2.1 Model for the dynamics of species-rich, interacting communities

We describe the dynamics of the rescaled abundances x_i(t) of S interacting species, labeled , in the framework of generalized Lotka-Volterra Equations. This is the simplest non-linear model capturing the effect of ecosystem composition on individual demographic rates, which are influenced by both intra-specific competition and inter-specific interactions. In order to focus on heterogeneity in the species’ interactions, we consider a dimensionless version of the equations, which is derived in the Appendix A in S1 Text from a more general formulation in terms of true population abundances.

The variation in time of the rescaled abundance of species i

(1)

depends on the interaction coefficients A_ij that account for the impact of species j on the instantaneous growth rate of species i. Moreover, a small immigration rate (set here to m = 10⁻⁸) is added to avoid species getting permanently excluded from the community [4,28]. Species that persist at the immigration level can therefore invade as soon as the community context becomes favorable. At any given time in the simulations, however, species whose abundance is smaller than a multiple of m are counted as extinct in order for immigration not to artificially inflate species diversity.

We model the interaction matrix A as the superposition of a structural and a random component (schematically illustrated in Fig 1). The first encodes our ecological knowledge of the community and the second all latent interactions that are too intricate to be accurately characterized. Other potential sources of stochasticity, such as demographic or environmental fluctuations, are not included in the model. When the matrix is completely unstructured, Eq (1) reduces to the so-called disordered gLVEs, whose dynamics has been extensively characterized under different assumptions on the statistics of the random interactions [4,22,28,29]. The structural part, on the other hand, is often modeled explicitly in various ways, e.g. with trophic or size classes, or the use of metabolic substrates and the production of public goods [8,30,31], but it is reduced here to its effect on pairwise interaction coefficients. The two terms of such decomposition can generally vary in their relative importance, bridging between two extreme cases that have been previously addressed with distinct mathematical approaches.

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Fig 1. Examples of community-level structure and species-level randomness in species interactions.

We model interactions between individual species abundances x_i resulting from the superposition of structured (left panels) and random components (right panel), whose relative magnitude can be tuned. Three examples of ecosystem structures (top row) produce different species interaction matrices A_ij (bottom row). Structure in each matrix follows Eq (3), where interactions between species (black disks) are mediated by a set of community-level processes (black arrows) involving collective functions (gray squares) defined in Eq (2). In our three examples, species interactions depend on: A. the total abundance of a small set of functional groups (e.g. plants, herbivores...), so that the block structure of the matrix reflects the group structure of the community (different colors indicate different values of interaction between pairs of groups); B. the availability or magnitude of a small set of public goods (e.g. resources or environmental variables), in which case the rank of the matrix is bounded by the number of public goods, resulting in highlighted lines and columns; C. traits that depend on phylogenetic distance, as encoded by the fractal structure of the interactions (larger blocks correspond to higher-order taxa, and block colour to the intensity of interactions between pairs of taxa), in which case are the total biomasses of subtrees of various depths. D. other unknown or “hidden” processes that affect every species independently are captured by the addition of interaction terms sampled at random from a distribution with given statistics.

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2.2 Structured interactions

We define structure as resulting from the existence of a set of n_F relevant community-level functions, which will be in limited number by construction. Functions are indexed with Greek letters, . Every species can contribute to a community function and can be affected by it (see examples in Fig 1). This relationship is characterized by two sets of species-specific functional traits (as illustrated in Fig B in S1 Text). The impact trait quantifies the per-capita contribution of species i to function . The sensitivity trait gives the impact of function on the per-capita growth rate of species i [32]. These can also be interpreted, along with [33], as the impact and requirement niches, respectively.

The total magnitude of the function at time t is for simplicity chosen as the sum of all species’ contributions, resulting in the linear combination of the abundances

(2)

Here and in the following, the overline defines an average over species for a fixed set of parameters (interaction coefficients). Due to the factor S⁻¹, each species’s contribution to a given function is small in comparison to that of the rest of the community, in accordance with our focus on collective functions – which cannot be achieved by one species alone. Similarly, the total impact of functions on a species’ growth rate is assumed to be the sum of the functional magnitudes, weighted by the sensitivity traits, i.e. .

Interactions mediated by collective functions are then entirely encapsulated in a structural matrix

(3)

where the effect of any species on another depends on their functional traits. Conversely, any interaction matrix with collective interactions can be written in the form of a product of trait vectors via its Singular Value Decomposition, even though the functional traits thus obtained need not correspond to known ecological processes. We show in Appendix B in S1 Text that, due to the scaling choices in Eq (3), we can always restrict ourselves to (i.e. our structural matrix is always low-rank) with important consequences for our results.

