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SUPPLEMENTARY INFORMATIONdoi:10.1038/nature10832

Stability criteria for complex ecosystemsSupplementary Information

Stefano Allesina1,2 & Si Tang1

1Dept. Ecology & Evolution, University of Chicago. 1101 E. 57th Chicago, IL 60637 USA.

2Computation Institute, University of Chicago.

Outline

The Supplementary Information is organized as follows: (1) We detail how the various matrices

presented in the main text are constructed; (2) We derive the stability criteria for unstructured

matrices (random; predator-prey; mixture of competition and mutualism; mutualism; competition);

(3) We extend the criteria to varying diagonal terms and combinations of interactions; (4) We

study the effect of realistic food web structure in the predator-prey case; (5) We test the effects of

nestedness and asymmetric interaction strengths on mutualistic webs; (6) We examine the effect of

the shape of the distribution of interaction strengths on stability.

1 - Construction of the matrices

In the main text, we analyze different types of matrices. Each matrix represents the Jacobian matrix

of an unspecified dynamical system evaluated at a feasible (i.e., where all species have positive

densities) equilibrium point for the system. The feasibility of the equilibrium is postulated, as in

other similar studies1–4, as unfeasible systems are clearly not ecologically interesting. Here we

detail how these matrices were constructed. In all cases, the parameters are: S, number of species,

C, desired level of connectance, −d, value of the diagonal coefficients (d > 0), σ, the standard

deviation of the random variable X from which the coefficients Mij take value. In the construction

algorithms, we use the normal distribution case (X ∼ N(0, σ2)) as an example. The algorithms

can accommodate different distributions, such as those examined in the main text and in the section

4 of the SI.

1

For each matrix, we also report the mean (λ) and the variance Var(λ) of the eigenvalues when

X ∼ N(0, σ2).

1.1 - Random Matrices

In the random case, we construct the matrices in the following way: i) For each off-diagonal

coefficient Mij,i�=j , we draw a random value p from a uniform distribution U [0, 1]. ii) If the value

is p ≤ C, we set coefficient Mij by sampling it from N(0, σ2). iii) Otherwise (p > C), Mij = 0.

iv) All diagonal terms, Mii, are set to −d. For these matrices, the eigenvalues have λ = −d and

Var(λ) = 0.

These matrices, for large S, generate a precise mixture of interaction types, with predator-

prey interactions being represented twice as frequently as mutualistic or competitive ones (Table

S1).

1.2 - Predator-Prey Matrices

In the predator-prey case, i) For each pair of interactions (Mij,Mji)i>j , we draw a random value

p1 from a uniform distribution U [0, 1]. ii) If p1 ≤ C, we draw a second random value p2 from

U [0, 1]. iii) If p2 ≤ 0.5, we draw Mij from a half-normal distribution |N(0, σ2)| and Mji from a

negative half-normal −|N(0, σ2)|, while if p2 > 0.5 we do the opposite. iv) If p1 > C, we assign

0 to both Mij and Mji. v) All diagonal terms, Mii, are set to −d. For these matrices, λ = −d and

Var(λ) = −2(S − 1)Cσ2/π.

1.3 - Mixture of Competition and Mutualism Matrices

i) For each pair of interactions (Mij,Mji)i>j , we draw a random value p1 from a uniform distri-

bution U [0, 1]. ii) If p1 ≤ C, we draw a second random value p2 from U [0, 1]. iii) If p2 ≤ 0.5,

we draw Mij and Mji independently from a half-normal distribution |N(0, σ2)|, while if p2 > 0.5

we draw them from a negative half normal distribution −|N(0, σ2)|. iv) If p1 > C, we assign 0

to both Mij and Mji. v) All diagonal terms, Mii, are set to −d. For these matrices, λ = −d and

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For each matrix, we also report the mean (λ) and the variance Var(λ) of the eigenvalues when

X ∼ N(0, σ2).

1.1 - Random Matrices

In the random case, we construct the matrices in the following way: i) For each off-diagonal

coefficient Mij,i�=j , we draw a random value p from a uniform distribution U [0, 1]. ii) If the value

is p ≤ C, we set coefficient Mij by sampling it from N(0, σ2). iii) Otherwise (p > C), Mij = 0.

iv) All diagonal terms, Mii, are set to −d. For these matrices, the eigenvalues have λ = −d and

Var(λ) = 0.

These matrices, for large S, generate a precise mixture of interaction types, with predator-

prey interactions being represented twice as frequently as mutualistic or competitive ones (Table

S1).

1.2 - Predator-Prey Matrices

In the predator-prey case, i) For each pair of interactions (Mij,Mji)i>j , we draw a random value

p1 from a uniform distribution U [0, 1]. ii) If p1 ≤ C, we draw a second random value p2 from

U [0, 1]. iii) If p2 ≤ 0.5, we draw Mij from a half-normal distribution |N(0, σ2)| and Mji from a

negative half-normal −|N(0, σ2)|, while if p2 > 0.5 we do the opposite. iv) If p1 > C, we assign

0 to both Mij and Mji. v) All diagonal terms, Mii, are set to −d. For these matrices, λ = −d and

Var(λ) = −2(S − 1)Cσ2/π.

1.3 - Mixture of Competition and Mutualism Matrices

i) For each pair of interactions (Mij,Mji)i>j , we draw a random value p1 from a uniform distri-

bution U [0, 1]. ii) If p1 ≤ C, we draw a second random value p2 from U [0, 1]. iii) If p2 ≤ 0.5,

we draw Mij and Mji independently from a half-normal distribution |N(0, σ2)|, while if p2 > 0.5

we draw them from a negative half normal distribution −|N(0, σ2)|. iv) If p1 > C, we assign 0

to both Mij and Mji. v) All diagonal terms, Mii, are set to −d. For these matrices, λ = −d and

2Var(λ) = 2(S − 1)Cσ2/π.

1.4 - Mutualistic Matrices

i) For each pair of interactions (Mij,Mji)i>j , we draw a random value p from a uniform distribu-

tion U [0, 1]. ii) If p ≤ C, we draw Mij and Mji independently from a half-normal distribution

|N(0, σ2)|. iii) If p > C, we assign 0 to both Mij and Mji. iv) All diagonal terms, Mii, are set to

−d. For these matrices, λ = −d and Var(λ) = 2(S − 1)Cσ2/π.

1.5 - Competitive Matrices

i) For each pair of interactions (Mij,Mji)i>j , we draw a random value p from a uniform distribution

U [0, 1]. ii) If p ≤ C, we draw Mij and Mji independently from a negative half-normal distribution

−|N(0, σ2)|. iii) If p > C, we assign 0 to both Mij and Mji. iv) All diagonal terms, Mii, are set to

−d. For these matrices, λ = −d and Var(λ) = 2(S − 1)Cσ2/π.

1.6 - Cascade Predator-Prey Matrices

In the cascade model5, species are ordered to form a hierarchy, and each species has a fixed prob-

ability of preying upon the preceding species. The produced networks do not contain cycles – al-

though cycles are observed in empirical networks6. In the cascade model, the species with highest

ranking functions as a top predator, while that with the lowest ranking as a producer. Accordingly,

the species with the highest ranking has negative column and positive row, while the opposite is

true for the species with the lowest ranking.

The construction algorithm for a cascade predator-prey matrix is: i) For each coefficient in

the lower-triangular part of M , Mij,i>j , we draw a random value p uniformly from [0, 1]. ii) If

the value p ≤ C, we draw the coefficient Mij from an half-normal distribution |N(0, σ2)| and the

coefficient Mji from a negative half-normal −|N(0, σ2)|. iii) Otherwise (p > C), we assign 0 to

both Mij and Mji. iv) All diagonal terms, Mii, are set to −d. For the cascade model, λ = −d and

Var(λ) = −2(S − 1)Cσ2/π.

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Var(λ) = 2(S − 1)Cσ2/π.

