Supporting Information (SI) Contents I Reference model ... - bioRxiv

Supporting Information (SI) Contents I

Reference model, community properties and I.1 Computing the generic parameters . . . . . . I.2 Analytical solution . . . . . . . . . . . . . . . I.3 Community properties . . . . . . . . . . . . . I.4 Saturating functional response . . . . . . . . I.5 Other structures . . . . . . . . . . . . . . . . I.5.1 Mixture model (functional groups) . . I.5.2 Nestedness . . . . . . . . . . . . . . . I.5.3 Local heterogeneity . . . . . . . . . . .

II Simulation models II.1 Resource competition . . . . . . . . . II.2 Plant-pollinator community . . . . . . II.3 Sweep . . . . . . . . . . . . . . . . . . II.3.1 Interaction types . . . . . . . . II.3.2 Trait distributions . . . . . . . II.3.3 Network structure . . . . . . . II.3.4 Specific examples in the sweep II.4 Different dynamics, same equilibrium .

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III Breakdowns of genericity

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12

Reference model, community properties and extensions

The reference model was solved by Bunin [5], and it is introduced and discussed in detail in lecture notes available online [3]. Thus, we only give a cursory description here. We start from Lotka-Volterra dynamics, which can be written in two ways:   S X d Ni = Ni ri Ni − Di Ni − AijNj  (S1) dt j6=i   S X = Di Ni Ki − Ni − αij Nj  (S2) j6=i

with ri the intrinsic growth rates or Ki the carrying capacities, Di = ri /Ki the self-regulation, and Aij the bare interactions (expressed as gain or loss of abundance per capita of both species i and j) or αij = Aij /Di the couplings, i.e.

1

how much the interaction with species j perturbs the independent, self-regulated dynamics of species i. Hence, at equilibrium, d Ni = 0 dt

I.1

⇒

0 = Ki − Ni −

S X

αij Nj

(S3)

j6=i

Computing the generic parameters

Once any model with linear functional response has been put under the form above, we can directly measure the following quantities on the matrices αij and Ki (where hxi stands for average of x)

 2 ζ 2 = Ki2 − hKi i , µ = S hαij i ,  

2 S 2 2 (S4) σ 2 = S( αij − hαij i ), γ = 2 hαij αji i − hαij i . σ Note that the factor S should be present only if the averages include zero elements in the matrix αij . Instead, it should be replaced by k the average number of interaction partners (k = SC with C the connectivity) if the average is taken only on nonzero matrix elements. We can see that hKi does not appear here. In fact, we can always rescale abundances by a constant factor to ensure hKi = 1. In other words, if hKi = 6 1, we must take

2 Ki ζ2 = (S5) 2 −1 hKi i and once predictions are obtained using these parameters, abundances should be multiplied by hKi.

I.2

Analytical solution

Given the parameters above, we explain in [3] the process of deriving the analytical solution. In short, we must solve a closed set of four coupled equations to obtain the abundance distribution P (N ), which we as a Gaus can  approximate 2 2 2 sian function with mean hN i and variance σN = N − hN i . These equation are the following: Z ∞ φ= dN0 P0 (N0 ) (S6) 0 Z 1 ∞ dN0 P0 (N0 ) N0 (S7) hN i = φ 0 Z

2 1 ∞ N = dN0 P0 (N0 ) N02 (S8) φ 0 1 v= (S9) 1 − φσ 2 γv 2

and they are coupled through P0 (N0 ), a different Gaussian function with mean 1 − φµ hN i , 1 − φvγσ 2

(S10)

 ζ 2 + σ2 φ N 2 = 1 − φvγσ 2

(S11)

hN0 i = and variance σ02

These equations are transcendental (if we make them more explicit, they involve the error function) and therefore the solution cannot be expressed as a simple formula. However, it is immediate to find the solution numerically, using a root  solver with φ, hN i, N 2 and v as variables.