Our approach can in principle be extended beyond linear community functions. Linearity can be relaxed either in the definition of the functions, or in their contribution to the species growth rate, thus encompassing interaction structures that depend nonlinearly on traits, e.g. trophic niche models based on body size differences [34], as well as saturating functional responses or higher-order interactions. Some of these possibilities can be captured by straightforward modifications of our analysis, following existing work [5,35–37], while others may require a more complex treatment, e.g. [38].

Fig 1 illustrates the structural matrices derived in three examples of communities structured by functional traits. When functional or phylogenetic groups exist [39], all species of a group have an even impact on its biomass, i.e. if the species is part of the group and 0 otherwise, and we assume that species are sensitive to the total biomass of the groups. When community functions are public goods or resources, species impact and sensitivity are based on their consumption (or production) of those compounds, and on the yield, respectively [15]. Most of our later examples will be illustrated in the more intuitive case of non-overlapping functional groups, but the generality of our analytical approach is demonstrated in Appendix I in S1 Text.

2.3 Combining structure and randomness

Structure being thus defined, we now turn to the unstructured part of the interactions, reflecting sources of between-species variation not described by community functions. These non-observable features are modelled as random terms [1,22,40], and are defined as a random matrix of entries (Fig 1D). For simplicity, the z_ij’s are chosen as independent, identically distributed variables with zero mean and unit variance. The specific details of the probability distribution do not matter, as long as it has finite variance and doesn’t depend on S. A generalization to z_ij with non-identical variances is provided in Appendix G in S1 Text.

The total interaction matrix A_ij is defined as the sum of the structured and of the disordered components

(4)

The S^−1/2 scaling factor, again, ensures that single interaction terms are small compared to the aggregated effect of the whole community. Given that (see above and Appendix B in S1 Text), the total interaction matrix is the superposition of a low-rank and a random term.

The parameter balances the relative importance of structure and randomness. Without structure, Eq (1) reduces to the classical disordered gLVEs studied in statistical physics. Accordingly, we will consider typical ecosystem outcomes, i.e. those that do not depend on the specific realization of randomness. The expectation is that, when structure is added, the statistical properties of the random term should be sufficient to characterize behaviours that are independent of the unknown species-level details. In the spirit of macroecology, moreover, such generic predictions are expected to be connected with recurrent empirical patterns.

3.1 Effective community dynamics reflects both structure and disorder

In the absence of randomness, , the Lotka-Volterra system’s behaviour is entirely determined by the structured relations between species. We can rewrite Eq (1) as

(5)

and thus express the dynamics in terms of the functional variables . In this case, the equilibrium abundances of species are exclusively set by the community-level variables, with multiple species (that share the same interaction coefficients) collapsing onto the same abundance (Fig 2A, left). For instance, when modeling how the total abundance of plants depends on that of herbivores, or how pH depends on water availability, one considers that all plants are the same, or that all species respond to the same environmental variable. Such strict correspondence between the microscopic and macroscopic scales also holds in the dynamics (Fig 2A, right), where equivalent species asymptotically follow the behaviour of their average abundance (thus determining macroscopic variations of the functional variables), even when these cancel out when averaging over the whole community. Community-level static or dynamical relations yield various ecological patterns such as biomass pyramids, trophic cascades or regime shifts, see e.g. [43,44], which directly trickle down to species scale as long as species are held equivalent.

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Fig 2. Theoretical expectations from either structure or randomness in species interactions.

With the modeling choices in Eqs (3) and (4), both purely structured and purely random species interactions allow species-level predictions, such as Species Abundance Distributions (SAD) and stability, but also a low-dimensional effective description of the whole community. A. Interactions structured by a small number (compared to the number of species) of community-level degrees of freedom define classes of functionally equivalent species. Community-level descriptors are then recapitulated by the collective functions and their relations, both at equilibrium (left) and out-of-equilibrium (right, where functions in colour are, for instance, the mean x_i(t) over all species of the same functional class). Functionally equivalent species keep a coherent dynamics that emerges at the macroscopic level. B. When interactions are randomly assigned, the static community-level patterns (abundance distribution, left) are accounted for by statistical metrics, e.g. the degree of dispersion in species interactions . Out of equilibrium, complex dynamics can manifest at the species level, but average out in macroscopic observables (right, note that here macroscopic functions represented in gray are averages over the whole unstructured community).

