finite time. For this reason, an immigration parameter $\lambda_i$ is often introduced to avoid the progressive erosion of diversity [22–24].

In keeping with May’s representation of communities with unknown interactions, the elements $\alpha_{ij}$ of the disordered GLV interaction matrix are assigned as realizations of a random sampling distribution or quenched noise. In a physics perspective, universality means here that only a few statistics of this distribution are important: the mean $\langle\alpha_{ij}\rangle$, the variance $\langle\alpha_{ij}^2\rangle_c = \langle(\alpha_{ij} - \langle\alpha_{ij}\rangle)^2\rangle$, and the covariance $\langle\alpha_{ij}\alpha_{ji}\rangle_c = \gamma\langle\alpha_{ij}^2\rangle_c$ (where $\gamma \in [-1, 1]$) (see footnote <fnref n="1" />). These first moments are often assumed to scale as the reciprocal of $S$, so that the total interaction strength stays finite for infinitely species-rich communities [22,25]. Despite many microscopic features being thus neglected, this model displays the main emergent, collective phases that also occur under more complex topologies of the interaction network and more general distributions of the interactions [26–28].

When interactions are sufficiently homogeneous and weak, the dynamic admits a single, globally attractive fixed point where at least some abundances are positive (and the others are zero, indicating that species went extinct). The degree of species coexistence in such Unique Equilibrium regime is quantified by the fraction $\phi$ of species with positive abundance —the ratio between richness, which denotes the number of surviving species, and total community size $S$— which is extensive in $S$ [25]. When inter-specific competition is strong compared to intra-specific competition, multiple low-diversity solutions exist, where the majority of species are pushed to extinction by competitive exclusion [29–31]. Analytical characterization of alternative fixed points has been made possible through replica and cavity methods [22,23,25], Dynamical Mean-Field Theory (DMFT) [32], Kac-Rice formalism [33], and random matrix theory [34]. Symmetric interactions between pairs of species ($\gamma = 1$) and finite demographic noise, for instance, result in locally stable coexistence equilibria whose number increases

exponentially with $S$ [23]. Furthermore, eq. (1) displays rich out-of-equilibrium dynamics, ranging from multiple out-of-equilibrium attractors when $\gamma \neq 1$ [31], to aging —akin to glassy systems with an associated energy landscape [35–37]— and generic slowdown of the dynamics in the vicinity of marginally unstable fixed points [38], to chaos [30,37,39]. These complex phases have been studied analytically under specific hypotheses about the interaction matrix [40], or through semi-analytical approaches involving numerical solutions of the equations for the temporal correlation functions [37] or approximations with an effective stochastic focal species model [30]. The same qualitative dynamical phases occur also in the presence of immigration and/or stochastic perturbations, and when eq. (1) is generalized by including further nonlinearities, like sublinear growth [12,31], time fluctuations in interaction strengths (annealed interaction parameters) [41] or in growth rates [29,42].

One of the most remarkable outcomes of the application of statistical physics methods is the independence of such dynamical regimes from species-level details, at least as long as species are statistically equivalent under changes in species identity and/or time averaging [22,25,29,30,32, 43,44]. Phase diagrams can then be drawn as a function of few effective parameters, such as the rescaled average and mean interactions (see, for instance, fig. 2 in [25]). The phase boundaries, derived exactly in the thermodynamic limit of infinitely large $S$, are robust when community richness is finite.

In the GLV model, species interactions are typically pair-wise and constant. In reality, however, they recapitulate multiple ecological relations, e.g., competition for shared resources or cross-feeding. This additional complexity can be captured by coupling the equations for the species’ dynamics to that of their shared compounds.

These additional variables are described in the classical MacArthur Consumer-Resource (CR) model [45]. This posits that interactions among species are mediated by consumption of limited shared resources, as reflected by simple mass-action laws<fnref n="2" />. In well-mixed communities, the corresponding mean-field-like dynamics of species abundances $N_i$, with $i = 1, \dots, S$, and resources $R_\nu$, with $\nu = 1, \dots, M$, read

$$ \begin{aligned} \frac{dN_i}{dt} &= N_i \left( \sum_{\nu} \beta_{i\nu} R_{\nu} - \chi_i \right), \\ \frac{dR_{\gamma}}{dt} &= h(R_{\gamma}) - \sum_{i} \beta_{i\gamma} N_i R_{\gamma}. \end{aligned} \qquad (2) $$

The coefficients $\beta_{i\nu}$ quantify the preference of consumer species $i$ towards resource $\nu$, and $\chi_i$ accounts for the minimal resource abundance necessary for self-sustainment at steady state. The function $h(R_\gamma)$ represents the resource production rate. Typically, for biotic resources, it is chosen as a logistic function $\frac{R_\gamma}{K_\gamma}R_\gamma(K_\gamma - R_\gamma)$, where $K_\gamma$ is