I.3

Community properties

From this solution, we obtain our community properties, shown in Fig. S1 and S2. φ the fraction of survivors is directly one of the variables we solved for. Then, we get total biomass X T = Ni = φS hN i , (S12) i

total productivity P =

X

ri Ni = φS hN Ki ,

(S13)

i

and Simpson index D

−1

=I=

X  Ni 2 i

T

2

=

2 / hN i 1 + σN . S

(S14)

Finally, for stability metrics, variability in response to environmental noise is a commonly used one. We show in [3] that it is given by X Var(Ni (t))  K  Venv = = N V (S15) S r i where V is given by 1 σ2 φS ≈1− (γ + 1) . 2V 2 S−µ

(S16)

For all comparisons we therefore used V , which is a more fundamental quantity and corresponds to the variability due to demographic noise. Finally, we also show in Fig. S1 and S2 the order parameter for multistability. Assembly dynamics have two main phases: one where there is a single global attractor reached asymptotically by any sequence of invasions, and another where multiple attractors exist and the result is history-dependent. The transition from one regime to another as σ increases is signalled by the divergence of the order parameter (see [3] for details and discussion). 3

Figure S1: Coexistence, abundance and stability properties for the assembled state of the random Lotka-Volterra model, in the space (µ, σ) for ζ = 0.3 and γ = 1. The uniform area in the left of each graph signals the parameter region where interactions cause abundances to diverge.

4

Figure S2: Coexistence, abundance and stability properties for the assembled state of the random Lotka-Volterra model, in the space (µ, σ) for ζ = 0.3 and γ = 0.

5

I.4

Saturating functional response

The model can be adapted to exhibit a saturating functional response   X ˜ij Nj d A  Ni = Ni Ri − Di Ni + P ˜ 1 dt 1 + A N ik k ˜ k j

(S17)

ANc

where we clarify the meaning of the usual saturation half-rate [10] by decomposing it into the average interaction coefficient A˜ and the population threshold Nc . This way, the interaction term saturates when the total population of all partners of a species exceeds Nc . Hence, the sum is bounded P X k Aik Nk ˜ P (S18) Ni Aij Nj = Ni 1 1 + k Aik Nk ANc j X ≈ ANi Nc if Aik Nk  |ANc | (S19) k

Analytically, it is possible to deal with the saturating functional response by a simple approximation. For each species, we replace it with a piecewise linear function: !  X X x |x| < |ANc | ˜ Aij Nj ≈ F Aik Nk , F (x) = (S20) ANc |x| ≥ |ANc | j

k

then at the community level, we use the real saturating function to evaluate the expected fraction f of species whose interactions are saturated. Details are provided in [3]. The situation is the same as above except we now get: hN0 i =

hKi − µ(f Nc /S + (1 − f )φ hN i) 1 − (1 − f )φσ 2 γv

(S21)

and σ02

 ζ 2 + (1 − f )2 σ 2 φ N 2 = (1 − (1 − f )φσ 2 γv)2

(S22)

The rest of the calculations follow as above, except we additionally need to know the fraction f of species with saturated interactions. This is where the real saturation function reappears (slightly rearranged): ! Z ∞ X 1 f= dxP x = α0k Nk (S23) µNc −∞ k 1+ Sx where, as usual, we can approximate P (x) by a normal distribution with hxi = |µ|φ 6

(S24)

 Var(x) = σ 2 φ N 2

(S25)

Note that if we want to make an equivalent model without saturation, we can apply the equivalence   Nc γ µ ˜=µ f + (1 − f )φ , σ ˜ = (1 − f )σ, γ˜ = (S26) S hN i 1−f which is possible only if γ˜ ∈ [−1, −1]. Then, a Lotka-Volterra system with parameters µ ˜, σ ˜ and γ˜ will have identical equilibrium properties to the system with a saturating functional response studied here. Of course, this equivalence can only be computed once we know f and hN i, i.e. after having solved the calculation above for this system.