1.4 - Mutualistic Matrices

i) For each pair of interactions (Mij,Mji)i>j , we draw a random value p from a uniform distribu-

tion U [0, 1]. ii) If p ≤ C, we draw Mij and Mji independently from a half-normal distribution

|N(0, σ2)|. iii) If p > C, we assign 0 to both Mij and Mji. iv) All diagonal terms, Mii, are set to

−d. For these matrices, λ = −d and Var(λ) = 2(S − 1)Cσ2/π.

1.5 - Competitive Matrices

i) For each pair of interactions (Mij,Mji)i>j , we draw a random value p from a uniform distribution

U [0, 1]. ii) If p ≤ C, we draw Mij and Mji independently from a negative half-normal distribution

−|N(0, σ2)|. iii) If p > C, we assign 0 to both Mij and Mji. iv) All diagonal terms, Mii, are set to

−d. For these matrices, λ = −d and Var(λ) = 2(S − 1)Cσ2/π.

1.6 - Cascade Predator-Prey Matrices

In the cascade model5, species are ordered to form a hierarchy, and each species has a fixed prob-

ability of preying upon the preceding species. The produced networks do not contain cycles – al-

though cycles are observed in empirical networks6. In the cascade model, the species with highest

ranking functions as a top predator, while that with the lowest ranking as a producer. Accordingly,

the species with the highest ranking has negative column and positive row, while the opposite is

true for the species with the lowest ranking.

The construction algorithm for a cascade predator-prey matrix is: i) For each coefficient in

the lower-triangular part of M , Mij,i>j , we draw a random value p uniformly from [0, 1]. ii) If

the value p ≤ C, we draw the coefficient Mij from an half-normal distribution |N(0, σ2)| and the

coefficient Mji from a negative half-normal −|N(0, σ2)|. iii) Otherwise (p > C), we assign 0 to

both Mij and Mji. iv) All diagonal terms, Mii, are set to −d. For the cascade model, λ = −d and

Var(λ) = −2(S − 1)Cσ2/π.

31.7 - Niche Predator-Prey Matrices

The niche model6 allows trophic cycles and cannibalism. The species are ordered and “niche

value”, ηi. A “niche radius”, ri, proportional to ηi, is drawn for each species along with a “niche

center” ci. Each species i preys upon all the species j whose niche value ηj are included in the

range [ci−ri/2, ci+ri/2]. Thus, the generated networks are interval (i.e., each predator preys upon

species with consecutive niche values). Empirical networks, however, are not perfectly interval6–8.

To build a niche predator-prey matrix, we first generate an adjacency matrix A for a com-

munity of S species and desired connectance C, using the niche model (Aij = 1 if i is a prey of

j). Then, we form a “sign matrix” P by −A + AT , describing the negative or positive effect of

each species on the others. Finally, Mij is obtained by multiplying Yij , drawn from an half-normal

distribution |N(0, σ2)|, and Pij . The diagonal elements are set to −d. For the niche model, λ = −d

and Var(λ) = −2(S − 1)Cσ2/π.

1.8 - Mutualistic Bipartite Matrices

For the bipartite case, we divided the species into two groups of equal size (S/2, when S is even).

For each Mij where i belongs to the first group and j to the second, we draw Mij and Mji from

an half-normal distribution |N(0, σ2)| with probability C ′ = 2C(S − 1)/S (so that the expected

connectance is matched). The diagonal is −d. For the bipartite mutualistic model, λ = −d and

Var(λ) = 2(S − 1)Cσ2/π.

1.9 - Mutualistic Nested Matrices

Nestedness is a property of the incidence matrix B (typically rectangular) in which the columns

represent the species belonging to the first group (e.g., plants) and the rows represent those be-

longing to the second group (e.g., pollinators). Suppose that, to match the desired connectance C

in the matrix M , we want to arrange L links in B. We arrange the links in the following way: i)

We fill the first row (to guarantee connectedness). ii) We fill the first column. iii) We arrange the

subsequent links so that the matrix is perfectly nested. For example (using a squared incidence

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1.7 - Niche Predator-Prey Matrices

The niche model6 allows trophic cycles and cannibalism. The species are ordered and “niche

value”, ηi. A “niche radius”, ri, proportional to ηi, is drawn for each species along with a “niche

center” ci. Each species i preys upon all the species j whose niche value ηj are included in the

range [ci−ri/2, ci+ri/2]. Thus, the generated networks are interval (i.e., each predator preys upon

species with consecutive niche values). Empirical networks, however, are not perfectly interval6–8.

To build a niche predator-prey matrix, we first generate an adjacency matrix A for a com-

munity of S species and desired connectance C, using the niche model (Aij = 1 if i is a prey of

j). Then, we form a “sign matrix” P by −A + AT , describing the negative or positive effect of

each species on the others. Finally, Mij is obtained by multiplying Yij , drawn from an half-normal

distribution |N(0, σ2)|, and Pij . The diagonal elements are set to −d. For the niche model, λ = −d

and Var(λ) = −2(S − 1)Cσ2/π.

1.8 - Mutualistic Bipartite Matrices

For the bipartite case, we divided the species into two groups of equal size (S/2, when S is even).

For each Mij where i belongs to the first group and j to the second, we draw Mij and Mji from

an half-normal distribution |N(0, σ2)| with probability C ′ = 2C(S − 1)/S (so that the expected

connectance is matched). The diagonal is −d. For the bipartite mutualistic model, λ = −d and

Var(λ) = 2(S − 1)Cσ2/π.

1.9 - Mutualistic Nested Matrices

Nestedness is a property of the incidence matrix B (typically rectangular) in which the columns

represent the species belonging to the first group (e.g., plants) and the rows represent those be-

longing to the second group (e.g., pollinators). Suppose that, to match the desired connectance C

in the matrix M , we want to arrange L links in B. We arrange the links in the following way: i)

We fill the first row (to guarantee connectedness). ii) We fill the first column. iii) We arrange the

subsequent links so that the matrix is perfectly nested. For example (using a squared incidence

4

matrix), suppose that B is 6× 6:

B =

1,1 1,2 1,3 1,4 1,5 1,6

2,1 2,2 2,3 2,4 2,5 2, 6

3,1 3,2 3,3 3, 4 3, 5 3, 6

4,1 4,2 4, 3 4, 4 4, 5 4, 6

5,1 5, 2 5, 3 5, 4 5, 5 5, 6

6,1 6, 2 6, 3 6, 4 6, 5 6, 6

M =

0 B

BT 0

and suppose we want to include 17 links. First, we fill the first row (1, 1) to (1, 6), so that we placed

6 links. The next five links are used to fill the first column (2, 1) to (6, 1). Finally, the last six links

are placed in (2, 2), (2, 3), (3, 2), (2, 4), (3, 3), etc. Note that the sum of the x and y coordinates

for the links is growing. In fact, ordering the potential links by their coordinate sums, and giving

precedence to those with smaller row numbers in the case of ties, guarantees the maintenance of

perfect nestedness. This is the filling algorithm we used in the simulations. Once we obtain B, we

use it (along with its transpose), to determine the interactions in the matrix M . All the nonzero

values of Mij are taken from the half-normal |N(0, σ2)|. The diagonal is −d. For the bipartite

nested mutualistic model, λ = −d and Var(λ) = 2(S − 1)Cσ2/π.

2 - Derivation of the Stability Criteria

To derive the stability criteria for all types of interactions, we start from random networks and

extend the work of May.

2.1 - Random

In the random case, the community matrix M is constructed in the following way: with probability

C, an off-diagonal term Mij takes the value of a random variable X , whose mean and variance

are E(X) = 0 and Var(X) = σ2 respectively, while Mij = 0 with probability 1 − C. Thus,

E(Mij)i �=j = 0 and Var(Mij)i �=j = Cσ2. The diagonal terms are all set to zero. According to

Girko’s circular law9 and the generalization by Tao et al.10, for any distribution of X with mean

zero and bounded variance σ2 < ∞, the eigenvalues of M/(σ√SC) (i.e., where we divide each

5

1.7 - Niche Predator-Prey Matrices

The niche model6 allows trophic cycles and cannibalism. The species are ordered and “niche

value”, ηi. A “niche radius”, ri, proportional to ηi, is drawn for each species along with a “niche

center” ci. Each species i preys upon all the species j whose niche value ηj are included in the

range [ci−ri/2, ci+ri/2]. Thus, the generated networks are interval (i.e., each predator preys upon

species with consecutive niche values). Empirical networks, however, are not perfectly interval6–8.

