Many of you will be familiar with the Red Queen from Lewis Carroll’s “Through the Looking Glass” (not to be confused with the “Off with her head!” Queen of Hearts). In the early 1970’s, Leigh Van Valen of the University of Chicago developed a theory of species evolution within a community-context in which species are engaged in a zero-sum game for survival, and hence never really make any progress, in a relative sense. He named his idea the Red Queen’s Hypothesis, in analogy with what the Red Queen actually said, “All the running you can do to keep in the same place.” Less well known than either the Red Queen or Van Valen’s hypothesis is Stephen Hubbell’s Unified Neutral Theory of Biodiversity. This theory is a mathematical tour de force, wherein Hubbell argues that it is the individuals of multiple species that are engaged in zero-sum dynamics. In an upcoming paper entitled, “Red Queen for a day: Models of symmetry and selection in paleoecology” (to be published in Evolutionary Ecology), I argue that these two seemingly different hypotheses are actually different versions of a more general theory of historical biodiversity, which I view from the perspective of the CEG hypothesis (to which many of this blog’s postings are dedicated). Following is the abstract for the paper, along with a short excerpt from the appendix for the more mathematically-inclined and curious reader.
Abstract:The Unified Theory of Biodiversity (UNTB), the Red Queen’s Hypothesis (RQH), and the Cascading Extinctions on Graphs hypothesis (CEG) are explored as members of a spectrum describing the ecological partitioning of species richness. All are models of historical biodiversity, but fare differently in explaining observed features of Phanerozoic biodiversity. The models treat species as symmetric, asymmetric, or partially symmetric respectively. Symmetry in the UNTB is broken by the generation and selection of variation of ecological performance, while the robustness and hence longevity of RQ communities are subject to selection. The CEG model reconciles some of the differences, demonstrating the importance of functional partitioning to both species evolution and selection at the community level. It is concluded that the UNTB explains communities partially on the shortest of evolutionary time scales, while RQ communities would be, at best, geologically ephemeral yet conditionally important.
Let the number of species in the community/metacommunity be . How many ways can be partitioned ecologically? Reserving a minimum of one partition for primary producers (photosynthetic or otherwise), this leaves at most species. All species could be assigned to a single partition, implying that they are either neutral in the UNTB sense, or share basic characteristics of their ecological interactions, as in the CEG model. Alternatively, each species could be considered as truly individual, yielding a maximum of partitions. In reality, however, it is far likelier for the species to occupy a number of partitions between one and . The number of different ways in which these species can therefore be partitioned is calculated by considering the number of partitions possible at each integer ranging from one up to , and summing over the entire range,
The symbol M is used to represent the number of partition schemes because this formula in fact yields the Mersenne Number for when is an integer, and M may be calculated for any integer as
Given the reservation of at least one partition for producers, the maximum number of partition schemes possible given is therefore
The number of ways in which S can be partitioned thus increases as . Results for ranging from 1 to 10 are illustrated in Figure 1.