I.5

Other structures

In the main text, we present three types of network structure: multiple functional groups, nestedness, and degree distribution (obtained by generating the network from a certain theoretical ensemble). Rather than arbitray choices, these correspond to three distinct ways of going beyond the assumption of a community where heterogeneity is uniformly distributed (i.e. random or more generally “disordered”). One such way, functional groups, assumes that species exhibit some clustering. Another way, nestedness, implies some sort of ordering or ranking between species, e.g. a niche axis or a hierarchy. These two options add structure in the form of correlations in the matrix of species interactions. By contrast, a degree distribution simply adds more “local” heterogeneity, i.e. heterogeneity at the level of individual species attributes1 . Local heterogeneity should not invalidate the disordered approach, which precisely assumes a lack of correlations between species (e.g. for any species, its interactions and the abundances of its partners can be drawn uniformly from the general community). However, to still use simplifying Gaussian approximations in the calculations, it can be useful to replace σ by two distinct metrics of interaction heterogeneity: how much species differ in the mean interaction they perceive, and how much variance there is around this species-specific mean. I.5.1

Mixture model (functional groups)

The equations for the reference model can be extended to any structure comprised of discrete groups, with disordered interactions within and between groups, but different statistics for each set of interactions. Coming back to the equilibrium equation, we can write for species i in group x (which contains S x species) as X xy y 0 = Kix − Nix − αij Nj (S27) j,y 1 Note however that network generated from ensembles such as Barabasi-Albert also come with non-local properties such as clustering, or degree correlations.

7

Thus, we now have vector ζ and matrices µ, σ, γ, defined by xy αij =

µxy σ xy + √ axy ij S S

(S28)

with

xy yx  aij aji = γ xy .

2 a = 1,

hai = 0,

(S29)

The equations to solve are the same as above, except there are now four equations per group, all coupled: for each group we solve Z ∞ x φ = dN0 P0x (N0 ) (S30) 0 Z ∞ 1 x dN0 P0x (N0 ) N0 (S31) hN i = x φ 0 Z ∞

2 x 1 N = dN0 P0x (N0 ) N02 (S32) φ 0 1 (S33) vx = x u with hN0 i = σ02 =

P

1−

x

ζ2 +

y

µxy ϕy hN i

,

(S34)

,

(S35)

ϕy v y γ xy σ xy σ yx .

(S36)

ux  xy 2 2 y y (σ ) ϕ N

P

(ux )2

ux = 1 −

y

X y

where ϕx = φx I.5.2

Sx . S

(S37)

Nestedness

We now assume we can characterize a species by its position x on a single axis of possible niches or roles. We can take the continuous limit of the group approach. Let us assume that interactions depend only on rank difference: µ(x, y) = µ + c˜µ (x − y) 2

2

(S38)

σ (x, y) = σ + c˜σ (x − y)

(S39)

γ(x, y) = γ + c˜γ (x − y)

(S40)

hKi (x) = K + c˜K (x)

(S41)

8

If furthermore we choose linear functions c˜∗ (z) = c∗ z we have K + cK x − (µ + cµ x) hϕN i + cµ hϕN xi , u ˜(x)



 ζ 2 + (σ + cσ x) ϕN 2 − cσ ϕN 2 x σ02 (x) = , u ˜2 (x)

hN0 i (x) =

u ˜(x) = 1 − ((γ + cγ x) hϕvi − cγ hϕvxi)σ 2 .

(S42) (S43) (S44)

Hence there is an additional set of unknowns: for Ψ = N, N 2 , v we must compute Z Z ∞ hϕΨi = dxP (x) dN P0 (N, x)Ψ (S45) 0

and

Z hϕΨxi =

∞

Z dxP (x)x

dN P0 (N, x)Ψ

(S46)

0

where P (x) is the probability distribution of x. I.5.3

Local heterogeneity

The effect of local heterogeneity – e.g. a degree distribution – can almost entirely be captured by stating that interactions advantage species differentially. In other words, let us assume that σ0 A0i = hAi0 + √ a0i S

(S47)

where the mean and the variance both depend on species 0. ∗

S σ0 X 0 = K0 − S hAi0 hN i − N0 − √ a0i Ni S i ∗

∗

(S48)

where we still assume that hN i over partners of 0 does not differ from the community average. Then, Ni ≈ Ni∗ −

dNi σi √ ai0 N0 dKi S

(S49)

hence 1X dNi 1− σi σ0 ai0 a0i S i dKi

!

