Large extinctions in an evolutionary model: The role of innovation and keystone species

Abstract

The causes of major and rapid transitions observed in biological macroevolution as well as in the evolution of social systems are a subject of much debate. Here we identify the proximate causes of crashes and recoveries that arise dynamically in a model system in which populations of (molecular) species coevolve with their network of chemical interactions. Crashes are events that involve the rapid extinction of many species, and recoveries the assimilation of new ones. These are analyzed and classified in terms of the structural properties of the network. We find that in the absence of large external perturbation, “innovation” is a major cause of large extinctions and the prime cause of recoveries. Another major cause of crashes is the extinction of a “keystone species.” Different classes of causes produce crashes of different characteristic sizes.

Major transitions in biological and social systems are attributed primarily to some combination of external perturbation, selection, novelty, and a complex internal dynamics and structure of the system (1–8). Although empirical studies attempt to identify proximate causes for individual events, most modeling efforts have tended to focus on the statistics of the events taken as a whole (see, e.g., refs. 9 and 10 for reviews of models in macroevolution and finance, respectively). Here we discuss an evolutionary model that permits a precise dissection of the (often multiple) causes of individual events. Although the model is abstract, highly simplified and motivated by chemical evolution, the structures and processes that arise in it seem to have the flavor of phenomena in biological and social evolution. These include the appearance of a “core” and “periphery” in the network structure of the system, a shifting balance between cooperative and competitive processes as structures evolve, “core-shifts,” history dependence, “keystone species,” “innovations” that are “core-transforming” or “dormant,” and others. Their precise mathematical formulation and analysis of their role in major transitions in this simplified context may help in constructing and analyzing more realistic models.

The Model

The system is an idealized prebiotic pond containing a set of s chemical species. A given species j can be a catalyst for the production of another species i with some small a priori probability p. Then the presence of j in the pond causes the population of i to increase according to the rate equation for catalyzed chemical reactions. The catalytic relationship is represented graphically by an arrow from node j to node i in a directed graph representing the chemical network. The graph is completely specified by its adjacency matrix C = (c_ij), i, j ∈ {1, … , s}. c_ij is unity if there is a link from node j to node i, i.e., if species j catalyzes the production of species i, and zero otherwise. Each species has a population y_i in the pond. x_i ≡ y_i/Σ_jy_j is its relative population; by definition 0 ≤ x_i ≤ 1, Σ x_i = 1. On a certain time scale T the relative populations of the species reach a steady state denoted X ≡ (X₁, … , X_s) that depends on the catalytic network. We imagine the pond to be subject to periodic external perturbations on a time scale greater than T in the form of tides, storms, or floods. Such a perturbation can flush out existing molecular species from the pond (the ones with the least X_i being the most likely to be eliminated) and bring in new chemical species from the environment whose catalytic relationships with existing species in the pond are quite different from the eliminated ones. Then, after the perturbation, the populations in the pond will evolve to a new steady state, be subject to another perturbation, and so on.

The precise rules are as follows: Initialization: Start with a random graph with s nodes and “catalytic probability” p. That is, for every ordered pair (i,j) with i ≠ j, c_ij is unity with probability p and zero with probability 1 − p. c_ii ≡ 0 for all i (to forbid self-replicating species). Fig. 1a is an example. Assign each x_i a random number between 0 and 1 and uniformly rescale all x_i so that Σ x_i = 1. Dynamics: First, keeping C fixed, evolve x from its initial condition according to

1

for a time T large enough to reach its attractor. Denote x_i(T) ≡ X_i. Eq. 1 follows from the rate equation ẏ_i = Σ_j c_ijy_j − φy_i for the populations, which in turn is an idealization of rate equations in a well-stirred chemical reactor.¶ Find the set of nodes with the least X_i. Second, pick a node (denoted k) from this set at random and remove this node from the graph along with all its links. Add a new node (also denoted k) to the graph, which is connected randomly to the existing nodes according to the same catalytic probability p. Mathematically, this means that for every i ≠ k, c_ki and c_ik are reassigned to unity with probability p and zero with probability 1 − p, irrespective of the value they had earlier, and c_kk = 0. Set x_k = x₀ (a small constant), perturb all other x_i about their existing value X_i by a small amount, and uniformly rescale all x_i to preserve the normalization Σ x_i = 1. This procedure provides a new graph and a new initial condition for x. Now return to the first step of the dynamics and iterate the procedure several times.‖

Figure 1.