<footnote n="1">¹The subscript $c$ stands for the connected part of the correlation function.</footnote>

<footnote n="2">²The depletion rate of the resource $\nu$ by consumer species $i$ is directly proportional to the concentration of the resource and the number of individuals of that species, reflecting their encounter rate.</footnote>
the carrying capacity and $r_\gamma$ the growth rate of the re-source $\gamma$, so that $r_\gamma/K_\gamma$ sets the timescale to equilibrate. When preferences $\beta_{i\nu}$ are randomly assigned and resources change much faster than species, eqs. (2) reduce to the functional form of eq. (1), with symmetric interaction co-efficients that now depend on the production and use of re-sources [46]. By further imposing that all surviving species are at equilibrium ($dN_i/dt = 0$), one obtains the condition for competitive exclusion, according to which *M* resources can sustain at most *M* species. In the thermodynamic limit $S, M \gg 1$, statistical physics was used to study key ecological properties, such as the stationary abun-dance distributions or the number and stability of fixed points [47,48]. Several studies have shown that, under ap-propriate constraints on species preferences [47–50], more species than resources can coexist, though these findings appear to be sensitive to higher-order nonlinearities [47, 51]. Similar conclusions are moreover obtained with the addition of self-limitation terms, which effectively capture the influence of spatial structure [52]. From a dynamical point of view, eqs. (2) do not display all the rich out-of equilibrium regimes of the GLV equation (1). In particu-lar, when heterogeneity is large, the latter diverges, while the former more realistically admits a single, globally sta-ble fixed point. Other scenarios, like limit cycles and per-sistent fluctuations, can nonetheless emerge when foraging preferences and resource depletion rates, which in eqs. (2) are both equal to the coefficients $\sigma_{i\nu}$, are not identical [53].

**Comparison with empirical data.** — The three classes of models discussed above translate into generic equations the intuition of how ecosystems work. Since their founding hypotheses differ in what species-level pro-cess dominates the community assembly, these classes have been often presented as alternative explanations of ecological patterns. Among such patterns, the community feature that is most directly obtainable from observational data is the Species Abundance Distribution (SAD) (see fig. 1). Although empirical SADs have been widely com-pared to the stationary solutions of population dynamics models like those discussed before, this approach can be problematic for several reasons [20,54]. First, the distribu-tions predicted by different models are sometimes theoret-ically or practically indistinguishable: either their asymp-totic distributions converge [55] or their differences depend on taking fluctuations due to sampling into account [56]. The shape of empirical SADs can be affected by the scale of description (whether abundances are aggregated over multiple samples, whether they are normalized), sample size and, notably for microbes, on the sequencing depth and bioinformatics pipeline. By using models that do not realistically account for space, sampling, or genomic struc-ture, it is difficult to determine whether SAD regularities have a common ecological underpinning.

Further insight can be gained by studying samples that have been collected and analyzed using the same methodology. Instead of identifying universal laws that

qualitatively describe the data, it is thus possible to quantitatively assess how patterns vary across different samples and environments —such as those from distinct biomes, size classes, healthy or diseased hosts, various geo-graphic locations, or different observational times. Studies on the sample-to-sample variation of parameters fitting SADs suggest that abundances of free-living organisms decay at highly consistent rates across multiple samples of the same type of community, irrespective of the envi-ronmental context [56,57]. Moreover, additional patterns have been revealed by looking at how the abundance of single species varies across samples, notably unveiling a Gamma-like distribution of species fluctuations, whose pa-rameters follow a universal scaling [60]. In microbiomes, statistical physics has been used to distinguish commu-nity states associated with disease and health [46], thus highlighting the potential value of local variations from universality.

To decide which class of models explains empirical ob-servations best, one can integrate other sources of infor-mation than just instantaneous SADs.

First, thanks to high-throughput sequencing becom-ing more affordable, observations of microbial abundances have ever increasing spatiotemporal resolution [61]. In longitudinal data, for instance, a “same” local ecosystem is sampled at relatively short time intervals, offering an additional comparison of deterministic and stochas-tic models. Even when these models produce similar SADs, they differ in the timescales of fluctuations and, consequently, in the trajectories of species abundances. A major obstacle in comparing models with data has

