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Local and collective transitions in sparsely-interacting ecological communities [1]

['Stav Marcus', 'Department Of Physics', 'Technion-Israel Institute Of Technology', 'Haifa', 'Ari M. Turner', 'Guy Bunin']

Date: 2022-07

2.1 Constant sparse interactions

2.1.1 The model. We work within the framework of species assembly, where species migrate from a species pool, and interact inside a community. The abundances change in time according to the standard multi-species Lotka-Volterra equations. There are S species in the pool. The abundance of the i-th species, N i , follows the equation (1) where α ij are the interaction coefficients, r i the growth rates, and λ i the migration rates. In this paper the matrix α ij , called the interaction matrix [25, 26], is always assumed to be symmetric, α ij = α ji , with equal intraspecific competition for all species, α ii = 1. The symmetry ensures that the dynamics in Eq (1) always reaches an equilibrium [27]; There may be one or more such equilibria. Here we only consider competitive interactions, α ij ≥ 0, and assume that all growth rates are positive, r i > 0; other than that the values of the r i ’s have no effect on the set of stable equilibria. In simulations we take all r i = 1, and run Eq (1) until changes in the N i -values are small. The migration strengths λ i are taken to be small, λ i → 0+, ensuring that at an equilibrium (i.e. a stable fixed point), all species that could invade do so. We use a migration rate of λ i = 10−10, and species are considered extinct when N i < 10−5. To ensure a true equilibrium has been reached, we separate the species considered extinct from those considered present. We then verify that the present species satisfy 1 − ∑ j α ij N j = 0 among themselves for all i, and each extinct species i cannot invade, dN i /dt < 0. Further details of the simulation procedure appear in Section F in S1 Text. We are interested here in sparse interactions, where many of the pairs of species do not interact (α ij = 0). The network of interactions forms an undirected graph, with vertices representing species and edges representing pairs of interacting species, sometimes called the community graph [28]. It is common to use random interactions sampled from different distributions, which capture different interaction characteristics. In this section we will consider the following model: (1) Each species interacts with exactly C other species, with the interacting pairs chosen at random so that the community graph is a random C-regular undirected graph. (2) The interaction strength is equal for all interacting pairs. Therefore, the interaction matrix can be written as α ij = δ ij + αA ij , where α is the interaction strength, and A ij is the symmetric adjacency matrix of the community graph. We consider C ≪ S, and more precisely the limit of large S at constant C. We will see that this simplified model already yields dramatically different results as compared with a fully-connected system with all-equal interactions. Extensions to varying interaction strength and number of interaction per species are then discussed in Section 2.2. From a physics perspective, this model is related to models of antiferromagnetic interactions on a tree [29–32]. However, many of our results arise from the distinct features of interacting populations, with species abundances described by continuous, non-negative variables, with zero being special (extinct species). We limit the discussion to properties of the system’s equilibria, and not the dynamics towards the equilibria, or under additional noise, which are very interesting (some already discussed in [33]) but beyond the scope of this work.