The structure of the evolving graph at various time instants for a run with s = 100, p = 0.0025. Node numbers i from 1 to 100 are shown in the circles representing the nodes. Nodes with zero relative population in the attractor configuration for the graph (X_i = 0) are shown as white circles; the rest (s₁ in number) have nonzero X_i. In the graphs where an ACS exists, black circles correspond to nodes in the “core” of the ACS, and grey to the “periphery,” defined in the text. (a) n = 1, the initial random graph. (b) n = 2,854, where the first ACS, a 2-cycle between nodes 26 and 90, appeared. (c) n = 3,880, the beginning of the organized phase when the ACS first spanned the entire graph. (d) n = 4,448, when the core reached a peak in the number of loops it contained. (e) n = 5,041, just before a “core-shift.” (f) n = 5,042, just after the core-shift caused by a “keystone” extinction in the presence of a “dormant innovation.” (g) n = 6,061, just before another core-shift. (h) n = 6,062, just after the core-shift caused by a “core-transforming innovation.” (i) n = 6,070, when the old core stages a come-back as a disconnected component after node 32 becomes a singleton. (j) n = 6,212, when the new core strengthens itself and depopulates the recently resurgent old core. (k) n = 8,232, just before the first “complete crash.” (l) n = 10,000, between the first complete crash and the subsequent “recovery.”

This model, introduced in ref. 11, was inspired by the work in refs. 4, 5, and 12–15. The removal of the least populated species implements selection (5), and its replacement by another randomly connected species implements the introduction of novelty into the system.

Three Regimes of Behavior and Transitions Between Them.

The system exhibits three regimes or phases of behavior. This is illustrated in Fig. 2, which shows the number of populated species in the attractor, s₁, (those species for which X_i > 0) vs. time (n, the number of graph updates) for a run with s = 100 and p = 0.0025. In the “random phase,” s₁ stays low with small fluctuations. In the “growth phase,” s₁ typically rises exponentially with occasional drops. Finally, in the “organized phase,” s₁ stays close to s, the maximum value it can take. The random and growth phases were discussed in refs. 11, 16, and 17. As is evident from Fig. 2, the organized and growth phases exhibit occasional discontinuous transitions or “crashes” in which a number of species suddenly go extinct (their X_i become zero in a single time step). At the end of a crash the system is in the random or growth phase. This is followed by a recovery in which the system moves again toward the organized phase. In ref. 18 it is shown that crashes are primarily “core-shifts,” a specific kind of change in the structure of the graphs (discussed below) and recoveries are the result of the growth of “autocatalytic sets.” The main purpose of this paper is to elucidate the mechanisms which cause core-shifts.

Figure 2.

The number of populated species, s₁ (continuous line), and the largest eigenvalue of C (whose significance is discussed later in the text), λ₁ (dotted line), vs. time, n for a run with s = 100 and p = 0.0025. The λ₁ values shown are 100 times the actual λ₁ value. The first 10,000 time steps are enlarged (Inset). The run is the same for which the graph snapshots are shown in Fig. 1. The impact of the events described in Fig. 1 is clearly visible in this curve. At n = 2,854, λ₁ jumps from 0 to 1 and s₁ exhibits the first sustained upward movement. At n = 3,880, s₁ hits its maximum value, 100, and then fluctuates mainly between 99 and 100; and at n = 4,448, λ₁ reaches a local maximum. s₁ drops from 100 to 3 as a result of the “core shift” at n = 5,042, and from 89 to 32 in the core-shift at 6,062. At n = 6,070, a large recovery event is seen as the old core and the still intact part of its periphery get repopulated, only to be trounced again at n = 6,212 when the new core strengthens itself to a λ₁ value greater than 1. At n = 8,233, s₁ crashes from 100 to 2 when the ACS is completely destroyed and λ₁ drops from 1 to 0.