∗

S σ0 X N0 = K0 − S hAi0 hN i − √ a0i Ni∗ (S50) S i ∗

∗

≈ (1 − φσ0 σγv)N0 where

 v=

1 1 − φσ0 σγv 9

 ≈

1 1 − φσ 2 γv

(S51)

(S52)

and finally ∗

hN0 i = v(hKi − φµ hN i ) (S53)

2

∗ 2 ∗2 2 2 ∗2 2 ∗ 2 N0 = hN0 i + v (σK + S (σhAi ) hN i − 2S (hK0 A0 i − hKi hAi) + σ φ N 2 ) (S54) where it turns out that the variance on σ0 is irrelevant. So in the end we only need two metrics of variance: σ and σA the standard deviation of hAii .

II II.1

Simulation models Resource competition

In this discretized version of a classic ecological model [11, 8], S species compete over R abiotic resources which are steadily resupplied into the ecosystem. It is generally known from competitive exclusion theory that R > S is necessary to at least potentially allow S species to survive – however, these many resources need not all differ by their nature, they could simply be distinguished by spatial or temporal availability within the ecosystem, with some species having an advantage at capturing certain patches rather than others. We define the net growth rate of species i as gi =

R X

ρα (t)ξiα − Mi

(S55)

α=1

with consumption rate ξiα for resource α, whose abundance is given by X ρα (t) = ρα − ξiα Ni (t)

(S56)

i

where ρα is the steady influx of the resource. On the other hand, Mi represents mortality from the energy costs associated with resource acquisition. To avoid favoring specialists or generalists a priori, we make these costs proportional to the total ability of a species to acquire resources: X Mi = ρmi ξiα (S57) α

where we factor out the average amount ρ of resource in the system so that mi is now a dimensionless number representing the intrinsic lack of fitness of species i. Hence, we get the dynamical equations   X X X d Ni = gi Ni = Ni  ξiα (ρα − ρmi ) − Nj ξiα ξjα  (S58) dt α α j

10

It is easy to see that they map onto the Lotka-Volterra equations (1equation.0.1) with X Ri = ξiα (ρα − ρmi ), (S59) α

Di =

X

2 ξiα

(S60)

ξiα ξjα .

(S61)

α

Aij =

X α

Let us assume a disordered pool where consumption rates, resource influxes and mortality are all drawn as independent random variables with mean ξ = hξiα i ,

ρ = hρα i ,

m = hmi i

(S62)

and standard deviation σξ , σρ and σm . Among other things, this implies that all consumers are generalists who can consume any resource but perform better at some. Without loss of generality, we can always set hRi i = Rρξ(1 − m) = 1

 hDi i = R ξ 2 = R(ξ 2 + σξ2 ) = 1.

(S63) (S64)

using a rescaling of time and of Ni (see [3]). The second equation means that there is a tradeoff between the effective intensity ξ and variability σξ of consumption rates.

II.2

Plant-pollinator community

This community of S = 300 species was divided into two functional groups, one given no intrinsic growth i.e. Ki = 0, while the other had Ki drawn with mean 1 and variance ζ. We then drew two types of interactions αij : mutualistic and competitive, both as exponential distributions with mean −0.01 and 0.01 respectively (larger mutualistic interactions caused population explosions since we did not use a saturating functional response here). The ordering parameter ω decided the probability that mutualistic interactions were exclusively assigned to inter-group links and competitive interactions to intra-group links. For ω = 0, both types of interactions were assigned at random, ignoring the group labels.

II.3

Sweep

In this section, we give concrete values and modelling choices for the vast combinatorial sweep through a class of simulation models. II.3.1

Interaction types

For competition and mutualism, we could directly draw Aij (see “trait distributions” below). For a predation model, interactions were slightly more structured. 11

For a pair (i, j) of species where j is the predator, interactions can be written as Aij = Pij , Aji = −Pij (S65) with Pij > 0 the predation intensity, and  the biomass conversion efficiency of the trophic interaction, generally estimated in the range  ≈ 0.1 − 0.2. These quantities were drawn with the mean and variance m = hPji i ,

v = Var(Pji ) .