To build a niche predator-prey matrix, we first generate an adjacency matrix A for a com-

munity of S species and desired connectance C, using the niche model (Aij = 1 if i is a prey of

j). Then, we form a “sign matrix” P by −A + AT , describing the negative or positive effect of

each species on the others. Finally, Mij is obtained by multiplying Yij , drawn from an half-normal

distribution |N(0, σ2)|, and Pij . The diagonal elements are set to −d. For the niche model, λ = −d

and Var(λ) = −2(S − 1)Cσ2/π.

1.8 - Mutualistic Bipartite Matrices

For the bipartite case, we divided the species into two groups of equal size (S/2, when S is even).

For each Mij where i belongs to the first group and j to the second, we draw Mij and Mji from

an half-normal distribution |N(0, σ2)| with probability C ′ = 2C(S − 1)/S (so that the expected

connectance is matched). The diagonal is −d. For the bipartite mutualistic model, λ = −d and

Var(λ) = 2(S − 1)Cσ2/π.

1.9 - Mutualistic Nested Matrices

Nestedness is a property of the incidence matrix B (typically rectangular) in which the columns

represent the species belonging to the first group (e.g., plants) and the rows represent those be-

longing to the second group (e.g., pollinators). Suppose that, to match the desired connectance C

in the matrix M , we want to arrange L links in B. We arrange the links in the following way: i)

We fill the first row (to guarantee connectedness). ii) We fill the first column. iii) We arrange the

subsequent links so that the matrix is perfectly nested. For example (using a squared incidence

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matrix), suppose that B is 6× 6:

B =

1,1 1,2 1,3 1,4 1,5 1,6

2,1 2,2 2,3 2,4 2,5 2, 6

3,1 3,2 3,3 3, 4 3, 5 3, 6

4,1 4,2 4, 3 4, 4 4, 5 4, 6

5,1 5, 2 5, 3 5, 4 5, 5 5, 6

6,1 6, 2 6, 3 6, 4 6, 5 6, 6

M =

0 B

BT 0

and suppose we want to include 17 links. First, we fill the first row (1, 1) to (1, 6), so that we placed

6 links. The next five links are used to fill the first column (2, 1) to (6, 1). Finally, the last six links

are placed in (2, 2), (2, 3), (3, 2), (2, 4), (3, 3), etc. Note that the sum of the x and y coordinates

for the links is growing. In fact, ordering the potential links by their coordinate sums, and giving

precedence to those with smaller row numbers in the case of ties, guarantees the maintenance of

perfect nestedness. This is the filling algorithm we used in the simulations. Once we obtain B, we

use it (along with its transpose), to determine the interactions in the matrix M . All the nonzero

values of Mij are taken from the half-normal |N(0, σ2)|. The diagonal is −d. For the bipartite

nested mutualistic model, λ = −d and Var(λ) = 2(S − 1)Cσ2/π.

2 - Derivation of the Stability Criteria

To derive the stability criteria for all types of interactions, we start from random networks and

extend the work of May.

2.1 - Random

In the random case, the community matrix M is constructed in the following way: with probability

C, an off-diagonal term Mij takes the value of a random variable X , whose mean and variance

are E(X) = 0 and Var(X) = σ2 respectively, while Mij = 0 with probability 1 − C. Thus,

E(Mij)i �=j = 0 and Var(Mij)i �=j = Cσ2. The diagonal terms are all set to zero. According to

Girko’s circular law9 and the generalization by Tao et al.10, for any distribution of X with mean

zero and bounded variance σ2 < ∞, the eigenvalues of M/(σ√SC) (i.e., where we divide each

5element by the constant σ

√SC) are uniformly distributed on a unit circle centered at (0, 0), as

S → ∞. It follows that, when S is sufficiently large, the eigenvalue distribution of M is uniform

on a circle of radius approximately σ√SC (Figure 1, main text).

Now we can introduce the effect of diagonal strength (density dependence). We start by

setting all the diagonal elements of M to be −d: this shifts the circle so that it is now centered

at (−d, 0). We relax this constraint in Section 3 below. To achieve stability, the circle must be

fully contained in the left half-plane (as all eigenvalues must have negative real parts). Hence, to

achieve stability, the radius of the circle must be smaller than d, i.e., σ√SC < d. Because the

values relevant to stability are the network complexity√SC and the ratio between d (magnitude

of the diagonal terms) and σ (standard deviation of the distribution of X), we write θ = d/σ. The

stability criterion for a random community matrix is√SC < θ, irrespective of the distribution of

X . For example, suppose that X follows a distribution with σ = 0.5, the diagonal terms Mii = −1,

and set C to be 0.1: then the stability criterion is violated whenever S ≥ 41.

2.2 - Predator-Prey

To derive other stability criteria, we need to formulate a new conjecture. Sommers et al.11 proved

the following theorem: take a matrix A, whose elements are follow a Gaussian distribution with

mean E(Aij) = 0, variance Var(Aij) = 1/S, and correlation E(AijAji) = τ/S, for all the elements

Aij . Then, when the size S → ∞, the eigenvalues of A, λ = x + iy, are uniformly distributed on

an ellipse (x/a)2 + (y/b)2 ≤ 1, where a = 1 + τ and b = 1 − τ . Here we conjecture that this

result also holds for the non-Gaussian case (as supported by extensive simulations, see main text

and below).

In the predator-prey case, matrices are constructed as in the random case, but with the con-

straint that if Mij > 0 then Mji < 0, i.e., the interaction is beneficial for one species and detrimen-

tal for the other. In the construction, with probability C, we randomly set one of the coefficients in

the pair (Mij,Mji)i �=j to be positive and the other to be negative. This can be achieved by sampling

one coefficient from the distribution of |X| and the other from −|X|. Both coefficients are zero

with probability 1 − C. The diagonal terms of M are initially set to be zero. Numerical simula-

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element by the constant σ√SC) are uniformly distributed on a unit circle centered at (0, 0), as

S → ∞. It follows that, when S is sufficiently large, the eigenvalue distribution of M is uniform

on a circle of radius approximately σ√SC (Figure 1, main text).

Now we can introduce the effect of diagonal strength (density dependence). We start by

setting all the diagonal elements of M to be −d: this shifts the circle so that it is now centered

at (−d, 0). We relax this constraint in Section 3 below. To achieve stability, the circle must be

fully contained in the left half-plane (as all eigenvalues must have negative real parts). Hence, to

achieve stability, the radius of the circle must be smaller than d, i.e., σ√SC < d. Because the

values relevant to stability are the network complexity√SC and the ratio between d (magnitude

of the diagonal terms) and σ (standard deviation of the distribution of X), we write θ = d/σ. The

stability criterion for a random community matrix is√SC < θ, irrespective of the distribution of

X . For example, suppose that X follows a distribution with σ = 0.5, the diagonal terms Mii = −1,

and set C to be 0.1: then the stability criterion is violated whenever S ≥ 41.

2.2 - Predator-Prey

To derive other stability criteria, we need to formulate a new conjecture. Sommers et al.11 proved

the following theorem: take a matrix A, whose elements are follow a Gaussian distribution with

mean E(Aij) = 0, variance Var(Aij) = 1/S, and correlation E(AijAji) = τ/S, for all the elements

Aij . Then, when the size S → ∞, the eigenvalues of A, λ = x + iy, are uniformly distributed on

an ellipse (x/a)2 + (y/b)2 ≤ 1, where a = 1 + τ and b = 1 − τ . Here we conjecture that this

result also holds for the non-Gaussian case (as supported by extensive simulations, see main text

and below).