Definitions and Notation

Autocatalytic set (ACS). An ACS is a subgraph, each of whose nodes has at least one incoming link from a node of the same subgraph. (By a subgraph we mean a subset of nodes together with all their mutual links.) Thus, an ACS contains a catalyst for each of its members (19–21). In Fig. 1 b–k, the subgraphs formed by the set of all black nodes or all the black and grey nodes are ACSs. Fig. 1 a and l do not have an ACS. For any subgraph A, let λ₁(A) be the largest eigenvalue of the adjacency matrix of A. We denote λ₁(C) ≡ λ₁. It can be shown (16) that if the graph does not have an ACS, then λ₁ = 0, and if it does, then λ₁ ≥ 1. λ₁ therefore represents a topological property of the network.

Dominant ACS. It can be shown that if λ₁ ≥ 1 the subgraph comprising the populated species (X_i > 0) must be an ACS (11, 16), which will be referred to as the dominant ACS. The dominant ACS is uniquely determined by the graph and does not depend on the initial condition for x (except for special initial conditions forming a set of measure zero, which we ignore). The subgraph formed by the set of all black and grey nodes in Fig. 1 b–k is the dominant ACS for that graph. In addition to its topological significance, λ₁ also has a dynamical interpretation as being the “population growth rate” of the dominant ACS.**

Core and periphery of the dominant ACS. The core of the dominant ACS of a graph C (sometimes also referred to as the “core of C”) is the maximal subgraph, Q, from each of whose nodes all nodes of the dominant ACS can be reached along some directed path. The rest of the dominant ACS is its periphery. The subgraph of all black nodes in Fig. 1 b–k constitutes the core of the dominant ACS in the graph.‡‡ Every periphery node has an incoming path that originates from the core, but no outgoing path that leads to the core. The periphery can contain loops within itself (e.g., the 2-cycle between nodes 36 and 74 in Fig. 1e). One can prove that λ₁(Q) = λ₁(C) ≡ λ₁.

An irreducible subgraph is one that contains two or more nodes, each of which has a directed path to every other node in the subgraph (22). An irreducible subgraph is always an ACS but the converse is not true. However, the core (of each component of a dominant ACS) is an irreducible subgraph. Because λ₁(A) ≥ 1 for any irreducible subgraph A, it follows that the latter is a “self-sustaining” structure in the sense that if no other links were present in the graph, the nodes of A would still have nonzero X_i by virtue of their mutual links. λ₁ measures the “strength” of the core in two ways: one, its intrinsic population growth rate, and two, its multiplicity of internal pathways. To see the latter, compare the increasing and decreasing pattern of λ₁ between n = 2,854 and 5,042 in Fig. 2 Inset with the sequence of Fig. 1 b–f. When the core (of every disjoint component of the dominant ACS) has exactly one cycle (Fig. 1 b, f–i, k), then λ₁ = 1, and vice versa. Such a core is fragile because of the absence of any redundancy in its internal pathways; the removal of any link from such a core will cause the ACS property (of that component) to disappear.

Crash. A crash is a graph update event, n, in which a significant number (say > s/2) of the species go extinct, i.e., Δs₁(n) ≡ s₁(n) − s₁ (n − 1) < − s/2.

Core overlap. Given any two graphs C and C′ whose nodes are labeled, the core overlap between them, denoted Ov(C,C′), is the number of common links in the cores of C and C′. If any one of them does not have an ACS, Ov(C,C′) ≡ 0.

Core-shift. A core-shift is a graph update event in which Ov(C_n−1, C_n) = 0, i.e., there is no overlap between the cores of the dominant ACS before and after the event (C_n is the graph at time step n).

Keystone Species.