(S66)

whose relationship to the parameters µ, σ and γ defined in (3equation.0.3) above is given by m( − 1) (S67) µ=− 2 p 1+ σ= √ m2 + v (S68) 2S µ2 + (m2 + v) γ=− (S69) Sσ 2 In our simulatons, we set v = 0, meaning that all interactions have equal intensity. This allows us to show that even in this seemingly non-random case, our reference random model can still succeed in predicting the equilibrium. II.3.2

Trait distributions

All species first were given a carrying capacity Ki drawn with mean 1 and variance ζ, and a self-regulation coefficient Di = 1. Given Di = 1, we could directly draw αij = Aij . The sweep encompassed three main types of ecological interactions: competition, mutualism and predation. All of these constrain the sign of the interaction coefficients, hence we drew their magnitudes from exponential distributions. We wished to avoid difficulties caused by very strong interactions (such as strict competitive exclusion causing all species but one to go extinct), so these magnitudes were always individually small compared to intra-species competition (|αij | < 1), although the sum interaction of a species with all others was gener P ally large ( j αij > 1). II.3.3

Network structure

Nestedness was controlled by varying pn the probability that Aij (for mutualism and competition) or Pij (for predation) was set to zero for i ≤ j. When that probability was set to pn = 1, the remaining matrix was upper-triangular. Bipartition was controlled by assigning each species a group index. Then, intragroup links were deleted with probability pb . Finally, to control the degree distribution, we generated a scale-free network using the Barabasi-Albert algorithm [1], and then rewired the edges randomly with probability 1 − pd . The “Ordering” parameter shown in figures in the main text was thus one of pn , pb or pd depending on the structural property. 12

II.3.4

Specific examples in the sweep

Here are instances where models generated as part of the sweep are similar to some model from the literature. Cascade predation The combination of predatory interactions and a nested structure is reminiscent of the cascade model [6], which has been used in many simulations although superseded by other models such as the niche model [4]. The main difference here is that we did not impose that only basal species have positive carrying capacities. The detailed study of plausible trophic systems was not our intent here, and is in fieri. C-R mutualism The sweep included mutualistic interactions. However, it is often thought [10] that Lotka-Volterra dynamics fail to represent this sort of interactions, especially by allowing for boundless population growth. The most common change introduced in the description of mutualistic interactions is a saturating functional response. This represents the simplest case considered in consumer-resource mutualism [9]. Spatial competition We used a similar functional response for competition in space, to represent antagonistic interactions with at most a finite number Nc of neighbors (e.g. competition for light between plants). This could be used in a qualitative comparison to models that have an explicit spatial dimension [12], including individual-based simulations of spatial competition.

II.4

Different dynamics, same equilibrium

We give here a more detailed version of the illustration, made in Fig. 3Equivalences and differences between communities. To explain how different models typically make distinct predictions, but can nevertheless overlap, we show where they lay once translated into the parameter space of the reference random model, following the comparison scheme outlined in Fig. 1Comparison scheme used throughout this article. On the left, various specific ecological models from the literature, some of which are used as examples below. They correspond to various choices of model ingredients, among the main categories detailed in Methods (not all connections are represented here). Numerical simulations of these models (and other combinations of ingredients) allow us to find their properties at equilibrium. On the right, the reference Lotka-Volterra model with its four control parameters, whose values can be obtained by “translation” from the ingredients of other models. This reference model is easily simulated, and it is also solved analytically, so that its equilibrium features are known and can be compared to simulations.figure.caption.1. We illustrate in the (µ, σ) plane the effective parameter values corresponding to: the spatial competition model (yellow) with neighbor threshold Nc ∈ [1, 100] and mean interaction A˜ ∈ [1, 20]; predation (magenta) with intensity m ∈ [0.1, 25] and biomass conversion efficiency  ∈ [0, 1]; and resource competition (blue) with number of 13

resources R ∈ [102 , 104 ] and consumer heterogeneity σξ ∈ [0.1, 0.6]. See SI for details on model-specific parameters. Inset: An example where a competitive community and a predator-prey community display identical species abundance distributions, corresponding to the parameter values marked by the red cross. figure.caption.3, that different models can create identical communities if they correspond to the same parameters µ, σ, γ and ζ. First, we chose a competitive Lotka-Volterra model with S = 100 species and interactions P (Aij ) = 10 e−10 Aij