In the predator-prey case, matrices are constructed as in the random case, but with the con-

straint that if Mij > 0 then Mji < 0, i.e., the interaction is beneficial for one species and detrimen-

tal for the other. In the construction, with probability C, we randomly set one of the coefficients in

the pair (Mij,Mji)i �=j to be positive and the other to be negative. This can be achieved by sampling

one coefficient from the distribution of |X| and the other from −|X|. Both coefficients are zero

with probability 1 − C. The diagonal terms of M are initially set to be zero. Numerical simula-

6tions suggest that this type of network is more likely to be stable than expected at random3. Also,

qualitative stability can be achieved only for predator-prey matrices2, 3.

As in the random case, predator-prey matrices have E(Mij)i �=j = 0, Var(Mij)i �=j = Cσ2

by construction. However, the constraint on pairwise interactions (Mij,Mji)i �=j makes the corre-

lation E(MijMji)i �=j negative, since Mij and Mji always have opposite signs (or both are zero).

More specifically, in predator-prey case E(MijMji)i �=j = −CE2(|X|), while in the random case,

this correlation is zero because of independence. For large S, the eigenvalues of the matrix

M/(σ√SC) are distributed in the ellipse (x/a)2 + (y/b)2 ≤ 1, where a = 1 + τ , b = 1 − τ

and τ = SE(MijMji/(σ

2SC))

= −E2(|X|)/σ2. Rescaling, for large S, we obtain eigen-

values of M that are approximately uniformly distributed in an ellipse with horizontal half-axis

a = σ√SC(1− E2(|X|)/σ2) and vertical half-axis b = σ

√SC(1 + E2(|X|)/σ2) (Figure 1, main

text). To include effects from the diagonal coefficients Mii = −d, we proceed as in the random

case. The system is stable whenever a < d, i.e., σ√SC(1 − E2(|X|)/σ2) < d, which becomes

√SC < θ/(1−E2(|X|)/σ2). Suppose that X comes from a normal distribution N(0, σ2), σ = 0.5,

−d = −1 and C = 0.1. In this case, E(|X|) = σ√

2/π. Accordingly, the stability criterion is√SC < θπ/(π− 2) ≈ 2.75 θ. We expect predator-prey systems to be unstable whenever S ≥ 303

(it was 41 for the corresponding random case).

If the distribution of X is symmetric around zero, such as in X ∼ N(0, σ2) and X ∼U [−

√3σ,

√3σ], then, we can take a predator-prey matrix and shuffle its interaction coefficients to

produce a random community matrix. In the same way, we can take a large random community

matrix and pair the interactions to produce a predator-prey matrix. In other words, the distributions

of interaction coefficients are the same in both cases, but the arrangements of the interaction coef-

ficients differ among matrix types. Thus, the increase in stability of the predator-prey case arises

from arranging the coefficients in pairs with opposite signs (which yields a negative τ ). This can

be confirmed from a different point of view: the eigenvalues have mean −d in both cases (the trace

being −dS), while their variance12, for large S, is Var(λ) = (S−1)E(MijMji)i �=j = (S−1)Cτσ2.

The variance is thus zero in the random case and negative in the predator-prey case. Note that the

variance can be negative since the eigenvalues can be complex conjugates. Having negative vari-

7

W W W. N A T U R E . C O M / N A T U R E | 7

SUPPLEMENTARY INFORMATION RESEARCH

tions suggest that this type of network is more likely to be stable than expected at random3. Also,

qualitative stability can be achieved only for predator-prey matrices2, 3.

As in the random case, predator-prey matrices have E(Mij)i �=j = 0, Var(Mij)i �=j = Cσ2

by construction. However, the constraint on pairwise interactions (Mij,Mji)i �=j makes the corre-

lation E(MijMji)i �=j negative, since Mij and Mji always have opposite signs (or both are zero).

More specifically, in predator-prey case E(MijMji)i �=j = −CE2(|X|), while in the random case,

this correlation is zero because of independence. For large S, the eigenvalues of the matrix

M/(σ√SC) are distributed in the ellipse (x/a)2 + (y/b)2 ≤ 1, where a = 1 + τ , b = 1 − τ

and τ = SE(MijMji/(σ

2SC))

= −E2(|X|)/σ2. Rescaling, for large S, we obtain eigen-

values of M that are approximately uniformly distributed in an ellipse with horizontal half-axis

a = σ√SC(1− E2(|X|)/σ2) and vertical half-axis b = σ

√SC(1 + E2(|X|)/σ2) (Figure 1, main

text). To include effects from the diagonal coefficients Mii = −d, we proceed as in the random

case. The system is stable whenever a < d, i.e., σ√SC(1 − E2(|X|)/σ2) < d, which becomes

√SC < θ/(1−E2(|X|)/σ2). Suppose that X comes from a normal distribution N(0, σ2), σ = 0.5,

−d = −1 and C = 0.1. In this case, E(|X|) = σ√2/π. Accordingly, the stability criterion is

√SC < θπ/(π− 2) ≈ 2.75 θ. We expect predator-prey systems to be unstable whenever S ≥ 303

(it was 41 for the corresponding random case).

If the distribution of X is symmetric around zero, such as in X ∼ N(0, σ2) and X ∼U [−

√3σ,

√3σ], then, we can take a predator-prey matrix and shuffle its interaction coefficients to

produce a random community matrix. In the same way, we can take a large random community

matrix and pair the interactions to produce a predator-prey matrix. In other words, the distributions

of interaction coefficients are the same in both cases, but the arrangements of the interaction coef-

ficients differ among matrix types. Thus, the increase in stability of the predator-prey case arises

from arranging the coefficients in pairs with opposite signs (which yields a negative τ ). This can

be confirmed from a different point of view: the eigenvalues have mean −d in both cases (the trace

being −dS), while their variance12, for large S, is Var(λ) = (S−1)E(MijMji)i �=j = (S−1)Cτσ2.

The variance is thus zero in the random case and negative in the predator-prey case. Note that the

variance can be negative since the eigenvalues can be complex conjugates. Having negative vari-

7

ance means that the variance of the imaginary part of the eigenvalues is larger than that of the real

part (while the two variances are equal in the random case). This agrees with our derivation that

yields a circle for the random case and a vertically-stretched ellipse for predator-prey networks

(Figure 1, main text). Given that imaginary parts influence cycling, this result is consistent with

the tendency of predator-prey systems to oscillate13.

2.3 - Mixture of Competition and Mutualism

If stability is driven by having negative τ , reversing its sign would decrease stability. Matrices in

which pairs of species are interacting as mutualists or competitors, with equal probability, yield a τ

with the same magnitude found in the predator-prey case but with the opposite sign. These mixture

matrices are constructed as in the predator-prey case, except that we choose the coefficient signs

differently: we either sample both Mij and Mji from the distribution of |X|, or both from −|X|,with equal probability. This guarantees that the two coefficients have the same sign: half of the

species interactions are mutualistic and half are competitive. In the mixture matrices, E(Mij)i �=j =

0 and Var(Mij)i �=j = Cσ2, as in the random and predator-prey cases, but the correlation becomes

E(MijMji)i �=j = CE2(|X|). Following the same procedure described for the predator-prey case,

we find that the stability criterion for mixture networks is√SC < θ/(1 + E2(|X|)/σ2) (Figure 1,

main text). For X following a normal distribution we have√SC < θπ/(π + 2) ≈ 0.61 θ. If we

set σ = 0.5, −d = −1 and C = 0.1, the criterion is violated for S ≥ 15.

2.4 - Mutualism and Competition in Isolation

Consider any mutualistic community matrix M ′, i.e., M ′ij ≥ 0, ∀i, j, with 0 on its diagonal. Then,

−M ′ is a competitive community matrix, since −M ′ij ≤ 0. For any eigenvalue λ of M ′, −λ is

an eigenvalue of −M ′. This means the eigenvalues of any zero-diagonal community mutualistic

matrix are linked to those of the corresponding competitive matrix: λ(M ′) = −λ(−M ′) (Figure

S1).

