The ground model

To illustrate the ideas we use a well-known model and introduce two variants to that model. The canonical model is MacArthur’s resource consumer model (MCRM) [15], that will be referred below as the “ground model”. The variants introduce small changes to its dynamical evolution.

The MCRM describes the dynamics of S species abundances N_i(i = 1, ‥, S) competing over M types of resources R_β(β = 1, ‥, M). The MCRM system evolves according the following set of coupled differential equations (1) where c_iβ describes the consumption preference of species i for resource β. m_i is a minimum maintenance cost that must be met by species i for it to grow. K_β is the carrying capacity of resource β. The first equation includes a migration term η_i from a species pool. It will taken to be small, allowing species to invade if they have positive growth rates.

Following [15], we consider the case where resources dynamics equilibrate quickly compared to the species dynamics, so that dR_β/dt ≈ 0, and therefore R_β = K_β − ∑_j c_jβ N_j. Substituting this into the first equation yields (2) where , and the superscript (r) denotes resource-mediated interactions. This equation is now in the form of generalized Lotka-Voltera equations.

In going from Eqs (1) to (2) there is a risk, that the dynamics will pass through negative values of R_β, which are biologically unrealistic. As shown below, all the phenomenology described using Eq (2) is also present in a model starting from Eq (1). See discussion in Sec. Theory for the onset of non-equilibrium dynamics, and S1 Text (Appendix I).

Demonstrating the sensitivity

A key result by MacArthur [15] is that the model in Eq (2) exhibits globally stable dynamics, reaching a single fixed point independently of the system’s initial conditions. In this section we show that by a small addition of other interactions on top of the resource competition interactions described above, the system is no longer guaranteed to approach a fixed point. Instead, for a broad region of control parameters, the species’ abundances fluctuate indefinitely, as shown in Fig 1(C).

To demonstrate this phenomena we introduce the first variant of the MCRM, which includes additional “direct” species interactions, , so that the total interaction coefficients read , with ω controlling the strength of the perturbation. These direct interactions may come as a result of many mechanisms that lie beyond the unperturbed MCRM. The important point will be to find when such additional interactions have a significant effect on the dynamics, even when they are small.

To quantify the size of the perturbation, we use the ratio of the Frobenius norms (sum of squared interaction coefficients) of the interaction matrices, setting ‖ω ⋅ α^(d)‖_F/‖α^(r)‖_F = 0.05 throughout. For any given model parameters, ω is chosen to satisfy this condition, allowing for a comparison between results with different model parameters.

The quantities c_iβ, K_β, m_i, that define the interactions are drawn at random, representing a generic diverse community, without any additional structure beyond that already incorporated into the resource-competition model. The parameters c_iβ, m_i, K_β are drawn independently for each value, and are drawn independently, except possibly a correlation between and controlling the symmetry of the direct interactions. All quantities are drawn from Gaussian distributions, parameterized by their first two moments. The results depend only on combinations of these first two moments with S, for example, and μ_c = S mean(c_iβ); for sparse networks S is replaced by the number of non-zero links per species. The definitions of all these control parameters are given in S1 Text (Appendix A).

The analytical treatment requires that S is large. Very strong competitive interactions, which would lead to mutual exclusion between pairs of species, are assumed to be rare in the community. A significant number of mutually exclusive pairs can lead to a different phenomenology [24].

As seen in Fig 1(C), the variant with even these small additional interactions shows persistent abundance fluctuations, even while the ground model reaches equilibrium as it always does.

In this section we provided a proof-of-principle demonstration of the sensitivity of the equilibrium to variations in the model dynamics. We now turn to show that this is in fact a robust and general phenomenon, found in a broad regime of parameters, and explain where it is expected and why.

Theory for the onset of non-equilibrium dynamics

To understand more generally whether and when the variant of the model will reach a fixed point or a non-equilibrium state, we assume a fixed point is reached and ask whether it is stable. The basic idea is that systems are sensitive to perturbations, if the fixed point in the ground model is close to marginal stability.