(S70)

with a covariance tuned to ensure the proper value of γ. Second, a predatory system with S = 1600, where pairs (i, j) are drawn randomly and their interactions set exactly to: Aij = 50/S Aji = −30/S (S71) As discussed above, these are characteristic of trophic interactions in that Aij and Aji have opposite sign, but also that the gain term Aji is smaller than the loss term Aij , corresponding to a biomass conversion efficiency of 60%. For the sake of comparison, we add a third system, a completely random community whose interactions are drawn from a normal distribution  √  (S72) P (Aij ) = N 10/S, 1/ S with a pool size made to vary from S = 200 to S = 1000. Interaction terms can be positive or negative; due to γ = −1, most pairs of species have opposite signs for Aij and Aji , but there are some strictly competitive (if asymmetrical) interactions as well. We have seen that the various models above can often be translated into the reference model for wide parameter ranges, although each with their caveats. In the region when they are translatable, it is possible for different models to correspond to the same values of the reference model’s four parameters. Then, we predict that their assembled states should be similar or even identical. We demonstrate this very counter-intuitive property in Fig. S3.

III

Breakdowns of genericity

The following figures showcase various structures that can or cannot lead to a breakdown of genericity, in the sense that a specific model generates equilibria that differ from that of the reference model. In Fig. S4 and S5, we consider the mutualistic model, and see that its behavior is always well predicted by the reference model. No change of the network structure (nestedness, bipartition) had any notable impact on the assembled state, except indirectly by changing µ and σ the mean and heterogeneity of interactions. A different picture is found in the case of the competitive model, in Fig. S6 and S7. Here, clearly, nestedness and partitioning actually affect our ability to 14

Figure S3: Abundance distributions: numerical results and theory in communities with different interaction types, corresponding to the same parameters µ = 10 and σ = 1, see Sec. II.4. The first system with S = 100 is the resource competition model, the second with S = 600 is the reference model, and the third with S = 1600 is the intraguild predation model. The solid line is the analytical prediction for the reference model. predict the results. For instance, partitioning the network in two (i.e. removing edges until the graph becomes bipartite) does not affect the success of the reference model for mutualistic interactions Fig. S5, but it does in the case of competitive interactions, Fig. S7. This may be a callback to the “universal” dynamics that are specific to mutualistic systems [7, 2]. Fig. S8 to S10 display the predictions for the intra-guild predation model. Predation intensity is generally found to be a negative factor for stability, diversity and productivity. We find that these equilibrium properties are wellpredicted by the reference model for any parameter values (Fig. S8 and any network degree distribution (Fig. S9), even quantitatively so, as long as ω is small enough, meaning that interaction nestedness is low. If we now allow for cascade structure, Fig. S10 shows that predictions diverge as nestedness ω increases: simulation results show reduced diversity, biomass and variability compared to the reference model, although the trends mostly remain qualitatively similar. Finally, we discuss one limit in resource competition, Fig. S11. As seen in the main text in Fig. 2Various community properties predicted by the reference model: (a) Total biomass T , (b) Fraction of surviving species φ, (c) Simpson diversity D, (d) Temporal variability V . Simulation results (dots) were obtained from the resource competition model, varying the number of resources R and the heterogeneity of consumption rates σξ (details in SI). They were then compared to theoretical predictions from the reference model (solid lines) parametrized only by the four generic properties of the species pool, see Fig. 1Comparison scheme used throughout this article. On the left, various specific ecological

15

Figure S4: Mutualistic interactions with saturating functional response: simulation results (dots) and theoretical predictions from the reference model (solid line) as a function of the functional response’s threshold population Nc , for different values of the nestedness.

16

Figure S5: Mutualistic interactions, partitioning and functional response: simulation results (dots) and theoretical predictions from the reference model (solid line) as a function of the threshold Nc , for different values of the partitioning (0 is a complete graph, 1 a bipartite graph).

17

Figure S6: Competitive interactions, modularity and functional response: simulation results (dots) and theoretical predictions from the reference model (solid line) as a function of the functional response’s threshold population Nc , for different values of the modularity.

18

Figure S7: Competitive interaction, partitioning and functional response: simulation results (dots) and theoretical predictions from the reference model (solid line) as a function of the functional response’s threshold population Nc , for different values of the partitioning.

19

Figure S8: Predation intensity and efficiency: simulation results (dots) and theoretical predictions from the reference model (solid line) as a function of predation intensity m for various values of biomass conversion efficiency . Here, nestedness ω = 0.1. Even though we set all trophic interactions have equal value, i.e. v = 0, and their disorder lies entirely in their structure (i.e. whether a given species is predator or prey in each of its interactions), the reference model with random interactions still can produce equilibria with identical properties.