One can consider the impact of the hypothetical removal of any species i from the graph, irrespective of whether i is the least populated or not. For example one can ask for the core of the graph C − i that would result if species i (along with all its links) were removed from C. A species i will be referred to as a keystone species if Ov(C,C − i) = 0. Thus, a keystone species is one whose removal modifies the organizational structure of the graph (as represented by its core) drastically. In the ecological literature a keystone species is regarded as one whose elimination from the ecosystem would cause a significant fraction of species in the ecosystem to go extinct (23–26). We will see that likewise in the present model the removal of keystone species causes large crashes. [Indices of keystoneness based on Ov(C,C − i) or on the change in s₁ caused by the removal of a species can also be defined.] Note that if λ₁ = 1 and the ACS has a single connected component (e.g., Fig. 1 b, f–h, k), every core node is necessarily a keystone species because its removal would destroy the cycle that constitutes the core.

Innovations.

In the present model the new species k at each time step together with its set of new links may be regarded as a “novelty” introduced into the system. In the new attractor the new species may go extinct, i.e., X_k may be zero, or it may survive, i.e., X_k is nonzero. Let us define an innovation as a novelty in which the relative population of the new species in the new attractor just after the novelty occurs is nonzero. This definition has the feature that an innovation always involves new connections. It does not use any exogenously defined notion of fitness. The only performance criterion it requires is that the new links should enable the new node to survive until the next graph update. Even this minimal requirement has nontrivial consequences. For instance when the new species receives an incoming link from an existing dominant ACS, it typically has a nonzero population in the new attractor. Each recovery process, which occurs because of the expansion of the dominant ACS during the growth phase (18), is an accumulation of just such innovations.

We will be interested in a special class of innovations in which the novelty creates a new populated irreducible subgraph. Such innovations create or add to a “self-sustaining structure” in the graph, in the sense discussed earlier. For instance the appearance of the first ACS at n = 2,854 (see Fig. 1b) is such an innovation. There, species 90, which was a singleton before the event, went extinct and was replaced by a new species 90 that had an incoming and outgoing link to node 26. The two formed an irreducible subgraph whose λ₁ value was 1. This innovation in fact triggered the self organization of the network around this ACS. By n = 3,880 (see Fig. 1c), the core had grown to 18 nodes as a result of several events in which the new species was an addition to the core. Every event in which the core is strengthened by the addition of a new node is also an innovation in which a new populated irreducible subgraph is formed. When such an innovation arises, denote the new irreducible subgraph which includes the new species as N (or N_n at time step n.) N or N_n will stand for the maximal irreducible subgraph of which the new species k is a member. It follows from the Perron–Frobenius theorem (22) that such an innovation necessarily increases the λ₁ value of some substructure in the graph, i.e., λ₁(N) > λ₁(N − k). N_n becomes the new core of the graph if it is “stronger” than the old core. More precisely, one can show that N_n will become the new core of the graph, replacing the old core Q_n−1, whenever any one of the following two conditions hold:

(i)  λ₁(N_n) > λ₁(Q′_n) or

(ii)  λ₁(N_n) = λ₁(Q′_n) and N_n is “downstream” of Q′_n.††

Such an innovation may be referred to as a core-transforming innovation. If Q_n−1 ⊂ N_n, such an innovation enlarges the existing core. However, if Q_n−1 and N_n are disjoint, we get a core-shift.

In the present model, the generation of novelty does not depend on existing structure because the links of a new node are chosen randomly from a fixed probability distribution: novelty is “noise” in this model. (Variants of the model that depart from this are easily constructed.) However, whether the novelty constitutes an innovation is “context-dependent” (i.e., dependent on the structure of the existing network). Also, the short- and long-term impact of an innovation depends on the (historical evolution of the) “context,” as will be seen below.

Classification of Core-Shifts

In a set of runs with s = 100, p = 0.0025 totaling 1.55 million iterations we observed 701 crashes. Of these, 612 were core-shifts (18). Fig. 3 differentiates between the 612 core-shifts we observed. They fall into three categories: (i) complete crashes (136 events), (ii) takeovers by core-transforming innovations (241 events), and (iii) takeovers by dormant innovations (235 events).