In a more general case, where the community matrix M has −d, d > 0 along its diagonal,

the eigenvalues are shifted to the left by d compared to the zero-diagonal case (Figure S2). Thus,

8

SUPPLEMENTARY INFORMATION

8 | W W W. N A T U R E . C O M / N A T U R E

RESEARCH

ance means that the variance of the imaginary part of the eigenvalues is larger than that of the real

part (while the two variances are equal in the random case). This agrees with our derivation that

yields a circle for the random case and a vertically-stretched ellipse for predator-prey networks

(Figure 1, main text). Given that imaginary parts influence cycling, this result is consistent with

the tendency of predator-prey systems to oscillate13.

2.3 - Mixture of Competition and Mutualism

If stability is driven by having negative τ , reversing its sign would decrease stability. Matrices in

which pairs of species are interacting as mutualists or competitors, with equal probability, yield a τ

with the same magnitude found in the predator-prey case but with the opposite sign. These mixture

matrices are constructed as in the predator-prey case, except that we choose the coefficient signs

differently: we either sample both Mij and Mji from the distribution of |X|, or both from −|X|,with equal probability. This guarantees that the two coefficients have the same sign: half of the

species interactions are mutualistic and half are competitive. In the mixture matrices, E(Mij)i �=j =

0 and Var(Mij)i �=j = Cσ2, as in the random and predator-prey cases, but the correlation becomes

E(MijMji)i �=j = CE2(|X|). Following the same procedure described for the predator-prey case,

we find that the stability criterion for mixture networks is√SC < θ/(1 + E2(|X|)/σ2) (Figure 1,

main text). For X following a normal distribution we have√SC < θπ/(π + 2) ≈ 0.61 θ. If we

set σ = 0.5, −d = −1 and C = 0.1, the criterion is violated for S ≥ 15.

2.4 - Mutualism and Competition in Isolation

Consider any mutualistic community matrix M ′, i.e., M ′ij ≥ 0, ∀i, j, with 0 on its diagonal. Then,

−M ′ is a competitive community matrix, since −M ′ij ≤ 0. For any eigenvalue λ of M ′, −λ is

an eigenvalue of −M ′. This means the eigenvalues of any zero-diagonal community mutualistic

matrix are linked to those of the corresponding competitive matrix: λ(M ′) = −λ(−M ′) (Figure

S1).

In a more general case, where the community matrix M has −d, d > 0 along its diagonal,

the eigenvalues are shifted to the left by d compared to the zero-diagonal case (Figure S2). Thus,

8

Interaction Type Signs Mij, Mji Frequency

Non Interacting (0, 0) (1− C)2

Commensalism (0,+) or (+, 0) C(1− C)

Amensalism (0,−) or (−, 0) C(1− C)

Competition (−,−) C2/4

Mutualism (+,+) C2/4

Predator-prey (+,−) or (−,+) C2/2

Table 1: Frequency of interactions in large random networks. The frequencies hold for a

large, S × S matrix M in which the off-diagonal coefficients are drawn from a normal dis-

tribution N(0, σ2) with probability C and are 0 otherwise. The type of interaction between

two species i and j is determined by the signs of the corresponding coefficients in the

matrix.

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Competition

Figure S1: Distribution of the eigenvalues for mutualism and competition. Plotted from 10

matrices (colors), with S = 250, C = 0.35, σ = 0.05 and d = 0. In both cases we observe an

extreme, real eigenvalue which equals the row-sum of the matrix (red arrow, larger points).

9

W W W. N A T U R E . C O M / N A T U R E | 9

SUPPLEMENTARY INFORMATION RESEARCH

Interaction Type Signs Mij, Mji Frequency

Non Interacting (0, 0) (1− C)2

Commensalism (0,+) or (+, 0) C(1− C)

Amensalism (0,−) or (−, 0) C(1− C)

Competition (−,−) C2/4

Mutualism (+,+) C2/4

Predator-prey (+,−) or (−,+) C2/2

Table 1: Frequency of interactions in large random networks. The frequencies hold for a

large, S × S matrix M in which the off-diagonal coefficients are drawn from a normal dis-

tribution N(0, σ2) with probability C and are 0 otherwise. The type of interaction between

two species i and j is determined by the signs of the corresponding coefficients in the

matrix.

−1 0 1 2 3 4

−0.4

0.0

0.4

Re

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Mutualism

−4 −3 −2 −1 0 1

−0.4

0.0

0.4

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Competition

Figure S1: Distribution of the eigenvalues for mutualism and competition. Plotted from 10

matrices (colors), with S = 250, C = 0.35, σ = 0.05 and d = 0. In both cases we observe an

extreme, real eigenvalue which equals the row-sum of the matrix (red arrow, larger points).

9

SUPPLEMENTARY INFORMATION

1 0 | W W W. N A T U R E . C O M / N A T U R E

RESEARCH

−6 −4 −2 0 2 4 6

−0.5

0.5

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λ2A

Competition

Figure S2: Distribution of the eigenvalues in competitive and mutualistic systems. Top: Distri-

bution of the eigenvalues for a mutualism matrix M . Bottom: Distribution of the eigenvalues for a

competition matrix A = −M−2dI . The red arrows mark the position of the dominant eigenvalues

for the two cases. The position of λM1 and λA

2 = −λM2 − 2d are the eigenvalues of the mutualistic

matrix M and the competition matrix A that are critical for the stability.

for any eigenvalue λ of the mutualistic matrix M ′ with 0 on the diagonal, λM = λ − d is the

eigenvalue of M = M ′ − dI . Note that M is mutualistic and its diagonal terms are −d. Similarly,

A = −M ′ − dI = −M − 2dI is a competitive matrix and its eigenvalues are all of the form

λA = −λ− d = −λM − 2d, where λ and λM are the eigenvalues of M ′ and M , respectively.

Let λM1 be the eigenvalue of a mutualistic matrix M that has the largest real part and λM

2 be

the one with the smallest real part. Because of the one-to-one mapping above, λA2 = −λM

2 − 2d

has the largest real part among all the eigenvalues of A = −M−2dI . Therefore, in the mutualistic

case, whenever λM1 < 0, the system is stable (Figure S2, top), while in the competition case

whenever λA2 = −λM

2 − 2d < 0 (Figure S2, bottom), the system is stable. If we are able to find the

expressions for λM1 and λM

2 , we can find the stability criteria for both mutualism and competition.

Suppose M is a mutualistic community matrix with S species (S is large) and connectance C,

which is randomly generated as described in Section 1 of the SI, with X following some statistical

10

W W W. N A T U R E . C O M / N A T U R E | 1 1

SUPPLEMENTARY INFORMATION RESEARCH

distribution such that E(X) = 0,Var(X) = σ2. For this matrix, Mii = −d, and

E(Mij)i �=j = CE(|X|)Var(Mij)i �=j = E(M2

ij)i �=j − E2(Mij)i �=j = Cσ2 − C2E2(|X|)

Derivation of λM1 and the stability criteria for Mutualism

For large S, since Mij are i.i.d. from the distribution of |X|, the row sum of matrix M is

roughly a constant:

S∑

j=1

Mij ≈ −d+ (S − 1)E(Mij)i �=j = −d+ (S − 1) · C · E(|X|)

Then, λM1 is approximately equal to the row sum, with corresponding eigenvector close to 1,

because

M · 1 =

∑Sj=1 M1j

...∑S

j=1 MSj

≈ {−d+ (S − 1)CE(|X|)} · 1

The dominance of λM1 follows from the Gershgorin circle theorem14, 15. Thus, λM

1 = −d +

(S − 1)CE(|X|) for sufficiently large S, and the stability criteria for mutualism is λM1 < 0, which

is equivalent to (S − 1)CE(|X|)/σ < θ, where θ = d/σ. Note that in the case of mutualistic

networks, stability is equivalent to diagonal dominance15.

Derivation of λM2 and the stability criteria for Competition

Since M + dI is nonnegative and 1 is an eigenvector of M + dI when S is sufficiently

large, according to Rothblum & Tan16, for any real number a ∈ R, M + dI − a · 1 · 1T has

(approximately) the same eigenvalues as M + dI except that the dominant eigenvalue of M + dI ,

λM1 + d, is replaced by λM

1 + d − Sa. Given that adding aI(a ∈ R) to any matrix only shifts the

eigenvalues horizontally, the eigenvalues of M + (d + a)I and M + (d + a)I − a · 1 · 1T should

also coincide except λM1 + d+ a is replaced by λM

1 − (S − 1)a+ d.

