Communities as cliques

High-diversity species assemblages are very common in nature, and yet the factors allowing for the maintenance of biodiversity remain obscure. The competitive exclusion principle and May’s complexity-diversity puzzle both suggest that a community can support only a small number of species, turning the spotlight on the dynamics of local patches or islands, where stable and uninvadable (SU) subsets of species play a crucial role. Here we map the question of the number of different possible SUs a community can support to the geometric problem of finding maximal cliques of the corresponding graph. This enables us to solve for the number of SUs as a function of the species richness in the regional pool, N, showing that the growth of this number is subexponential in N, contrary to long-standing wisdom. To understand the dynamics under noise we examine the relaxation time to an SU. Symmetric systems relax rapidly, whereas in asymmetric systems the relaxation time grows much faster with N, suggesting an excitable dynamics under noise.

To get the large N asymptotic of this sum we define α ≡ ln(1/p).
As a first approximation one may assume that the logarithm of the sum (1) is equal to the contribution from S * alone. Plugging this into Eq. (3) one finds, In principle, we have to include also the contribution from a Gaussian integral in S − S * around S * . However, the coefficient of (S − S * ) 2 in this integral is given by, so in fact the correction decays so quickly around S * that only the leading term (7) contributes (note that the sum is discrete). This leads to Eq. (3) of the main text.
For asymmetric links, we found that (Eq. (4) of the main text) Again, writing things in terms of α ≡ ln(1/p), we have The dominant S, S * , obeys The dominant balance is now between the terms αN e −αS * and −αS * , giving Using this, the O(ln 2 N ) terms cancel, leaving us with and this yields the result that appears in Eq. (5) of the main text. Note that the convergence of this expression to the exact sum over S in Eq. (9) is quite slow and nonmonotonic, see Figure 1. Again, the decay of SU (N, S) is so rapid, that only the S * term contributes to the sum asymptotically.
In the main text we discussed the connection between SUs and cliques in the limit where the competition matrix elements are either zero or infinite, so two species i and j may be noninteracting (c ij = c ji = 0), mutually exclusive (c ij = ∞, c ji = ∞) or in dominance relationships (c ij = ∞, c ji = 0 or c ij = 0, c ji = ∞). This presentation leads us immediately to the notion of cliques and to the conclusions we drow from the equivalence between cliques and SUs. Here we would like to discuss the relationships between this extreme limit and the "standard" GLV description of the system, in which the c ij s are picked at random from a continuous distribution with finite moments. To be specific we will use, as in [2], Gamma distributed c i,j s and denote this version of the generalized Lotka-Volterra as the Gamma model.
To bridge the gap between the zero-infinite and the Gamma model, we present here an intermediate scenario, the binary model, which allows us to obtain a few analytic results that establish the relevance of the clique-based analysis. At the same time, this model, which has finite σ and C, facilitates a numerical comparison with the standard GLV model. These three types of interaction matrices -Gamma, binary and zero-infinite -are illustrated in Figure 2. For simplicity our discussion is presented for the symmetric case, but the results are general.
FIG. 2: Three types of interaction matrices ci,j. The left matrix corresponds to a four species community in which the niche overlap between species is assumed to be a random number taken from a Gamma distribution. To the right one sees an interaction matrix that corresponds to the zero/infinity limit considered in the main text, where every two species are either non-interacting or are mutually exclusive. The interaction matrix in the middle (C and A are constants, see below) exemplifies our binary model: like the infinite σ case is has only two types of interactions, but unlike it, it admits a finite value for σ, allowing for a comparison with GLV systems with continuous ci,js that have the same parameters. For simplicity we present examples for the symmetric case, where the level of competition between every two species is characterized by a single number that corresponds to the niche overlap between these two species. In the non-symmetric case cij = cji, but all other features of the three models are the same.
To begin, let us rewrite the GLV equation used in the main text, where now (following [2]) we assume that the c i,j s are normalized such that their average is unity and C sets the overall scale of the interaction. In the binary model, the c i,j s are either A or zero, so to fix their average to unity we must choose A = 1/(1 − p), where p is the probability of c i,j = c j,i = 0. It follows that the variance of the c i,j s is Thus, our binary model allows us to control, as in the standard Gamma distribution model, both the overall competition strength C and the variance of the competition, σ 2 , and to compare Gamma and binary competition matrices with the same mean and variance. The binary model is similar in many respects to the infinite σ model discussed in the main text. Indeed, its maximal cliques are precisely the SUs for C greater than some threshold value C t < 1. It is trivial to see that a maximal clique is a solution of the GLV equation (15), since for that subset of species the model is noninteracting, and the state with x i = 1 for all species i represented in the maximal clique is feasible and stable. The nontrivial task is to find under what conditions the solution is also uninvadable, rendering it an SU. The equation of invadability for an absent species α isẋ where the sum is over all present species i ∈ clique. In the binary model c i,j is either A or zero, so i∈S c α,i is simply A times the integer m S α , the number of species (in the clique S) with nonzero c α,i element with the spececies α. Thus, the solution is uninvadable (and so is an SU) as long as, the maximum is taken from the values for all the species α which are not in the clique. Since any species out of the clique has at least one enemy in the clique (otherwise, the clique is not maximal) for any C > 1 − p no clique is invadable.
For example, let us consider the following competition matrix: This is a 4-species realization of the binary model, and since there are 6 possible competition terms from which only one is active, it follows that p = 5/6 hence A = 6. Clearly, species {1, 2, 3} constitute a maximum clique: they all do not compete with each other, while species {4} suffers from competition with {1}. Eq. 15 for x 4 (when the island is occupied by species {1, 2, 3}, each with abundance one) now reads, so x 4 may invade (its linear growth term is positive) only if C < 1/6, but above this value the maximal clique is indeed an SU. This agrees with Eq. (18) as A = 6 and m S 4 = 1. However, the condition C > 1 − p turns out to be too restrictive for large systems. On average, not only one but a fraction 1 − p of the c α,i are nonzero, so a naive estimate of the threshold C, C t , is where S is the size of the maximal clique. Since the typical size of a clique grows (slowly) with N , we expect that C t → 0 as N → ∞, and so our clique picture should become exact in that limit -any maximal clique of the infinite-σ model will be an SU of the corresponding binary system. We have verified these calculations by comparing the set of maximal cliques and SUs for the binary model and seeing that indeed the two sets are identical down to some threshold C satisfying our bound. Within the binary model, the general features of the Gamma distribution model are preserved. For example, if C is smaller than some threshold value C < C 1 (using the notations defined in fig 1 of [2]) there is exactly one SU, namely the state with all species present. This is clearly not a maximal clique for general p. Then, up to a second critical point C 2 , there is still only a single SU, with however, some species are missing. The general trend is that below C t the number of SUs falls below the number of maximal cliques, and decreases monotonically as C decreases. This is demonstrated in Fig. 3, where we have plotted the number of SU's in the binary model (averaged over a number of realizations) as a function of C for N = 20 and two different values of σ.
Together with the results of the binary model we have plotted the average number of SUs in the Gamma distribution model for the same values of N , C and σ. One sees that the number of SUs in the Gamma distribution model is significantly smaller. In the limit C → ∞ every pair of species in the Gamma model is mutually exclusive so every species is an SU, meaning that the number of SUs in the Gamma model approaches N for large C. This numerical evidence, together with those presented in Fig. 4 of the main text and in Fig. 4 here, reinforce our claim that the number of SUs in the Gamma distribution case also grows slower than exponentially with N .