<figure><img src="image_6.png" /></figure>

Fig. 1: An example of a Species Abundance Distribution for a community of marine plankton protists in one station of the Tara oceans expedition (see [57] for the analysis of the full dataset and for the explanation of the fitting algorithm). The histogram of species with a given abundance (circles) is com-pared with a theoretically predicted distribution (continuous line). Here, this distribution is a negative binomial Beta dis-tribution —whose main trend is a power law— derived from a semi-neutral model where *per capita* birth and death rates are density-dependent [58]. A similar power-law decay is pre-dicted by other models, for instance GLVs with strong in-teractions [30] or with time-averaged neutral environmental fluctuations [44], and models describing transport by chaotic advection [59]. More generally, distributions with extended power-law trends match empirical SADs in many different ecosystems [56].
been the intricacy of distinguishing measurement errors, noise, and chaotic fluctuations. New techniques for non-parametric time-series analysis are now lifting these limitations [62]. Additionally, longitudinal data allow to assess the degree of ergodicity of the ecological dynamics by comparing abundance distributions for single species over time to that of all species at a given time, as one typically does in theoretical modeling [30,60]. Even in species-symmetric models, where species are statistically equivalent, these distributions do not necessarily coincide for finite-size systems, and they are expected to diverge even more for realistic heterogeneity [44]. Broken ergodicity manifests as well in multistability. Sudden changes in ecological dynamics, with a consequent catastrophic shift of attractor, are nowadays of major concern for ecologists and decision-makers, and statistical physics techniques can be used for their timely detection. For instance, critical slowdown is a dynamical signature of tipping points between alternative community states [63].

Second, the *curse of dimensionality* —that is, the problem of fitting the huge number of parameters that define species-level dynamics— can be mitigated by identifying community-level degrees of freedom of ecological relevance. Unlike in physics, where established principles guide coarse-graining, in community models relevant macroscopic variables can be selected by integrating ecological knowledge, e.g., on productivity, metabolic activity and trophic position. For instance, abundances aggregated at the level of *functional groups* (ensembles of species that share the same role in the community), are more regular across samples than species composition. This suggests that redundancy may allow ecological dynamics to be captured by a few, functional degrees of freedom [64,65].

Related to the previous two, a third observation is that microbial communities can be found in multiple alternative steady states, and shift among them through external perturbations [66]. As multistability appears to reflect distinct properties and structures of the community [67,68], there is hope that community-level functional landscapes inferred from data [69] could aid in predicting species compositions associated with given ecosystem services.

**Theoretical perspectives.** — In this last section, we provide a perspective view of some major directions that promise significant progress in the near future.

*Role of different sources of randomness.* Since real ecosystems are neither deterministic nor have constant parameters, it is important to understand how to bridge between purely stochastic and purely deterministic models while retaining the description of species interactions. Due to its simplicity, the Lotka-Volterra model equation (1) is particularly well suited to explore the dynamical effects of different sources of noise. Formally, environmental fluctuations are often represented by multiplying the growth rate by a white noise. When added to

single-species models, such as the Stochastic Logistic Model [60,70], multiplicative noise yields realistic macroecological patterns of microbial communities. In combination with just a self-limitation term, such noise produces a Gamma steady-state distribution for the abundance fluctuations of a particular community member across different sites, allowing to collapse multiple datasets [60]. A similar focal-species model, where noise is instead time-correlated [30], produces species distributions with power-law tails, together with intermittent chaotic fluctuations that align with empirical observations [56,57]. These results suggest that low-dimensional models capture the essential macroecological signatures of many different microbial communities when they are complemented with appropriately chosen environmental fluctuations. However, it is still unclear what general property of all (or a class of) ecosystems allows for such a drastic dimensionality reduction. Multiplicative noise can also be added to the full eq. (1), either as fluctuations in the growth rate [42], or with annealed interaction coefficients [71]). In both cases, population abundances can bounce back even if species are on the verge of extinction, a phenomenon that has been termed *stickiness of rarity* [72] and that generates highly skewed abundance distributions. The space of possibilities in terms of noise scaling (e.g., environmental-multiplicative vs. demographic) and time correlation (white vs. coloured) is still not entirely explored, but the effective equivalence of some modelling choices appears to underpin their shared approximation with a single-species, stochastic description [44].

*Inferring and modeling species interactions.* Several inference and interpolation methods have been proposed to quantify the strength of microbial interactions either between species or aggregated degrees of freedom [73]. On the theoretical side, the role of *higher-order interactions* remains to be clarified [74]. Species interactions involving three or more species are gathering growing attention for their biological implications [75], and have the potential to overturn classical results on ecosystem stability. According to May's stability criterion, indeed, when the number of species *S* exceeds a critical threshold, random pairwise interactions that scale like $1/S$ destabilize the ecosystem. The scaling with richness of higher-order interactions, however, depends on interaction order [76]$^3$. Communities therefore appear to be more resilient to sufficiently strong three-way and four-way interactions.