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Once randomness is introduced, the connection between community-level and single-species dynamics is less straightforward (Fig 2B). DMFT provides an appropriate framework for studying aggregated properties that are shared by typical realizations of the disordered matrix [4]. This approach describes aggregated, community-level variables, usually called “self-averaging”, that are the same for any (large) community whose fixed set of interactions is randomly sampled from a distribution with given variance. Contrary to such coarse-grained quantities (represented by in Fig 2B left), microscopic variables such as the abundance of a single species (whose distribution is represented in Fig 2B left) generally change from one to another realization (sampling) of randomness.

DMFT can be generalized to encompass structure and disorder in the interaction matrix Eq (4). We summarize in the following the essential steps of the derivation of such Structured DMFT, and leave the details for Appendix D in S1 Text. A key feature is that, due to the large number of species, a formal equivalence can be established between the original dynamical equations with random interaction coefficients and a set of S uncoupled Stochastic Differential Equations (SDE)

(6)

where the dynamics of species i depends, via the species-specific sensitivity traits, on the magnitude of functions and on a Gaussian colored noise . The noise term in Eq (6) depends on the abundances of all species in the community through the relation , where is the two-time average of abundance, or community correlator,

(7)

Here, brackets indicate averages over a single species in multiple realizations of the random matrix, while the overline is an average over species for a given realization, as defined in Eq (2). Owing to the law of large numbers, averages over species equal averages over species and randomness. This implies that, as for the ordinary DMFT, the correlator doesn’t depend on the matrix realization.

Similarly, for a fixed value of randomness intensity , functional variables are self-averaging and can thus be directly computed from the abundances averaged over realizations of the randomness:

(8)

Such averaging makes them independent of species-specific details represented by the random contributions z_ij, while they retain the structural features through the effect traits . The most significant difference with prior studies is that, while DMFT in the usual setting yields a low-dimensional model, i.e. a single equation representing the distribution of possible trajectories of any species picked at random in the system, here Eq Eq (6) are still as numerous as in the initial model, because species are intrinsically different from each other due to their distinct traits ’s and ’s; what has been gained is that the equations are now uncoupled.

Due to the non-linear nature of Eq (1), the simplicity of the equations does not entail simplicity in the dynamics. Every species is influenced by n_F  +  1 community-level, self-averaging quantities: the functional magnitudes Eq (8) and the correlator Eq (7). Just as the functions are sufficient to capture the effect of structure in the absence of disorder, the effects of disorder are summarized by the sole correlator and are not species-specific.

The statistical equivalence between the original dynamics and trajectories obtained from the coarse-grained equations hinges upon such community-level observables being self-consistent, that is matching the microscopic, species-level stochastic processes. Their computation can in principle be achieved numerically [28], as in the unstructured case. The algorithm however would have important computational costs, exacerbated in our case by the fact that different species have different statistics. We therefore concentrate here on exploiting analytical results that can be obtained at equilibrium, and use them to gain also insight on the onset of out-of-equilibrium regimes.

3.2 A closed set of community-level variables determines equilibrium patterns

We show that, when the community is at equilibrium, the previously defined coarse-grained variables obey closed equations, which in turn allow us to characterize species abundance distributions. In the next section, we will discuss the conditions for such an equilibrium to be stable.

Here, we focus on the Species Abundance Distribution (SAD), i.e. the fraction of species of at a given equilibrium abundance . A long tradition in ecology looks for shared patterns in such distributions, and compares them with the predictions of theoretical models that typically encompass randomness but not structure [45–47] (but see [48]).

At equilibrium, the n_F functional magnitudes and the correlator attain the values

(9)

As discussed in Sect 3.1, the equilibrium abundances of individual species are not self-averaging, and therefore depend on the realization of randomness. Still, it is useful to investigate how macroscopic structure biases species towards smaller or higher abundances. We do so by defining the modal abundances:

(10)

which we will later prove to correspond to the most likely abundance at equilibrium of extant species (see Eq (13)). Unlike species abundances x_i, modal abundances depend only on the magnitude of ecological functions, and are therefore self-averaging. Finally, let us introduce the parameter

(11)

that increases with the intensity of disorder and is also a self-averaging quantity.

When interactions are dominated by structure (), the equilibrium values of species abundances coincide with the modal abundance, and are readily obtained from Eq (5) as . Equilibrium functional magnitudes then obey the closed set of n_F equations:

(12)

that only involve functional traits.