2.1.2 Overview of different regimes. To get a bird’s eye view of the different behaviors, we follow the diversity at the equilibria as a function of the interaction strength α (recall that in this first model α is identical for all pairs). Denote by S the total number of species in the pool, S* the number of coexisting species at an equilibrium (species richness), and define the relative diversity ϕ = S*/S. Fig 1 shows simulation results for ϕ as a function of α. ϕ is estimated by running simulations of Eq (1) over many realizations of adjacency matrices A ij , starting from a few different initial conditions per realization, with each initial abundance N i (t = 0) sampled uniformly from [0, 1]. The variability in ϕ between simulations under the same conditions decreases with the diversity S, and for large S it is essentially set deterministically. In this limit ϕ also becomes independent of S, as illustrated in Fig A in S1 Text. For comparison, the case of a full interaction matrix, where all species interact with each other with strength α is also plotted. In this case the behavior is simple: For α < 1 there is a unique fixed point in which all species are persistent with equal abundances, so ϕ = 1, while for α > 1 there are S different fixed points, each with a single persistent species so that ϕ = 1/S, tending to zero at large S. For both the fully-interacting system and a sparse system on a random regular graph, the interactions are locally ordered (the neighborhood of any vertex is a tree with probability 1, see next section), and both admit a fixed point with all species present. The sparsely-interacting system, in contrast, is very rich and exhibits multiple different behaviors with sharp transitions between them. At values of α close to zero, the interactions are weak enough to allow all species to coexist with ϕ = 1, again with all equal abundances. This persists for larger α up to some critical value α UE where ϕ starts to decrease, above which this fixed point is no longer stable. At a higher value α perc there is a percolation transition, above which none of the components of persistent species scales with the system size S. The relative diversity ϕ keeps decreasing until it reaches another transition where there is a jump in ϕ, at a value we denote by . At , the relative diversity ϕ(α) consists of infinitely many plateaus punctuated by jumps, until the last jump at α = 1 and a single plateau above it. In the following sections, we discuss this behavior in detail, and explain the multiple changes in system behavior and the reasons behind them. We will show in the next sections that and , and provide analytical values for α of all jumps in ϕ(α) at α ≥ 1/2. In Subsection 2.1.4 we discuss the percolation transition, and in Subsection 2.1.5 the unique to multiple equilibria transition, and show that it coincides with α where ϕ first drops below 1.