*Finite-size communities.* In statistical physics, exact results are obtained in the limit when the number of microscopic degrees of freedom (species, resources) tends to infinity. Any real ecosystem is, however, composed only of a finite number of species. While natural microbial communities typically have hundreds to a few thousands species per sample, synthetic and non-microbial ones seldom

<footnote>$^3$When higher-order contributions are turned off, the critical strength of pairwise interactions decreases as $1/S$. In contrast, the critical threshold for three-body interactions is independent of *S*, enabling maximization of stability at intermediate system size.</footnote>
harbour more than a few tens. The robustness of
theoretical results to finite-size fluctuations (both for
parameter sampling and in species' number) is there-
fore critical to applications. Many works use numerical
simulations to illustrate analytical conclusions, therefore
implying that system size has no major effect on the con-
clusions, at least as long as it is sufficiently large. Com-
monly, richness can be reduced to a few tenths of species
without qualitatively altering results, and finite-size fluc-
tuations obey the usual 1/√s scaling. In some cases,
however, like for May's original work, reducing richness
yields qualitative regime changes [4,12]. How to assess
the validity of statistical descriptions of real communities
represents an exciting open question.

*Evolution*. The quenched and annealed heterogeneity discussed thus far are sampled from an assigned distribution, and abundances vary on the ecological time scale of demographic change. In natural systems, on the other hand, long-term dynamics is driven by natural selection acting on random changes in species' traits. The emergence of new species or types, and the extinction of others, thus produces biased variation in the parameters represented in a community. Neutral models can encompass speciation, but they assume that selection acts equally on all species. In niche models, the addition of an immigration term can partially mimic the effect of mutations, but it does not allow the number and types of species to expand beyond the initial diversity. Therefore, the inclusion of open-ended community evolution requires new hypotheses on how to implement the mutation-selection processes [51,77,78]. Particularly relevant to applications are nested models, which pose the double challenge of encompassing high species diversity and a high number of separate communities [79–81]. Extensions of niche models that include mutational stochasticity and multi-level selection are highly needed to interpret phylogenetic signatures of community evolution [82] and to fully characterize possible eco-evolutionary effects, such as eco-evolutionary tunneling [83,84] and host-microbiota interactions [85].

*Metacommunities, sparse and structured networks.* In natural habitats, species and community patterns are spatially distributed and shaped by spatiotemporal variations of the landscape. Such inherently patchy ecosystems can be modelled by local, well-mixed communities interconnected through dispersal or migration [86], whose rate is a new key parameter. Describing diffusion on graphs and directed dispersal is particularly challenging for non-ordinary topologies, such as river networks and heterogeneous landscapes [87,88].

Random matrix and network theories provide analytical
insights when edges are assigned based on a specific
probability distribution (e.g., Poisson for Erdős-Rényi
graphs) and offer a means to simplify complex spatial
arrangements. The stability of *metacommunities* is then
found to be affected by both the interaction nature
(competition, mutualism, parasitism) and the degree of
connectivity between distinct patches [89].

Dispersal networks often lead to the emergence of mul-
tiple, weakly connected communities, each consisting of
tightly linked species or clusters of species. Spatial het-
erogeneity then results in local differences among commu-
nities, eventually enhancing coexistence [88]. When the
interaction matrix has a sufficiently simple (sparse, low-
rank or modular) structure, as a result of its spatial
or functional organization, statistical physics approaches
can be used to inquire the stability and dynamics of
the community. For instance, DMFT has been recently
generalized to structured interaction matrices, provid-
ing a method for coarse-graining the community dynam-
ics [27,28,90–93].

*Effective equivalence classes.* Despite the variation in model structure and in their underlying hypotheses, the number of readily distinguishable dynamical regimes is not huge, and the ways the system transitions among them seem to follow standard routes. It is therefore natural to ask which general features of the various models are fundamental to each dynamical transition. For instance, what kind of symmetries make neutral or time-averaged neutral models sound descriptions of natural communities? How do specific observational choices and constraints (e.g., the fact of dealing with compositional variables, the existence of viability ranges, and finite sampling) affect ecologically relevant findings? What features are truly universal, and which ones can be exploited to partition communities into ecologically distinct equivalence classes?

**Conclusions.** – Mathematical models have brought important insights into the way we think of complex natural communities. The ever-increasing availability of highly resolved (in composition, time, and space) data paves the way to more quantitative comparisons between theory and observations. On the one hand, this requires learning how to use low-dimensional effective models to interpret empirical patterns. On the other hand, it implies developing strategies for extracting essential information from large datasets. Measuring every single parameter of ecological models of species-rich communities will remain practically unfeasible. This is where statistical physics can bring ideas, not only for efficiently describing currently available observations but also for planning future ones.

***

AA acknowledges the support of the Agence Nationale de la Recherche (ANR) of the French government, under the grant ANR-23-CE30-0012-01 (“SIDECAR”). SDM was supported by the French Government under the program Investissements d’Avenir (ANR-10-LABX-54 MEMOLIFE and ANR-11-IDEX-0001-02PSL)

*Data availability statement:* No new data were created or analysed in this study.

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