The distribution of species abundances in the absence of randomness can be easily visualized when the community is structured in homogeneous functional groups (Fig 1A). Every species belonging to a same group has the same traits, therefore exactly the same equilibrium abundance, which corresponds simply to the solution of the low-dimensional Lotka-Volterra equations for the functional groups. Panels A and B of Fig 3 illustrate the theoretical SAD for a community structured in four groups, where abundances of non-extinct species concentrate over just three modal values, with relative frequency equal to the fraction of species in each group. Species with negative modal abundance are driven towards extinction by their interaction with the community. Due to migration, they never go fully extinct and persist at small abundances . Since , they do not significantly contribute to the ecology of the community, and we count them as extinct.

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Fig 3. Simple interplay in the Species Abundance Distribution.

The equilibrium SAD is a convolution of the predictions of structure and disorder. Top row: schematic illustration of the theoretical predictions of Eq (16) when the abundances are at equilibrium. Bottom row: numerical SADs for a community structured in two functional groups of different size, whose interaction matrix A_ij, plotted in the inset, is either purely structured or has a random component of intensity . A. and C. In the absence of randomness (), species abundances are set by the sensitivity traits to collective functions (impact traits being identical within groups). B. and D. When increases, the distribution of species abundances within groups widens by a factor around each trait-based peak , but these peaks are also displaced due to changes in (see also Fig 4). The SADs for the two groups are separately plotted in C. and D. and their mean abundances (solid vertical lines) show that a signature of the underlying structure persists even when randomness is so strong that the total SAD would appear featureless.

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The addition of random interactions modifies this solution, and species will have a different equilibrium abundance even if they share the same functional traits. The effect of increasing the variance is to spread the abundances around the modal abundance. Species that share the same will have the same , consistent with the idea that they respond coherently to the function , despite individual heterogeneity. More precisely, in Appendix F in S1 Text we show that, when , the equilibrium abundance of species i is randomly distributed according to a truncated-Gaussian random variable,

(13)

where are independent standard Gaussian variables that reflect the diversity of traits among different species.

Remarkably, community-level variables are related by a closed set of relations. Despite accounting for most of the dimensionality of the interaction matrix, the random component thus modifies the system in a minimal way, equivalent to adding a single macroscopic relation on top of those provided by structure. They read

(14)

where the defined in Eq (10) only depend on collective variables and species traits, and we have defined the mathematical functions

(15)

i.e. the kth moment of a truncated Gaussian distribution centered at w. In particular, represents the area with positive abundance, and will be used below to define a species’ survival probability.

We can now exploit the self-consistent relations Eq (14) to derive the exact distribution of the microscopic abundances. The SAD turns out to be a convolution of a ‘structure-driven SAD’, reflecting the deterministic biases given by the modal abundances , and of a Gaussian distribution of typical width (Fig 3B):

(16)

Here, is the Probability Density Function of a standard Gaussian, is the Heaviside function and

(17)

is the diversity, that is, the total fraction of species that survive without being rescued by migration.

In Fig 3C,D we provide an example of a simulated simple community composed of two competing functional groups: each group contains a different number of species, and all species within one group share the same structural traits (interaction matrices are represented in insets). The SAD is trivial in the absence of randomness, as one group excludes the other at equilibrium (Fig 3C). When random variation is added, the excluded group is rescued, and the two peaks widen until they largely overlap (Fig 3D). Yet, the existence of an underlying structure remains detectable if the mean abundance of each group is plotted (solid vertical line). The structure-driven SAD can also feature more complex profiles that reflect the distribution of traits within the community or the functional groups. For instance, if the total abundance is the only community function and the sensitivity traits are distributed as a power-law (see appendix I.1 in S1 Text), the structure-driven SAD will have a power-law tail which progressively turns into a Gaussian shape as randomness is increased (Fig B in S1 Text). For the presence of structure to have no impact on the SAD, should have particular symmetries. This happens, for instance, when the structural interactions derive from phylogenetic proximity in a balanced binary tree (see Appendix I.2 in S1 Text).

The SAD is therefore usually very sensitive to the presence of structure in the interactions. Its exact shape is dictated by the interplay of structure and randomness, and it is not expected to follow universal rules, other than those possibly imposed by universal scalings of species traits.