Before turning to the present model, we review the results for the random Lotka-Volterra models, which in the terminology of Sec. Demonstrating the sensitivity have only “direct” interactions, . Their dynamics have been studied recently [8, 9, 12, 14] (see also related results in other models [6]). A number of sharply delineated regions in parameter space, or “phases”, have been found. In one phase the system reaches a unique equilibrium. The boundary of this phase is marked by loss of stability of these fixed points. Beyond this boundary (with a sharp transition at large S) lies another phase, where the dynamics fail to reach a fixed point and abundances fluctuate indefinitely [14]. In a special case where the interactions are symmetric, namely α_ij = α_ji, this phase is instead characterized by many possible alternative equilibria, all of which are close to marginal stability [12].

The behavior of the model variants defined here (above and in Sec. Non-linear resource intake below), bears many similarities to that of the random Lotka-Volterra models, see Fig 2. There is a unique-equilibrium phase, which is delineated by a boundary at which the equilibrium loses its stability. Beyond it, we find in simulations that the dynamics never reach a fixed point, see Fig 2(C). The case of symmetric is special and appears to follow the scenario in [12], see S1 Text (Appendix H). An important difference from random Lotka-Volterra models is the mechanism by which fixed points lose their stability, which we now discuss.

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Fig 2. The model exhibits three phases, i.e. regions with qualitatively distinct behavior.

In one, the system converges to a stable fixed point (FP), in another fixed points of the system are unstable yielding persistent dynamics (PD). In the third phase, unbounded growth (UG), species abundances grow without bound. ω is adjusted in order to maintain constant perturbation strength of 0.05. (A) Color map of the ratio S*/M, indicating how close the system is to competitive exclusion S*/M = 1, assuming an equilibrium reached. In the PD phase the calculation of S*/M is no longer valid and S*/M > 1 may be reached (see later sections). (B) The minimal real part of eigenvalues of the interaction matrix between coexisting species, λ_min. Fixed point stability is lost at λ_min = 0, where the solid line separates the FP and PD phases. The increase in S*/M reduces the stability of the equilibria, triggering a transition to persistent dynamics. (C) Probability of reaching persistent dynamics along a vertical cross section of the diagrams in panels A and B, in simulations with different pool richness S. The transition between equilibrium and non-equilibrium outcomes becomes sharper as system size increases, and matches the theoretically predicted transition point between the phases (dashed line).

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We describe a method of calculating the species richness and stability of the fixed points for the model variant described in Sec. Demonstrating the sensitivity. This method is exact when the system admits a unique fixed point; the loss of its stability marks the boundary of the phase. To highlight the relation between marginal stability and sensitivity to perturbations, we study the spectrum of the interaction matrix, and how it changes for the model variant described above. A different approach, using Dynamical Mean Field Theory, is possible and ultimately equivalent, and has been employed on a related model in [7].

Consider fixed points of the dynamics, i.e. abundance vectors for which in Eq (2). We are interested in the linear stability of these fixed points under perturbations. Specifically, we consider a perturbation which modifies the growth rates of the surviving species (species for which N_i → c > 0, even as the migration η_i → 0⁺). Eq (2) changes to dN_i/dt = N_i[… + ξ_i] + η_i, where is a perturbation, constant in time (press perturbation). The change in the fixed point abundances of the surviving species, due to a small perturbation and at small migration η_i → 0⁺, is given by Here α* is the reduced interaction matrix, comprised only from interactions between surviving species. This expression diverges when α* becomes singular, i.e. when the minimum of the real part of the eigenvalues, λ_min ≡ min{Re[Λ(α*)]}, touches zero. It signals the loss of stability of the fixed point. A fixed point is called marginally stable if λ_min → 0 when S → ∞.

While the MCRM only has stable or marginally stable fixed points, i.e. with λ_min ≥ 0, in the model variant λ_min can cross zero and become negative. In related models [6, 7, 14] this signals a transition to a phase with persistent dynamics. We show in simulations that this also happens here, see below (Fig 2(C)). This result can also be derived more formally, as has been done for other models [6, 7, 14].

Close to marginality, i.e. when λ_min is zero or close to zero, even a small perturbation is able to push parts of the spectrum to λ_min < 0. This is the case for a broad region in parameter space, as we now show. In Fig 3, we show the spectrum of such an α* matrix close to a marginal fixed point. As expected, the marginal case is characterized by non-vanishing density of eigenvalues arbitrarily close to zero. When applying a small random perturbation to the marginal interaction matrix α*, for example described in Sec. Demonstrating the sensitivity, the spectrum is broadened and may cross zero to give eigenvalues with negative real parts, resulting in a dynamically unstable fixed point. The properties of the fixed point depend crucially on the species richness (the number of species that survive), which is a result of a balance between competition that pushes species out of the system, and migration which allow them to try and invade.

