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

doi:10.1073/pnas.032618499

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

Sanjay Jain*,†,‡,§ and
Sandeep Krishna*

^*Centre for Theoretical Studies, Indian Institute of Science, Bangalore 560 012, India; ^†Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501; and ^‡Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560 064, India

Communicated by Robert Axelrod, University of Michigan, Ann Arbor, MI (received for review July 17, 2001)

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.

Footnotes

↵ § To whom reprint requests should be addressed. E-mail: jain{at}cts.iisc.ernet.in.
↵ ¶ The rate equation ẏ_i = k(1 + vy _j)n _A n _B − φy _i follows from the reaction scheme A + B $\text{[math]}$ i, where A and B are reactants with populations n _A and n _B, j and i are catalyst and product with populations y _j and y _i respectively, and φ is a death rate or dilution flux in the reactor (k is the rate constant for the spontaneous reaction and v is the catalytic efficiency). We assume the reactants are buffered (n _A, n _B are large and fixed), and the spontaneous reaction is much slower then the catalyzed reaction. Then the growth rate depends only on the catalyst population: ẏ_i = cy _j − φy _i, where c is a constant. A generalization of the latter equation is ẏ_i = Σ $\text{[math]}$ c _ij y _j − φy _i for the case where species i has multiple catalysts. Eq. 1 follows from this by taking the time derivative of x _i = y _i/Σ $\text{[math]}$ y _j. In the present model, we make the idealization that all catalytic strengths are equal. The second (quadratic) term in Eq. 1 is needed to preserve the normalization of the x _i under time evolution. Note that it follows automatically from the nonlinear relationship between x _i and y _i when the time derivative of x _i is taken.
↵ ‖ The attractor configuration X is determined in this article by its algebraic properties discussed later, not by numerically integrating Eq. 1. Hence we are effectively taking T = ∞.
↵ ** This follows from the fact that the attractor configuration X is always an eigenvector of C with eigenvalue λ₁, i.e., Σ_j c _ij X _j = λ₁ X _i [11]. Thus, when φ = 0, substituting y _i ∝ X _i in the population dynamics equation ẏ = C y, one gets ẏ = λ₁ y.
↵ †† We use the notation C′_n ≡ C _n−1 − k for the graph of s − 1 nodes just before the novelty at time step n is brought in (and just after the least populated species k is removed from C _n−1). Q′_n stands for the core of C′_n. A subgraph A is “downstream” of another subgraph B if there exists a directed path from some node of B to a node of A but none from any node of A to a node of B.
↵ §§ Analogues of innovations and core-shifts seem to be playing an important role in another related but quite different model (32) where rapid transitions are observed.
↵ ‡‡ Sometimes the dominant ACS consists of two or more disjoint subgraphs, as in Fig. 1 i. Then the definition applies to each component separately. There exist other ACS structures for which this definition is not adequate, e.g., two disjoint 2-cycles pointing to a single downstream node. Such structures arise rarely and can be treated by a more general definition of core and periphery without altering the main conclusions presented here.
Abbreviation:

ACS,

autocatalytic set