Even though randomness blurs the SAD, the effect of structure never vanishes in the self-consistent relations Eq (14), but the magnitude of functional variables is modified by heterogeneity (Fig 4). We see in these equations that increasing randomness has two effects. First, all else being equal, functional magnitudes tend to increase with , since abundances spread, but they are bounded only on the left by extinction (Fig 4B). Second, it causes all species to contribute more evenly to functions, due to the rescaling . This, notably, reduces the differences between species that deterministically would persist and those that would be excluded (positive or negative ). The probability, measured over realizations of randomness, that species i persists, as per Eq (17), is different from zero and one for . Hence, species that would be excluded based solely on structured interactions can be “rescued” by randomness and contribute to functions, leading to non-trivial effects, even in the simple case of two functional groups illustrated in Fig 4A. In that example, group 1, initially excluded, can take advantage of randomness and drive a community-wide increase in diversity (Fig 4C). This increases the competitive pressure on group 2, and the functional dominance in the community can eventually get reversed, with the less-competitive group 1 becoming more abundant.

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Fig 4. Shift in equilibrium relations between macroscopic observables.

Even as randomness increases, the collective variables remain meaningful observables connected by deterministic functional relations, but their balance can shift due to heterogeneity , following Eq (14). Applied to various interaction structures, these relations translate to many ecological patterns of interest, e.g. biomass pyramids [43,49] or balance between resources [50]. A. Similar to Fig 3, we consider an example community structured in two competing groups: group 1 (blue) is larger but less competitive, whereas group 2 (orange) is smaller but more competitive. B. The functional variables (total group biomasses) change with the magnitude of random interactions (dots, numerical simulations; solid lines, theoretical prediction). At , all species in a group have the same abundance, and group 1 remains at the migration threshold. As increases, heterogeneity starts to matter, and the larger group 1 can contain more “lucky” overabundant species, rescuing it from exclusion then surpassing group 2 in total abundance. C. This simple interplay of structure and randomness can yield non-trivial patterns. Increasing randomness initially increases total extant species diversity (grey: overall fraction of surviving species) by rescuing the blue group from exclusion (blue: fraction of surviving species within that group), but eventually reduces diversity in both groups and as a whole.

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In summary, the addition of randomness preserves the existence of deterministic relations between macroscopic functions, but it changes the magnitude of these functions through the nonlinearities in Eq (14), notably by determining which species are extinct or not, and equalizing their contribution to functions.

3.3 Randomness can drive fluctuating communities to stable equilibrium coexistence

The previous analytical results rely on the existence of a stable fixed point, yet Eq (1) with structured or random interactions is known to also display out-of equilibrium dynamics, including limit cycles, neutral cycles, and chaotic behaviour [51]. In particular, it has long been theorized that adding random interactions past some critical value of destabilizes communities that would otherwise be at a fixed point [1]. We show next that increasing can also have an opposite, stabilizing effect on communities that would otherwise be out-of-equilibrium [52].

As an illustration, we explore the effect of introducing random variability at microscopic scale in a case where the macroscopic variables exhibit chaotic dynamics [53]. As before, the smaller scale will be called ‘species’ and the larger one ‘groups’ (whose total abundances are the associated ‘functional variables’), but the argument applies for any pair of taxonomic levels, e.g. strains within species. The recent availability of data at high taxonomic resolution, as well as theoretical investigations, indeed indicate that both functional groups and species harbour a degree of genetic and trait variation, with dynamical consequences [54–57].

We consider a set of n_F groups and divide each of them into s species. The community is thus composed of a total S = s n_F species that constitute the microscopic variables of our model. In the purely structured (group-scale) model, detailed in Appendix I.3 in S1 Text, a combination of strong interactions and weak immigration causes a persistent turnover between a small set of dominant groups and a large set of rare groups, thus reproducing dynamic and static patterns that are characteristic of microbial and plankton communities when observed at a high taxonomic resolution [57,58].

Fig 5 displays the asymptotic regimes of a community with fixed structural interactions and for increasing values of within-group variability. When , all species within a group have the same chaotic trajectory as the group they belong to (Fig 5A).

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Fig 5. Dynamical interplay of complex interactions at microscopic and macroscopic scales.