2.1.3 Allowed subgraphs and their dependence on interaction strength. Here we begin to explain the different regimes described in Section 2.1.2, by analyzing properties of the equilibria of the model. In the limit of small migration (λ i → 0+) some of the species will persist (N i > 0) and others go locally extinct (N i = 0 as λ i → 0+). At an equilibrium, the extinct species must be unable to invade (dN i /dt < 0), and the abundances of the persistent species must return to the fixed point if perturbed away from it. These conditions will be referred to as uninvadability and stability, respectively. The persistent species can be grouped into connected subgraphs of the community graph, see Fig 1A. We begin in the limit of very large α, studied in [8, 22]. Under this very strong competition, the problem of finding equilibria reduces to choosing sets of coexisting species that satisfy two conditions. First, two interacting species cannot both persist (competitive exclusion). The connected subgraphs are thus individual species, see Fig 1A, rightmost illustration. Second, an extinct species cannot invade if and only if it interacts with one or more persistent species. Stability is automatically satisfied, as it involves isolated persistent species. Importantly, in this limit of strong interactions the exact values of α do not appear in these two conditions, and so finding an equilibrium point reduces to a discrete, combinatorial problem on the graph, of finding a maximally independent set [8]. In [22], the authors used this insight to calculate the diversity and number of equilibria on Erdős–Rényi graphs (where the pairs of interacting species are chosen independently with some probability). At lower values of α the connected subgraphs are no longer only isolated species, see Fig 1A. These subgraphs must satisfy certain “internal properties” in order for them to appear at a given α. As long as all of the neighboring species to the subgraph are extinct, the abundances at a fixed point of the subgraph are determined entirely by interactions within it. These abundances must be positive (a condition known as feasibility), and the fixed point must be stable. These conditions depend only on α. This allows us to understand much of the behavior by looking at individual subgraphs: each type of subgraph μ will have a critical value , above which it is either unstable or not feasible, and can therefore only appear at an equilibrium of a system in the “allowed” range . (This leaves out a possibility that a graph could switch back and forth between being allowed or not, see Section A in S1 Text). Thus the system is governed by discrete combinatorial conditions, which determine the entire set of possible equilibria of a given system. Here another important simplification enters. Sparse random graphs, including random-regular graphs and Erdős–Rényi graphs discussed in Sec 2.2, are locally tree-like, meaning that they have only a finite number of short cycles even when S is large. For example, in a large random regular graph with C = 3 the average number of triangles is 4/3 [34]. Thus, most connected subgraphs of finite size in the network will be trees, i.e., contain no cycles, and properties such as diversity and species abundance distribution that are averages over the entire community can be calculated by only considering trees, and specifically, the critical values need to be found only for trees. Examples of connected subgraphs within a local tree neighborhood are shown in Fig B in S1 Text. The trees can be divided into chains and other trees. We calculated for chains analytically, see Section A in S1 Text. For a chain with n species, (2) For chains of even length, is a decreasing series that converges from above to , which is also the critical value for all chains of odd length, . All other trees have , with the first ones appearing, coincidentally, exactly at 1/2, as we prove in Section A in S1 Text. Therefore, only for chains of even length, so only they can appear in communities at α > 1/2. In addition to these “intrinsic” considerations about the stability and feasibility of different connected components, uninvadability must also be considered. This is more complex since it depends on how the components fit together, and in principle this could lead to additional jumps in ϕ. However, in the α > 1/2 region, such jumps seem to be rare if they exist at all, and their size is so small that we have not detected them in simulations. See details in Section B in S1 Text. This means that for α > 1/2, in each range of α between the , the allowed types of trees will not change and so essentially the same set of equilibria will exist (since uninvadability does not seem to be important except at the transitions). As α is lowered below some , a new tree abruptly appears, leading to many new possible configurations and thus causing the diversity to jump. While the dynamical simulations used to obtain ϕ(α) do not necessarily reach all equilibria with the same probability, they clearly show jumps in ϕ at these values, with plateaus of approximately constant values of ϕ in between. Fig 1B shows the function ϕ(α), marking some of the critical from Eq (2) as dashed vertical lines, showing that the jumps in ϕ indeed happen exactly at . This also happens for trees that are not chains when α < 1/2, see an example in Section A in S1 Text. As α is lowered, infinitely many subgraphs of more complex structures become stable, so the values become more dense, and the jumps in ϕ(α) smaller (see Fig C in S1 Text). This makes it harder to observe them in numerics, but we expect that they exist in the entire range down to α UE , defined in the following. Once trees appear there are many interesting types of transitions that could happen. Just as at 1/2 arbitrarily long chains appear, there could be other points where there are qualitative changes in the properties of trees; see the Discussion section for further discussion. We note that properties of the system, such as the value of ϕ, cannot be determined without consideration of dynamics. Indeed, the dynamics of Eq (1) is more likely to reach some equilibria over others. For instance, for an Erdős-Rényi graph with mean degree C = 3 and interaction strength α > 1, the likeliest relative diversity calculated when assuming all equilibria are equally likely is ϕ ≃ 0.427 [22]; yet in dynamical simulations at α = 1.1 we find ϕ = 0.514±0.003. The transitions are also reflected in the possible abundances of species, as seen in rank-abundance curves, which show the abundances sorted in decreasing order, see Fig 2. At a given α, the abundance of a species depends only on the connected tree it belongs to, and its position within it; for example, species that belong to a chain of length two have . Therefore, as a tree μ becomes feasible and stable at , the abundances associated with it can appear at an equilibrium. As shown in Fig 2A, this causes the abundance graphs to smooth out as α is lowered, since the number of possible abundances increases. PPT PowerPoint slide