To calculate λM2 approximately for large S, which is also the subdominant eigenvalue of M ,

11

SUPPLEMENTARY INFORMATION

1 2 | W W W. N A T U R E . C O M / N A T U R E

RESEARCH

we need the center (c) and the radius (r) of the eigenvalue “plate” on the left, and then λM2 = c− r.

The center of the plate can be calculated by averaging all the eigenvalues of M but the dominant

one (λM1 ), i.e.

c = −d− CE(|X|)

If we choose a = C ·E(|X|), then the eigenvalue distribution of N = M+(d+a)I−a ·1 ·1T

should be centered at (0, 0), since Nii = 0, ∀i. The spectral radius of N , rN , should be the same

as the half horizontal radius of the ellipse in the eigenvalue distribution of M , i.e. rN = r.

For the distribution of the off-diagonal elements of N , we can easily get:

E(Nij)i �=j = E(Mij − a)i �=j = E(Mij)i �=j − CE(|X|) = 0

Var(Nij)i �=j = Var(Mij)i �=j = Cσ2 − C2E2(|X|)

E(NijNji)i �=j = C(1− C)E2(|X|)

Then, following Sommers et al.11 – although they only proved the Gaussian ensemble case

– we conjecture that the conclusion is also applicable to the non-Gaussian case. This conjecture

is enlightened by Tao et al.10 (who proved, among other things, that this holds for i.i.d. random

variables), and we performed numerical simulations to verify its correctness. Thus the real radius

of N , for large S, is approximately: rN =√S · Var(Nij)i �=j(1 + E(NijNji)i �=j/Var(Nij)i �=j) and

the subdominant eigenvalue:

λM2 = c− r = −d− CE(|X|)−

√S · Var(Nij)i �=j

(1 +

E(NijNji)i �=j

Var(Nij)i �=j

)

The stability criterion for a competition system, when S is large, is:

λA2 = −λM

2 − 2d =√SC (σ2 − CE2(|X|)) · σ

2 + (1− 2C)E2(|X|)σ2 − CE2(|X|) + CE(|X|)− d < 0

which is equivalent to√SC

(1− CE2(|X|)

σ2

)· 1+

(1−2C)E2(|X|)σ2

1−CE2(|X|)σ2

+ CE(|X|)σ

< θ. Specifically, if X

follows a normal distribution N(0, σ2), then E(|X|) = σ√

2π

, E2(|X|)/σ2 = 2/π, and the stability

12

W W W. N A T U R E . C O M / N A T U R E | 1 3

SUPPLEMENTARY INFORMATION RESEARCH

criterion for the competition case is√

SC(1− 2C/π)(1 + 2(1−C)

π−2C

)+ C

√2/π < θ , where the

term C√

2/π is negligible for large S.

3 - Extension to General Diagonal and Mixtures of Interactions

We have analyzed the case in which the diagonal elements of the matrix are all −d. However,

the results for the random, predator-prey and mixture matrices are robust to variable diagonal

terms such that E(Mii) = −d, provided that the variance Var(Mii) is small compared to the size

of the network11. In fact, in all these cases, the variance of the eigenvalues can be written as12

Var(λ) = Var(Mii) + (S − 1)E(MijMji). For large S, the effect of Var(Mii) becomes negligible,

yielding the criteria derived for Mii = −d. Note that if the trace of the matrix is positive the

system is trivially unstable. However, if the trace is negative, matrices in which some of the

diagonal coefficients are positive can be stable.

Mutualism and competition matrices are more sensitive to variable diagonal terms, with

consequences for stability: the higher the Var(Mii), the lower the probability of stability. This

is easy to see in the mutualism case: because all of the coefficients (except those on the di-

agonal) are non-negative, according to Gershgorin circle theorem14, 15 the dominant eigenvalue

λM1 ≤ maxi(

∑j Mij). Thus, variation in Mii is likely to be destabilizing compared to the case in

which all Mii = −d.

An interesting property of the three cases in which the interaction strengths follow a distribu-

tion with E(Mij)i �=j = 0 and Var(Mij)i �=j = Cσ2 is that we can combine them to find the stability

criterion for matrices following a given mixture of interactions, provided that E(Mij)i �=j = 0 and

Var(Mij)i �=j = Cσ2. For example, suppose that a network has connectance C and the interacting

pairs can be divided into random (with probability p = 0.2), mutualistic (p = 0.15), competitive

(p = 0.15), and predator-prey (p = 0.5), ensuring that E(Mij)i �=j = 0. Furthermore, assume

X ∼ N(0, σ2). For these matrices, τ = 0.2 · 0 + 0.3 · (2/π)− 0.5 · (2/π) = −0.2/π, which yields

a stability criterion√SC < θπ/(π − 0.2) ≈ 1.07 θ.

The derivation of the stability criteria for distributions with non-zero means (competition,

mutualism) suggests that other ecologically-interesting cases can be solved using the same strategy.

13

SUPPLEMENTARY INFORMATION

1 4 | W W W. N A T U R E . C O M / N A T U R E

RESEARCH

For example, it would be interesting to derive the stability criterion for predator-prey systems in

which the negative interactions are, in absolute value, larger than the positive ones, due to the

inefficiency in prey assimilation. This requires extending the findings presented here to the case in

which the distribution of positive coefficients differs from that of negative coefficients not only in

sign, but also in shape. We plan to follow the same method we used in the analysis of competitive

networks to achieve this goal.

4 - Effects of Realistic Food Web Structure

When we drew the eigenvalue distributions for the cascade and niche matrices, we found several

extreme eigenvalues with large imaginary part (Figure S3). Because the eigenvalues of these ma-

trices must have the same mean and variance observed in the unstructured predator-prey case12 (SI,

section 1), the ellipses are distorted, yielding larger real parts than in the unstructured case.

5 - Effects of Nestedness and Asymmetric Interactions in Mutualistic Systems

For mutualistic matrices, in both the bipartite and nested cases we observed a large, real eigenvalue

(as in the unstructured mutualistic case). However, in bipartite and nested bipartite matrices, when

the diagonal is zero, for each real positive eigenvalue we found an equally large negative one

(Figure 3, main text). The bipartite case yields row sums that are equal to the unstructured case,

and thus the same stability criterion, because rearranging interactions to be only between groups

does not alter the expected row sum of the community matrix. However, in nested structures

generalist plants and animals yield larger than average row/column sums, which hampers stability.

The stability profiles confirm these findings (Figure S4).

In mutualistic systems, asymmetric interaction strengths are believed to be stabilizing17. That

is, if each large Mij is coupled with a weak Mji, this should enhance stability. We have shown

that the stability criterion for the mutualistic case, for a given θ, depends exclusively on the row

sum ((S − 1)CE(|X|)− d). To test the effect of asymmetry on stability, we randomly constructed

three types of mutualistic community matrices, M (1), M (2) and M (3), with increasing levels of

asymmetry. Taking the uniform distribution X ∼ U [0, 1] as an example: 1) In the matrix M (1),

both coefficients in the pair (M (1)ij M

(1)ji )i>j are independently drawn from U [0, 1], with probability

14

W W W. N A T U R E . C O M / N A T U R E | 1 5

SUPPLEMENTARY INFORMATION RESEARCH

−5 0 5

−50

050

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−d●

Unstructured Predator−Prey

1.5 2.0 2.5 3.0 3.5 4.0

0.0

0.2

0.4

0.6

0.8

1.0

σ SC

P(St

abilit

y)

Unstructured Predator−Prey

−5 0 5

−50

050

Re

Im

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−d●

CascadePredator−Prey

1.5 2.0 2.5 3.0 3.5 4.0

0.0

0.2

0.4

0.6

0.8

1.0

σ SC

P(St

abilit

y)

CascadePredator−Prey

−5 0 5

−50

050

Re

Im

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−d●

NichePredator−Prey

1.5 2.0 2.5 3.0 3.5 4.0

0.0

0.2

0.4

0.6

0.8

1.0

σ SC

NichePredator−Prey

Figure S3: Eigenvalues and stability profiles for structured predator-prey networks. Top:

Distribution of the eigenvalues for the unstructured predator-prey case, the cascade model and

the niche model, with the same values used in Figure 1 (main text). In both cascade and niche

cases we observe extreme eigenvalues with large imaginary parts. Accordingly, the theoretical

ellipse derived for the unstructured predator-prey case (black) does not contain all the remaining

eigenvalues, and thus the probability of stability is reduced. This effect is more pronounced in

the niche case. Bottom: Stability profiles obtained using the same values as Figure 1 (main text).