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Fig 3. Spectrum of α*, the interaction matrix of persistent species, in the ground model (blue), and the variant with additional direct interactions (orange).

The perturbation spectrum is shown in red, to illustrate its size we normalize the area under the perturbation spectrum to the size of the relative perturbation strength (0.05). (Inset) Minimal eigenvalue real part of the reduced interaction matrix α*, when varying σ_c at fixed μ_c. Solid line is theoretical curve. A phase transition occurs when the minimal real part of the eigenvalues crosses λ_min = 0, from λ_min > 0 at which fixed points of the dynamical system are stable, to λ_min < 0 where all fixed points are unstable and persistent dynamics ensue. For example, here the transition happens at σ_c close to 1.

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The method for calculating the spectrum consists of the following main steps: first, we find the number S* of coexisting species using the cavity method. This follows similar calculations in previous works [9, 13], and is detailed in S1 Text (Appendix C). We then calculate λ_min, the minimal real part of the eigenvalues of α*, for a reduced interaction matrix with S* species. This is done using random matrix theory and detailed in S1 Text (Appendix D).

Following this method, we can predict the dynamical behavior as a function of the model parameters. We find three phases, shown in Fig 2. In the first, the system converges to a unique fixed point, independent of the initial conditions, as in Fig 1(B). In the second, the system fails to reach a fixed point, with abundances fluctuating indefinitely, as in Fig 1(C). In the third phase the abundances diverge, indicating that the model is no longer adequate in this parameter regime.

Notably, when S*/M ≈ 1 the unperturbed system is close to competitive exclusion and correspondingly close to marginality, and therefore the model variant with the direct interactions becomes unstable, i.e. λ_min < 0. As expected theoretically, the transition between the two behaviors is sharp when S, M are large, and happens at the theoretically predicted value of the parameters, see Fig 2(C). In less diverse systems, the transition is more gradual. Thus, our analytical method allows us to precisely predict the location of the transition between equilibrium and non-equilibrium phases, which is borne out by simulations.

The loss of stability of putative fixed points results in persistent dynamics where species invade but are then pushed back out by the instability of fixed points. This is clear in Fig 1(C).

As mentioned before, when going from Eqs (1) to (2), we have replaced the resource values R_β with their equilibrium values without making sure that resource values remain positive, R_β > 0, as expected for resource abundance. We show in S1 Text (Appendix I) that all the phenomenology described hitherto is also found in a model that starts directly from Eq (1), where resources are modeled explicitly, and prevented from going negative by migration. The main difference is that now equilibria loose their stability when S*/M* ≈ 1, where M* is the number of resources present in the community (that have not gone locally extinct). There we also show that the phase where abundances diverge can be removed by stronger limitation on growth rate at high abundances, with little effect on the transition between equilibrium and persistent dynamics.

Non-linear resource intake

So far, we have discussed the ground model with a small addition of other interactions. This allows us to identify regions in parameter space where the ground model is sensitive to perturbations. By adding interactions that are not mediated by resource competition, the model variant can no longer be strictly interpreted as a resource-competition model. Here we consider a second variant of the ground model, which belongs to the resource-competition class, but with non-linear resource intake. There are many situations in which growth rates depend on resource availability in ways that lie outside the linear, additive MCRM model, such as in competition over essential resources [19, 25]. Here the aim is not to study the consequences of a specific non-linear mechanism, but rather to demonstrate the sensitivity of the model to such variations in the dynamical rules.

We find that much like the model variant discussed in previous sections, here too the dynamics are sensitive to changes to the ground model, with fixed points turning into persistent abundance fluctuations, in much the same parameter regions as found previously.