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TIFF original image Download: Fig 2. Changes in feasible and stable trees are reflected in species abundances. At each value of the interaction strength α, certain trees are allowed, and the abundance of a species depends only on α and the position within a tree. (A) The rank-abundance curves at equilibria reached dynamically for S = 400, C = 3, at several values of α. As α decreases, the increasing number of feasible and stable trees generates more possible species abundances. (B-E) Species abundances on both sides of two transitions at for n = 2, 4, where new trees appear. (C) and (E) show the behavior of ϕ around the transitions associated with pairs of species and chains of length 4 respectively becoming feasible and stable. (B) and (D) show the abundances at equilibria at values of α on two sides of the transitions. The expected abundances are marked by dashed black lines, with thicker lines for the abundances of the species in the tree associated with the transition. Next to each abundance appears the tree that contains it, with the species that have this abundance in dark red (or gray in the case of the abundance 0 of extinct species). https://doi.org/10.1371/journal.pcbi.1010274.g002 To summarize, in this section we described how the interaction network breaks up into connected subgraphs, with changes in allowed subgraphs driving jumps in diversity and species abundances. These subgraphs are trees that are feasible and stable at that interaction strength. Finding the equilibria of Eq (1) reduces into a discrete graph theoretical problem on the community graph. Broadly speaking, for stronger competition there are fewer and typically smaller allowed trees. As α is lowered, the size of the allowed subgraphs grows until they span a finite fraction of the species, as discussed in the next section. The number of different types of allowed graphs quickly grows with their size, and the problem of classifying them becomes more difficult, and less useful. These very large connected graphs can include the rare but still existing cycles in the graphs, and so they are no longer trees.

2.1.4 Percolation transition. Percolation transitions are one of the canonical phenomena studied in graph theory, and appear in various contexts in community ecology (e.g., [1, 35, 36]). In site percolation, some vertices of a graph are removed. As the probability of vertex removal varies, on one side of the transition the remaining graph breaks into small (sub-extensive) pieces; on the other side, a finite fraction of vertices belong to a single connected component. Natural communities belonging to both regimes are known to exist [1]. We find that at some interaction strength α perc there is a percolation transition, below which the largest connected subgraph formed by surviving species is extensive, that is, includes a finite fraction of all the species. Fig 3B shows the fraction of species belonging to the largest connected component as a function of α, for several values S with C = 3. Above a certain α, which for this connectivity is at α perc (C = 3) ≈ 0.41±0.01 (marked by a dashed line), this fraction drops as S increases, indicating a sub-extensive largest component. Below α perc this fraction converges to a constant value. As expected, this value is smaller than 1/2, since at the only possible components are finite-length chains, as shown in Sec 2.1.3 above. Also, α perc ≥ α UE where all species persist, see Sec 2.1.5 below. The fact that the transition becomes sharper with growing S is a hallmark of a collective transition. PPT PowerPoint slide

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TIFF original image Download: Fig 3. Collective transitions. (A) Unique to multiple equilibria transition: the probability for a unique equilibrium as a function of α, for connectivity C = 3 and several pool sizes S. The probability is obtained by generating many realizations of interaction matrices and determining whether there is a unique equilibrium by the stability of the fully-feasible fixed point, as described in the text body. The exact value for the transition, calculated using Eq (3), is shown as a dashed black line. Inset: the same graph over a larger range of α. The transition becomes sharper with S grows, as expected from a collective transition. (B) Percolation transition: the fraction of species in the largest connected component as a function of α, for several values of S. The location of the transition is α perc ≈ 0.41 ± 0.01 (dashed black line). At α < α perc a finite fraction of species belongs to the largest component even when S grows. At α > α perc , this fraction decreases with S. Inset: the same graph over a larger range of α. Here too, the transition becomes sharper at larger values of S. https://doi.org/10.1371/journal.pcbi.1010274.g003 Fig 3B is qualitatively similar to that of a standard site-percolation transition [37], where vertices are randomly and independently chosen to be “present”. This similarity is used in order to estimate the value of α perc , as in random regular graphs this is the value in which the relative size of the largest component is proportional to S−1/3 [38, 39], see further details in Section D in S1 Text. However, the fraction of persistent species at α perc is around ϕ perc ≈ 0.64 ± 0.02, which is larger than the ϕ perc = 1/2 of a standard site percolation transition at C = 3 [37]. This is because in our model, the species that persist are not sampled independently; the higher value in our model is expected given that persistent species are correlated, tending not to be adjacent to one another. The percolation transition marks an abrupt change in the connectivity of the network. Unlike the other transitions discussed here, we have not observed any other sharp changes occurring at this transition, in terms of diversity, stability or other measures beyond the graph connectivity.

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[1] Url: https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1010274

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