Figure 3.

Frequency, f, of the 612 core-shifts observed in a set of runs with s = 100 and p = 0.0025 vs. the λ₁ values before, λ₁(C_n−1), and after, λ₁(C_n), the core-shift. Complete crashes [black; λ₁(C_n−1) = 1, λ₁(C_n) = 0], takeovers by core-transforming innovations [blue; λ₁(C_n) ≥ λ₁(C_n−1) ≥ 1], and takeovers by dormant innovations [red; λ₁(C_n−1) > λ₁(C_n) ≥ 1] are distinguished. Numbers alongside vertical lines represent the corresponding f value.

Complete Crashes.

A complete crash is an event in which an ACS exists before but not after the graph update. Therefore for a complete crash at time step n, λ₁(C_n−1) > 0 and λ₁(C_n) = 0. These events take the system back to the random phase. For example at n = 8,232, in Fig. 1k, node 54 is one of the species with the least X_i and is hit at n = 8,233. It is replaced by a new species that has a single outgoing link to node 50 and no incoming link, resulting in the complete disruption of the ACS. It is evident that complete crashes must always be caused by the elimination of a keystone species. Furthermore, Fig. 3 shows that λ₁(C_n−1) = 1 for every complete crash observed in the runs. Hence the core of the ACS is a single cycle when the event occurs and the species removed is a member of that cycle.

Takeovers by Core-Transforming Innovations.

An example of a takeover by a core-transforming innovation is given in Fig. 1 g and h. At n = 6,061 the core was a single loop comprising nodes 36 and 74. Node 60 was replaced by a new species at n = 6,062. The new node 60 created an innovation at n = 6,062, with N₆₀₆₂ being the cycle comprising nodes 60, 21, 41, 19, and 73, downstream from the old core. The graph at n = 6,062 has one cycle feeding into a second cycle that is downstream from it. [This is an example of condition (b) for a core-transforming innovation.] The population attractor of such a graph has the property that the X_i of the nodes in the upstream cycle are all zero and only the nodes of the second cycle (as well as nodes further downstream from it) have nonzero X_i. Thus, when the above innovation arises, the new cycle becomes the new core and all nodes that are not downstream from it get X_i = 0, resulting in a large drop in s₁ from 89 to 32. For all such events in Fig. 3, λ₁(Q′_n) = λ₁(C_n−1) because k happened not to be a core node of C_n−1. Thus, these core-shifts satisfy λ₁(C_n) = λ₁(N_n) ≥ λ₁(Q′_n) = λ₁(C_n−1) ≥ 1 in Fig. 3.

Takeovers by Dormant Innovations.

Fig. 1 e and f show an example of a takeover by a dormant innovation. At n = 5,041, the core has λ₁ = 1.24 and there is a cycle comprising nodes 36 and 74 in its periphery. Node 85 is hit, which results in a cycle (26 and 90) feeding into another cycle (36 and 74) at n = 5,042. Thus, at n = 5,042, for the same reason as in the previous paragraph, 36 and 74 form the new core with only one other downstream node, 11, being populated. All other nodes become depopulated, resulting in a drop in s₁ by 97. Such a core-shift is the result of an innovation that arose earlier (the cycle between 36 and 74 arose at n = 4,695) but lay dormant downstream of the existing core until one of the keystone species of the latter (node 85) was hit and made it weak (i.e., reduced its λ₁ to a value less than or equal to the λ₁ of the downstream innovation). In general, a takeover by a dormant innovation is a core-shift in which the new core existed as a subgraph before the present graph update. One can prove that a takeover by a dormant innovation can occur only after a keystone extinction, which weakens the old core. In such an event the new core necessarily has a lower (but nonzero) λ₁ than the old core, i.e., λ₁(C_n−1) > λ₁(C_n) ≥ 1. Note that at n = 5,041 if the downstream cycle between 36 and 74 were absent, 85 would not be a keystone species by our definition, because its removal would still leave part of the core intact (nodes 26 and 90). Node 85 becomes keystone, and the core of which it is a part becomes fragile and susceptible to a core-shift, because a self-sustaining innovation has occurred in the distant periphery. It is not difficult to see, given the definitions above, that this is an exhaustive classification of core-shifts.