The second variant to the ground model, Eq (2), is different from the ground model in the way that different resources translate into the growth rate of the consumer. Whereas in Eq (2) the growth rate is a linear combination of the resource values R_β, here we use a non-linear function. We choose a non-linear consumption function with control parameter w. With this consumption function, the dynamical equations read: (3) The parameter w allows us to tune the deviation from the ground model. For small values of w, h(R) ≃ R so the non-linear effects become small and the equations reduce to the ground model, Eq (2). For finite w non-linear effects may be significant. We quantify the deviation from linearity explored by the dynamics by the relative perturbation strength , where 〈‥〉_iβ is the average over species and resources, and denotes the time average of the resource abundance R_β (taken over a window of Δt = 1000). We find that even a rather small value of ρ is sufficient to induce a transition to dynamical persistence, see Fig 4, where the transition occurs at ρ ∼ 0.06. Again, this demonstrates how the system’s dynamics may be sensitive to small changes in the equations governing the model, in this case in a variant that is itself strictly a resource competition model.

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Fig 4. The transition between the equilibrium and persistent dynamics phases in a model with non-linear resource intake.

(A) As model parameters are changed (here varying σ_c) the fraction of systems exhibiting persistent dynamics varies. Communities reaching a fixed-point must satisfy S*/M < 1 due to competitive exclusion, but in the persistent dynamics regime can sustain more coexisting species, with S*/M > 1. (B) The relative perturbation strength, ρ, quantifies how much the growth-rate as a function of resource availability deviates from the weighted linear sum (see text for definition). ρ changes as model parameters are changed, and is measured in simulations. Persistent non-equilibrium dynamics already found for ρ ≳ 0.06 marked by dashed vertical line.

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Species abundance distribution and diversity

Above, we saw how communities near marginal stability are sensitive to small variations in the dynamical rules, either due to additional interactions, or changes to the functional form of the interactions. Here we show that these changes can allow the diversity to go well above the number of resources. This is made possible by the persistent dynamics, which are no longer bound by the competitive exclusion principle.

The competitive exclusion principle [26] states that for models describing an ecological community of S species relying on M limiting resources, no stable fixed points with M < S* exist. Briefly, the core of the argument is that any fixed point with M < S* would imply a degenerate Jacobian matrix with rank M or less. This kind of fixed point can be marginally stable, but not stable. The second variant of the model, Eq (3), satisfies the conditions for this principle to hold, so the diversity of stable equilibria is bound by M.

As an example we look at the second model variant, as defined in Sec. Non-linear resource intake. Long-time simulations of the persistent dynamics show that the species abundance distribution converges to a stationary form that can be decomposed into a power law at intermediate abundances, and other functions at the highest and lowest abundances, see Fig 5(A): (4) Here N_u is a constant, and c is set by the normalization ∫ P(N)dN = 1. From the simulations, ν is not far from zero when η → 0 (ν ≈ 0.02 in Fig 5, and similar for other parameter sets, see S1 Text (Appendix B)).

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Fig 5. Diversity above competitive exclusion in the non-linear resource competition model, see Sec. Non-linear resource intake.

(A) Abundance distribution for different values of migration. The area to the right of the vertical lines hold exactly M species. The rest of the distribution, below the line, accounts for species above competitive exclusion. Inset: the total number of coexisting species normalized by number of resources, with values above one indicate crossing of competitive exclusion. (B) Due to the abundance fluctuations, averaging abundances over a time window pushes the distribution of abundance upwards due to fluctuations. The cumulative abundance distribution is shown, defined as The dashed line is the competitive exclusion bound, M/S. For comparison, the distribution at a fixed point is given, showing that the number of species at high abundance(N ≳ 10⁻³ does not reach this bound, and the rest of the species are at low abundances, only supported by migration.

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We first ask about the instantaneous species richness, namely the fraction of species that are not at the migration floor (say, above 100η). By integrating P(N) in Eq (4) one finds that the fraction of the species above the migration floor approaches a finite number when η → 0⁺, for details see S1 Text (Appendix E). This number can be larger than M, and in fact is so in the example shown in Fig 5, reaching about 20% above M (see inset in Fig 5(A)). This is also found in results of many simulations at different parameters, see Fig 4(A). In other words, a finite fraction of the species coexist above the competitive exclusion limit even when migration is very small. This is possible since the community is not in a fixed point, and so is not bound by the competitive exclusion principle.

If the species abundance is measured by integrating over a finite-time window, see Fig 5, the abundances shift to higher values as the time window grows, indicating that species have periods of time with high abundance. This leads to a growth in the abundance N_CE above which there are exactly M species with higher abundances.