We include the unstructured predator-prey profile (blue dashed line) in the cascade and the niche

models for comparison. Note that, contrary to the unstructured model, in the cascade and niche

matrices increasing C (×) or σ (+) yields different effects (separation between the two lines).

15

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1 6 | W W W. N A T U R E . C O M / N A T U R E

RESEARCH

0.4 0.6 0.8 1.0 1.2 1.4

0.0

0.2

0.4

0.6

0.8

1.0

R+d

P(St

abilit

y)

MutualismBipartite Mut.Nested Mut.

Figure S4: Stability profile for mutualistic matrices. For S = 250, σ = 0.025 and d = 1, we

vary C so that the critical value (S − 1)Cσ√2/π = R + d (x-axis) spans [0.2, 1.2]. The critical

value is reached for (S − 1)Cσ√2/π = 1 (red line). Note that, as expected from Figure 3 (main

text), nested matrices are much less stable than the other two types of matrices.

C, and M(1)ij,i>j = M

(1)ji,i>j = 0 with probability (1 − C); 2) For the matrix M (2), with probability

C, we randomly assign one of the coefficient pair (M(2)ij ,M

(2)ji )i>j by sampling from U [0, 1/2]

and the other from U [1/2, 1]; while with probability (1 − C), M (2)ij,i>j = M

(2)ji,i>j = 0; 3) For the

matrix M (3), with probability C, we randomly assign one of the coefficient pair (M (3)ij ,M

(3)ji )i>j by

sampling from U [0, 1/3] and the other from U [2/3, 1]; while with probability (1 − C), M (3)ij,i>j =

M(3)ji,i>j = 0. In the three cases, the diagonals are all set to 0. By construction, M (1) would

represents the lowest level of asymmety because all M (1)ij,i�=j are independently drawn from the

same distribution, whereas M (2) and M (3) represent moderate and the highest levels of asymmetry,

respectively.

We expect the stability criteria not to change in the three cases, as the row sums of M (1), M (2)

and M (3) are approximately the same for large S, and the stability is exclusively determined by

the row sum for a mutualistic system. Numerical simulations confirm our prediction (black arrow

in Figure S5). The radii of the eigenvalue ellipses of M (1), M (2) and M (3), E(M (1)ij M

(1)ji )i �=j >

E(M (2)ij M

(2)ji )i �=j > E(M (3)

ij M(3)ji )i �=j (on the left of Figure S5) decrease with the skewness of the

16

W W W. N A T U R E . C O M / N A T U R E | 1 7

SUPPLEMENTARY INFORMATION RESEARCH

−5 0 5 10 15 20

−3−1

12

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M(1)

M(2)

M(3)

Figure S5: Distribution of the eigenvalues for mutualistic matrices, with different levels of

asymmetry of interaction strengths. M (1), M (2), and M (3) are randomly constructed in the way

described in Section 5 of SI, with parameters S = 200 and C = 0.20. The eigenvalues of M (1),

M (2) and M (3) with the largest real parts, which are approximately equal to the row sum of the

matrices when S is sufficiently large, are highlighted by the black arrow.

interaction strengths, provided E(Mij) is constant. This means that asymmetry can modify the

dynamic properties of the system, but not its stability.

The fact that we are finding many counterintuitive results can be accounted for noting that

previous studies focused on simulations4, 18–20. Typically, a system of differential equations is

parametrized and then integrated through time; in order to test the effects of network structure

the way the network is generated is modified. However, in such a framework it is very difficult

to isolate the effect of the structure from the interaction between the structure and the specific

functional form, parametrization and initial conditions used in the simulations.

Here we took a different approach and studied the effects of structural features for systems

in which all the other variables (size, connectance, distribution of interaction strengths) are kept

constant. Basically, each network corresponds simply to a specific arrangement of the same coef-

ficients. In this way, we can isolate the contribution of structural property, such as nestedness, to

stability.

17

SUPPLEMENTARY INFORMATION

1 8 | W W W. N A T U R E . C O M / N A T U R E

RESEARCH

6 - Distribution of Interaction Strengths and Weak Interactions

Our results hold for arbitrary distributions of interaction strengths. In the main text, we have

analyzed the difference between normal and uniform distributions. Here we derive the criteria for

the Gamma distribution, as it allows us to study the limiting cases directly.

We choose a symmetric distribution for X where the positive half distribution, X|X > 0, is

Γ(k, σ

√1/(k2 + k)

), k > 0 and the negative half distribution, X|X < 0, is a reflected gamma

distribution (see the first row of Figure S6 and Figure S7 for the shape of the entire distribution

of X). Therefore, we have E(X) = 0 and Var(X) = σ2 for arbitrary choice of k. Now we

can tune k to obtain any proportion of weak interactions. In fact, the expected magnitude of the

interactions is E(|X|) = kσ√1/(k2 + k) = σ

√k/(k + 1): choosing large k leads to E(|X|) ≈ σ

(rare weak interactions), while small k leads to E(|X|) ≈ 0 (preponderant weak interactions).

The stability criterion for any choice of k is√SC < (k + 1)θ in the predator-prey case and

√SC < k+1

2k+1θ in the mixture case, while the random case

√SC < θ irrespective of the choice

of k. For example, when k = 3, and thus E(|X|) =√3σ/2, we recover the same results found

in the uniform case (Figures S6-7). Note that the uniform distribution U [−√3σ,

√3σ] has very

different shape compared to the one we constructed by reflecting a gamma distribution. However,

the stability criteria for both distributions is the same because both yield the same E(|X|). If we

choose k = 0.7, we have E(|X|) =√7σ/

√17 ≈ 0.642 (weaker than normal) and accordingly

√SC < 1.7 θ for the predator-prey case and

√SC < 0.708 θ for the mixture case (Figures S6-7).

Higher probability of stability for predator-prey networks is achieved for larger k (and thus very

strong interactions), while lower probability of stability (approaching the random case,√SC < θ)

as k → 0. The criterion for stability is bounded by the random case from below, while it has no

upper limit. The opposite is true for the networks with a mixture of competition and mutualism,

where a higher probability of stability (√SC < θ, as in the random case) is obtained for k → 0

(weaker interactions on average), while for k → ∞ (stronger interactions on average) we have√SC < θ/2. Thus, the stability of mixture matrices is bounded from above by the random case.

These derivations, matched by numerical simulations (Figures S7) reveal that weak interactions,

for a given S, C, d and σ, are destabilizing for the predator-prey case, stabilizing for the mixture

18

W W W. N A T U R E . C O M / N A T U R E | 1 9

SUPPLEMENTARY INFORMATION RESEARCH

case (and for mutualism and competition in isolation as well) and have no effect on the random

case.

Thus, mixture and predator-prey networks can be stabilized by altering the distribution of

the off-diagonal elements of the community matrix while leaving the diagonal unchanged. To test

whether this is feasible in a practical context, we examine the case of so-called Generalized Lotka-

Volterra models21–23, which have been employed to study the relationship between stability and

complexity. We start with the system:

dXi

dt= ri

(1− Xi

Ki

)Xi +

S∑

j=1,j �=i

αijXiXj ∀i = 1, 2, . . . , S, (S1)

where Xi is the density of species i at time t, ri is the intrinsic growth rate, Ki is the carrying

capacity and αij combines encounter rate and the effect of j on i.

Suppose that the system is resting at a feasible equilibrium point:

0 = ri

(1− X∗

i

Ki

)+∑S

j=1,j �=i αijX∗j

X∗i =

(ri +

∑Sj=1,j �=i αijX

∗j

)Ki

ri

(S2)

such that X∗i > 0 for all species.