If we consider the set of all drops in s₁, large or small, where an ACS exists before the event (there are 126,454 such events in the above-mentioned runs), we find that the number of complete crashes remains the same, 136 (the mean size of the drop in such events is = 98.2 with an SD σ = 1.2), whereas core-shifts caused by dormant innovations go up to 359 (with = 62.2, σ = 25.9) and those caused by core-transforming innovations to 524 ( = 48.2, σ = 25.6). The rest of the events consist of 9,851 “partial core-shifts” (in which the core changes, but Ov(C_n−1, C_n) ≠ 0; in this category = 2.18, σ = 7.42), and 115,584 events where there was no change in the core but the periphery is affected ( = 1.05, σ = 0.99). Thus, different classes of proximate causes of drops arise dynamically with different frequencies and typically produce events in different size ranges. The ranges, however, overlap and some distributions have fat tails (e.g., of the 701 crashes with |Δs₁| > 50, there are 79 and 10 events, respectively, in the last two categories). Detailed distributions and their dependence on s,p are open questions (but see ref. 18).

Variants of the Model

Relaxing various idealizations of the model does not change the qualitative behavior. In particular simulations show this is true for variants of the model where, respectively: (i) c_ij can take values in the interval [0, 1], (ii) negative links are allowed (17), (iii) Eq. 1 is numerically integrated for a finite time T, and (iv) all species with X_i below a fixed threshold are removed at each graph update (thus more than one species can get deleted and the number of species is not constant). The virtue of the present idealized model is its analytical tractability.

A variant where the qualitative behavior does change drastically is one where Eq. 1 is replaced by the replicator equation, making it an evolving version of the hypercycle model (27). In this case it is found (28) that small (hyper)cycles do form but quickly get destroyed when a graph update creates a parasite. Thus, large complex networks cannot form in the first place. This is consistent with existing work on well-stirred hypercycles and replicator networks (29, 30). Unlike the hypercycle model, the ACSs that arise in the present model do not get destroyed by parasites (nodes belonging to the periphery). This robustness allows them to evolve into complex networks that are then subject to the more interesting kind of instabilities discussed above.

Discussion

The present model exhibits mechanisms by which innovations can play a major role in crashes and recoveries in a complex system.§§ It provides a mathematical example of “creative destruction” (31) at work in causing large upheavals. It distinguishes two processes involving innovations, both having analogues in the real world. One is exemplified by the appearance of the automobile, which made the horse-drawn carriage and its ancillary industries obsolete. This is like the example of the core-transforming innovation shown in Fig. 1 g and h where the graph update produced a self-sustaining structure that was more vibrant than the existing core within the context of the present organization. This structure became the new core, rendering many nodes drawing sustenance from the old core dysfunctional. The subsequent development of other industries that depended on the automobile mirrors the growth of the ACS around the new core. The second process is exemplified by the emergence of the body plans of several phyla that are dominant today. It is believed that while these body plans originated in the Cambrian era more than 520 million years ago (33), the organisms with these body plans played no major role until about 250 million years ago. They started flourishing only when the Permian extinction depleted the other species that were dominant until that time (34). This is similar to the events shown in Fig. 1 e and f where an earlier innovation had lain dormant for a while without disturbing the existing core, but when the latter became sufficiently weak, took over as the new core and flourished.

Recently there has been substantial progress in graph theoretic analyses of complex systems and in particular, “small-world” (35) and “scale-free” (36) properties have been found for several real networks (for reviews and references see refs. 37 and 38). It may be interesting to study whether certain classes of real networks also have some kind of a “core-periphery” structure. The citric acid cycle in the metabolic network of organisms might be an example of a core, and one that may be central to the origin of life (39).

Abbreviation

ACS

autocatalytic set