In [23], chaotic dynamics where studied in Lotka-Volterra equations with random interactions coefficients, and the existence of of time periods with high abundance have been reported, as well as a power law like in Eq (4) (albeit with a different exponent). The relation of these results to the present resource-competition model are an interesting question for future research.

Comparison with random Lotka-Volterra models

This picture bridges a gap to the behavior of high-dimensional models where interactions are sampled at random without a specified mechanism. In the notation of Sec. Demonstrating the sensitivity, this corresponds to having only. These models show a phase with persistent dynamics [6–9, 12, 14], in contrast to resource-competition which have thus far only shown relaxation to equilibrium in highly diverse communities [10, 13]. We find that the generic phase-diagram is in fact much more similar, with a transition to non-equilibrium dynamics when the variability in interaction strengths is high enough (compare, for example, Fig 2 with the phase diagram in [9]).

One difference is that here, the symmetry of the interactions can be very high and still lead to non-equilibrium dynamics. For example, in the model with added direct interactions (Sec. Demonstrating the sensitivity), the total interactions are very close to symmetric, with corr(α_ij, α_ji) = 0.997. In random Lotka-Volterra models, dynamics at a comparable level of symmetry would typically relax to equilibria. This highlights the importance of certain structures in the interaction network on dynamics.

Predictions

How can the behavior presented in this work be identified in natural or experimental communities? The dynamical outcome will depend on the following considerations:

• Is the community isolated; under migration from a regional species pool; or part of a meta-community?
• The ratio of realized species diversity to the number of resources (S*/M).
• Is the realized diversity S* high enough for high-diversity effects to show?

Consider first a single well-mixed system with continuous migration from a species pool, which was the focus of previous sections. In such a setting, dynamics either a relax to single uninvadable equilibria or reach persistent fluctuations. Which of these two possibilities is realized depends on the system parameters: the realized species diversity (S*) is set by the balance between extinctions due to competition and species able to invade. If fixed points at this diversity are unstable, persistent fluctuations will result—this is the Persistent Dynamics phase in Fig 2. As shown in Fig 2, it is attained when there is sufficient variability in the interactions, mediated for example by a broad distribution of consumption preferences (high σ_c).

We turn to a single well-mixed community that is isolated (no migration, η_i = 0). Here species may go extinct due to large abundance fluctuations, without being able to invade again. Extinctions may then lead to equilibria even when non-equilibrium dynamics are expected with migration, see S1 Text (Appendix G). The difference is that these equilibria can be invaded by species from the species pool. Importantly, in these conditions all fixed points are invadable, as uninvadable ones would translate to equilibria in the presence of migration. If there are now isolated migration events from the species pool that are well-separated in time (for example, at low migration rates, or in experiments where species are re-introduced) the equilibria will be punctuated by migration events that change the community composition [27].

An explicit spatial dimension, such as a meta-community in which several well-mixed systems are coupled by migration, again changes the phenomenology. A meta-community can exhibit persistent fluctuations, even if it is isolated from any outside species pool, thus allowing species to go extinct within it. Still, the remaining species might continue to fluctuate for extremely long times without inducing extinctions. This has been shown recently for many-species meta-communities with random Lotka-Volterra interactions in [22, 23]. An example simulation, provided as a proof-of-principle, is given in S1 Text (Appendix G). The conditions for non-equilibrium dynamics to persist depend on additional parameters including the migration rates and the number of communities in the meta-community. A fuller account of this effect is an interesting direction for future research.

Finally, we note that the non-equilibrium dynamics discussed in this work apply to communities with many species and resources or niches. Simulations indicate that dynamical fluctuations appear when there are tens of species in the community or more; communities with fewer species may instead relax to equilibria.

We have looked at interactions without any structure beyond resource competition, such as different functional groups or higher trophic levels. More complex interaction networks may possibly be approached by combining “fully random” interactions used here, in a modular fashion [11]; the implications to dynamics are an interesting subject for future research.

High-dimensional ecological dynamics are, in some respects, qualitatively different from their low-dimensional counterparts. Here we classified possible scenarios for the dynamics of resource-competition communities, and provided predictions for each scenario. We hope it may help in guiding future theoretical works, observations and experiments on high-diversity communities.