20

Figure S9: Network structures in a system with trophic interactions: we compare results for a random graph, a Watts-Strogatz small-world graph with rewiring probability p = 0.1, and a Barabasi-Albert scale-free network. Despite very different degree distributions, clustering coefficients and mean path lengths, these different structures give rise to the same assembled community properties, all of which are predictable from the reference model (solid lines).

21

Figure S10: Predation cascade: simulation results (dots) and theoretical predictions from the reference model (solid line) as a function of the average predation intensity m for various values of the ordering parameter ω in the cascade-like model, between the limits ω = 0 (fully disordered) where the theory applies, and ω = 1 (fully ordered) where it breaks down.

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models from the literature, some of which are used as examples below. They correspond to various choices of model ingredients, among the main categories detailed in Methods (not all connections are represented here). Numerical simulations of these models (and other combinations of ingredients) allow us to find their properties at equilibrium. On the right, the reference Lotka-Volterra model with its four control parameters, whose values can be obtained by “translation” from the ingredients of other models. This reference model is easily simulated, and it is also solved analytically, so that its equilibrium features are known and can be compared to simulations.figure.caption.1. figure.caption.2, predictions from the reference model are very accurate. However we see here that they break down when all resource supplies are equal. In that case, more species survive than we predict. This discrepancy arises as groups of species are correlated because they consume the same resources. If all resources are equally supplied, this intricate correlation structure allows more species to survive due to complex feedbacks: if species i competes with species j, it also competes with all the competitors of j in such a way that j is allowed to recover to some degree. However, if resource supplies differ, then the relative importance of losing a given resource is not the same for all competitors; this heterogeneity is enough to “blur out” the correlation structure and recover the predictions of the reference model.

References [1] R´eka Albert and Albert-L´aszl´o Barab´asi. Statistical mechanics of complex networks. Reviews of modern physics, 74(1):47, 2002. [2] Jean-Fran¸cois Arnoldi, Bart Haegeman, Tom´as Revilla, and Michel Loreau. Particularity of “universal resilience patterns in complex networks”. bioRxiv, page 056218, 2016. [3] Matthieu Barbier. Lecture notes: The cavity method for large ecosystem assembly. Unpublished paper, 2017. [4] Ulrich Brose, Richard J Williams, and Neo D Martinez. Allometric scaling enhances stability in complex food webs. Ecology letters, 9(11):1228–1236, 2006. [5] Guy Bunin. Ecological communities with lotka-volterra dynamics. Physical Review E, 95(4):042414, 2017. [6] JE Cohen and CM Newman. A stochastic theory of community food webs: I. models and aggregated data. Proceedings of the Royal Society of London B: Biological Sciences, 224:421–448, 1985. [7] Jianxi Gao, Baruch Barzel, and Albert-L´aszl´o Barab´asi. Universal resilience patterns in complex networks. Nature, 530(7590):307–312, 2016.

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Figure S11: Resource competition: simulation results (dots) and theoretical predictions from the reference model (solid line) as a function of the number of resources R, for different values of the heterogeneity of resources σρ . The prediction breaks down as σρ → 0.

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[8] M Gatto. A general minimum principle for competing populations: Some ecological and evolutionary consequences. Theoretical Population Biology, 37(3):369–388, 1990. [9] J Nathaniel Holland and Donald L DeAngelis. A consumer–resource approach to the density-dependent population dynamics of mutualism. Ecology, 91(5):1286–1295, 2010. [10] J Nathaniel Holland, Donald L DeAngelis, and Judith L Bronstein. Population dynamics and mutualism: functional responses of benefits and costs. The American Naturalist, 159(3):231–244, 2002. [11] Robert Mac Arthur. Species packing, and what competition minimizes. Proceedings of the National Academy of Sciences, 64(4):1369–1371, 1969. [12] Simone Pigolotti, Roberto Benzi, Prasad Perlekar, Mogens Høgh Jensen, Federico Toschi, and David R Nelson. Growth, competition and cooperation in spatial population genetics. Theoretical population biology, 84:72–86, 2013.

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