A matrix of group-group interactions is chosen as in Appendix I.3 in S1 Text to generate complex group dynamics. Each of n_F = 150 groups contains s = 40 species. The within-group variance of the added random interactions between species increases from top to bottom (). Left column: the trajectories of species and groups can be compared by projecting them onto a complex plane. This is achieved by plotting the Hilbert transform of the abundance against the abundance itself. Here, we consider the trajectory of the average abundance f₁ of the first group (solid line) and a snapshot of the abundances of individual species belonging to that group (blue dots). Middle column: stacked abundances of groups. Right column: abundance trajectories for a couple of individual species within a few groups (of corresponding colour). A. When there is no disorder, species overlap with their average (gray dot), illustrating the coherence of within-group dynamics. Groups and species undergo chaotic dynamics, and within-group differences are transient effects of distributed initial conditions. B. As within-group mismatch in interactions is introduced, species spread out, but they remain close to their group’s mean even if the qualitative dynamics changes – here it is a limit cycle. Species within groups have persistently distinct trajectories, reflecting their individual traits. C. For intermediate heterogeneity, oscillations are lost both at the group and at the species level, and species have different equilibria. D. Increasing further, species become unstable through a collective transition, whereby their trajectories start to fluctuate asynchronously and chaotically. The dispersion of species belonging to a same group results in averaged-out fluctuations, which are however discernible at the group level when the community has finite size. E. The amplitude of fluctuations at the level of species (blue) and groups (orange) varies non-monotonically with : before stabilization at , both are similar and decreasing, due to synchronization (as in rows A and B)); after destabilization (as in row D), species variance increases due microscopic chaos, but group variance remains small because the fluctuations at the species level are now asynchronous.

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When randomness increases, species remain largely synchronized within groups, so that the two levels of description have the same qualitative dynamics. Their trajectories get however progressively modified and the community crosses different dynamical regimes (Fig 5B), until it reaches a stable equilibrium (Fig 5C) where numerous groups coexist even in the absence of immigration. For even larger , the species start to fluctuate again chaotically, but are desynchronized – so that their incoherent oscillations cancel out when abundances are averaged within groups (Fig 5D). Unlike when , the dynamics at the species and group levels are now decoupled.

These results show that microscopic heterogeneity has different predicted effects depending on the scale of aggregation that is under scrutiny (Fig 5E). When one is interested in the microscopic scales (here, species), heterogeneity has a non-monotonic effect: it is first stabilizing and then destabilizing for strong-enough disorder (as already emphasized in many theoretical studies [1,4,22]), leading to complicated chaotic dynamics. On the other hand, the macroscopic scale (here, groups) is stabilized by intermediate heterogeneity, and fluctuates only little (and less and less if the number of species per group ) even after the microscopic dynamics looses stability.

We now turn to a theoretical explanation of the dynamical regimes highlighted in Fig 5. We focus on the stability of the intermediate-randomness equilibrium, and show how the transition towards synchronized out-of-equilibrium dynamics (for small and that towards microscopic chaotic dynamics (for large ) can be related, respectively, to the random and the structured part of the interactions. The species-level instability occurring for strong randomness (Fig 5C, 5D) is akin to the bulk-driven collective phase transition of gLVEs with purely random interactions [4]. Both the Jacobian matrix and the reduced interaction matrix (where only non-extinct species are considered) undergo a bulk instability when

(18)

where is the total fraction of surviving species, defined in Eq (17). While purely random systems obey the same condition, the location of the transition changes here with structure through the dependence of on . The bulk-driven transition to microscopic chaos, predicted for in the many-groups example, is confirmed by the numerical simulations.

On top of this classic collective transition, structure-driven transitions to equilibrium can occur as bifurcations of the coarse-grained variables. Instead of being related to the bulk of the Jacobian matrix, these bifurcations involve its outliers. Directly linked to the low-rank matrix structure , such eigenvalues are also modified by randomness. We show in Appendix H in S1 Text that structure-driven transitions can be located using the ‘pseudo-Jacobian’ matrix

(19)

which explicitly depends on the structural part of the interactions, but not on randomness. Nonetheless, randomness impacts indirectly through the equilibrium abundances , see Eq (13). Thus, structure-driven transitions are signaled by eigenvalues of Eq (19) having a vanishing real part. To illustrate this, we report the spectrum of for the system of Fig 5 in Fig A in S1 Text.

Whereas the large-S collective transition Eq (18) involves an extensive number of bulk eigenvalues becoming unstable at once, here the increase of microscopic heterogeneity can cause a single pair of conjugate outliers to cross the imaginary axis through a Hopf bifurcation. The mathematical underpinning of how microscopic randomness generically suppresses collective fluctuations is addressed elsewhere [52].

Here, we used this example to stress the fact that microscopic randomness and macroscopic structure may appear to impact fixed point stability through independent pathways (corresponding to bulk and outlier eigenvalues of the interaction matrix, respectively), yet microscopic randomness can actually impact and even stabilize complex structure-driven dynamics through its simple effect on equilibrium abundances, as shown in Fig 3.