At the equilibrium, the community matrix is defined as:

Mii = ri

(1− 2X∗

i

Ki

)+∑S

j=1,j �=i αijX∗j = − riX

∗i

Ki

Mij = αijX∗i

(S3)

To test whether we can modify the parameters (ri, Ki, αij) so that the distribution of the off-

diagonal elements is distorted while the diagonal elements are unchanged, we define the modified

matrix M ′ as M ′ij = βijMij with βii = 1 and βij > 0 (so that the sign pattern is not altered).

The matrix β represents the transformation we want to apply to the distribution of the off-diagonal

elements.

The goal is therefore to find another Generalized Lotka-Volterra system that has the same

equilibrium of the previous one, but that is described by the community matrix M ′ around the

equilibrium point. We define the desired system as:

19

SUPPLEMENTARY INFORMATION

2 0 | W W W. N A T U R E . C O M / N A T U R E

RESEARCH

dYi

dt= r′i

(1− Yi

K ′i

)Yi +

S∑

j=1,j �=i

α′ijYiYj ∀i = 1, 2, . . . , S, (S4)

The community matrix M ′ for the new system must satisfy:

M ′ij = βijMij

M ′ii = Mii

Y ∗i = X∗

i

(S5)

These constraints are satisfied by setting the growth rate r′i = βiri− riX∗i

Ki(βi−1), K ′

i =r′iriKi

and the interactions α′ij = βijαij , where βi is defined as a weighted mean:

βi =

∑Sj=1,j �=i βijαijX

∗j∑S

j=1,j �=i αijX∗j

(S6)

We now show that the three constrains illustrated above are satisfied. First, we need M ′ij =

βijMij = βijαijX∗j . In the new system, M ′

ij = βijαijY∗j . Second, we need M ′

ii = Mii =riX

∗i

Ki.

In the new system, M ′ii =

r′iY∗i

K′i

=riY

∗i

Ki. Thus, if the third condition X∗

i = Y ∗i is satisfied, all

conditions are met.

To confirm that the equilibrium is unaltered, we can plug back the values of r′i, K′i, α

′ij and

Y ∗i = X∗

i into the system of equations to test whether Y ∗ = X∗ is an equilibrium of the new

system.dYi

dt

∣∣Y=X∗ = X∗

i

(r′i − r′iX

∗i

K′i+∑S

j=1,j �=i αijβijX∗j

)=

= X∗i

(βiri − βi

riX∗i

Ki+ βi

∑Sj=1,j �=i αijXj

)=

= βi

{ri

(1− X∗

i

Ki

)X∗

i +∑S

j=1,j �=i αijX∗i X

∗j

}= 0

(S7)

Thus, the equilibrium for the new system is Y ∗ = X∗ and all conditions are satisfied. This means

that in Generalized Lotka-Volterra models we are able to arbitrarily alter the distribution of the

off-diagonal elements (and thus the stability of the system) without affecting the equilibrium point

or the diagonal elements of the community matrix. The same strategy can be applied to prove that

we can manipulate the off-diagonal coefficients in systems where species interact with Type II or

Type III functional response (simply assume that handling times and the equilibrium are the same

in the two systems).

20

W W W. N A T U R E . C O M / N A T U R E | 2 1

SUPPLEMENTARY INFORMATION RESEARCH

Half −4 −2 0 2 4

0.0

0.1

0.2

0.3

0.4

Den

sity

2 π σ− 2 π σ

−4 −2 0 2 4

0.0

0.1

0.2

0.3

0.4

Den

sity

3 4 σ− 3 4 σ

−4 −2 0 2 4

0.0

0.5

1.0

1.5

Den

sity

0.7 1.7 σ− 0.7 1.7 σ

−4 −2 0 2 4

0.0

0.1

0.2

0.3

Den

sity

3σ 2− 3σ 2

Distribution Half Normal(0,σ2

(1− 2

π

) )Γ(3,0.289σ) Γ(0.7,0.917σ) Unif(0,

√3σ)

−20 −10 0 10 20

−20

−10

010

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Imaginary

Real

−1●

Mixture

Figure S6: Distribution of the eigenvalues for random, predator-prey and mixture of competition and mutualism matrices

for different distributions. For S = 250, C = 0.25 and σ = 1, we plot the eigenvalues of 10 matrices (colors) with −1 on the

diagonal. For the off-diagonal elements, the magnitudes are taken from an half-normal (first column, main text), gamma (second

and third columns) or uniform (fourth column, main text) distributions, while the signs are assigned according to the types of

matrices. The distributions are parametrized in a way such that E(Mij)i �=j = 0 and Var(Mij)i �=j = Cσ2. The ellipses have

horizontal radius σ√SC(1 + τ) and vertical radius σ

√SC(1− τ), where τ is derived for arbitrary distributions in the text.

21

SUPPLEMENTARY INFORMATION

2 2 | W W W. N A T U R E . C O M / N A T U R E

RESEARCH

Half −4 −2 0 2 4

0.0

0.1

0.2

0.3

0.4

Den

sity

2 π σ− 2 π σ

−4 −2 0 2 4

0.0

0.1

0.2

0.3

0.4

Den

sity

3 4 σ− 3 4 σ

−4 −2 0 2 4

0.0

0.5

1.0

1.5

Den

sity

0.7 1.7 σ− 0.7 1.7 σ

−4 −2 0 2 4

0.0

0.1

0.2

0.3

Den

sity

3σ 2− 3σ 2

Distribution Half Normal(0,σ2

(1− 2

π

) )Γ(3,0.289σ) Γ(0.7,0.917σ) Unif(0,

√3σ)

0.6 0.8 1.0 1.2 1.4

0.0

0.2

0.4

0.6

0.8

1.0

σ SC

P(St

abilit

y)

0.6 0.8 1.0 1.2 1.4

0.0

0.2

0.4

0.6

0.8

1.0

σ SC

P(St

abilit

y)

0.6 0.8 1.0 1.2 1.4

0.0

0.2

0.4

0.6

0.8

1.0

σ SC

P(St

abilit

y)

0.6 0.8 1.0 1.2 1.4

0.0

0.2

0.4

0.6

0.8

1.0

σ SC

P(St

abilit

y)

Random

1.5 2.0 2.5 3.0 3.5 4.0

0.0

0.2

0.4

0.6

0.8

1.0

σ SC

P(St

abilit

y)

2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

σ SC

P(St

abilit

y)

1.0 1.5 2.0 2.5

0.0

0.2

0.4

0.6

0.8

1.0

σ SC

P(St

abilit

y)

2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

σ SC

P(St

abilit

y)

Predator-Prey

0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.0

0.2

0.4

0.6

0.8

1.0

σ SC

P(St

abilit

y)

0.3 0.4 0.5 0.6 0.7 0.8

0.0

0.2

0.4

0.6

0.8

1.0

σ SC

P(St

abilit

y)

0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.2

0.4

0.6

0.8

1.0

σ SC

P(St

abilit

y)0.3 0.4 0.5 0.6 0.7 0.8

0.0

0.2

0.4

0.6

0.8

1.0

σ SC

P(St

abilit

y)

Mixture

Figure S7: Stability profiles for the combinations of matrix type and distribution illustrated in Figure S6. For the random

and mixture cases, starting from S = 250, C = 0.5, σ = 0.1 and d = 1, we systematically varied C (×) or σ (+) in order

obtain σ√SC spanning [0.5, . . . , 1.0, . . . , 1.5] of the critical value for stability (indicated in red, 1 in the case of random matrices).

The profiles were obtained by computing the fraction of stable matrices out of 1000 realizations. The predator-prey case is as

the random but with σ = 0.3 for the half-normal and Γ(0.7, 0.917), while σ = 0.5 for the uniform and Γ(3, 0.289) cases. The

adjustment of σ is necessary as it would otherwise lead to C > 1.

22

W W W. N A T U R E . C O M / N A T U R E | 2 3

SUPPLEMENTARY INFORMATION RESEARCH

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