Alan Roberts

The checkerboard score and species distributions

SummaryThere has been an ongoing controversy over how to decide whether the distribution of species is “random” — i.e., whether it is not greatly different from what it would be if species did not interact. We recently showed (Roberts and Stone (1990)) that in the case of the Vanuatu (formerly New Hebrides) avifauna, the number of islands shared by species pairs was incompatible with a “random” null hypothesis. However, it was difficult to determine the causes or direction of the communitys exceptionality. In this paper, the latter problem is examined further. We use Diamonds (1975) notion of checkerboard distributions (originally developed as an indicator of competition) and construct a C-score statistic which quantifies “checkerboardedness”. This statistic is based on the way two species might colonise a pair of islands; whenever each species colonises a different island this adds 1 to the C-score. Following Connor and Simberloff (1979) we generate a “control group” of random colonisation patterns (matrices), and use the C-score to determine their checkerboard characteristics. As an alternative mode of enquiry, we make slight alterations to the observed data, repeating this process many times so as to obtain another “control group”. In both cases, when we compare the observed data for the Vanuatu avifauna and the Antillean bat communities with that given by their respective “control group”, we find that these communities have significantly large checkerboard distributions, making implausible the hypothesis that their species distributions are a product of random colonisation.

Competitive exclusion, or species aggregation?

SummaryThere is a long-standing dispute over whether the analysis of species co-occurrence data, typically on islands in an archipelago, can disclose the forces at work in structuring a community. Here we present and utilise three “scores” S, C and T. S gives the mean number of islands shared by a species pair in the presence/absence data under study. The scores C and T are based on the way that a pair of species occurs on a pair of islands. When each species occurs on a different island, this adds to the “checkerboard score” C; if they occupy the same island, this increases the “togetherness score” T.In judging whether observed values of S, C and T are compatible with a null hypothesis assuming no species interaction, we follow Connor and Simberloff (1979) in generating a “control group” of (constrained) simulated incidence patterns.Presence/absence matrices can have paradoxical features, in combining a high mutual exclusion by species (checkerboardedness) with a degree of species aggregation that is also high. We show that this is in fact inevitable — that, given the usual contraints, C and T can differ only by a constant. This means that extreme checkerboardedness can be produced by forces making for species aggregation, just as well as by those making for avoidance.If we restrict our attention to a subset of species, the constraints are less rigid and the S, C and T scores are somewhat freer to vary. We consider the confamilial subsets in the Vanuatu archipelago as likely candidates for revealing any competition forces at work. Calculating the actual S, C and T scores for these subsets, we compare them with the corresponding scores in a sample of simulated colonization patterns.The actual species-distributions differ significantly from what we would expect if the colonization choices of different species were uncorrelated (save for some biological constraints). The confamilial species of the real world share more islands, and occur in a pattern less checkerboarded, and more aggregated, than their simulation counterparts. This suggests that competition pressures, if they exist, are overcome by countervailing factors.The method used is applicable in other ways, and to a wider class of problems, in analysing the forces behind community structure.

Complex systems which evolve towards homeostasis

It is still not well understood how large complex systems (such as ecosystems) come into being or how they persist over long periods of time, although recent studies1–5 have clarified certain aspects of the associated stability problems. The occurrence of such systems presents no problem if a ideological assumption is valid—that is, if the variables may be presumed (or compelled) to adjust their mutual interactions in advance, to achieve the desired persistence. But when no such ‘foresight’ is credible, a real problem emerges; how a given variable can interact will generally be constrained by factors internal to it, independent of the ‘needs’ of other variables or of system stability. This consideration has prompted studies of ‘randomly-interacting’ system models, and interesting results have emerged1–4 concerning the stability of such models—that is, their capacity to return to equilibrium after a fluctuation. However, the question remains: how did such a collection of randomly-interacting entities arrive at an equilibrium in the first place? We present here results from a simple model using mutually interacting variables. We show how, even on assumptions which make large complex and persistent systems highly improbable initially, they can nevertheless be expected to arise in profusion in the normal course of time development.

Island-sharing by archipelago species

SummaryDiamond (1975) formulated “assembly rules” for avian species on islands in an archipelago, which made a successful colonisation depend essentially on which other species were present. Critically examining these rules, Connor and Simberloff (1979) maintained that, in the Vanuatu (New Hebrides) archipelago, the field data on species distribution was quite compatible with a null hypothesis, in which species colonise at random with no species interaction. Their work was in turn criticised (Diamond and Gilpin (1982), Gilpin and Diamond (1982)) and a vigorous controversy has ensued.Here we contribute a method in which a simple but hitherto neglected statistic is used as a probe: the number of islands shared by a pair of species, with its first and second moments. The matrix of these sharing values is given as a simple product of the incidence matrix, and its properties are examined — first, for the field data, and then in the random ensemble used by Connor and Simberloff (1979). It is shown that their constraints hold constant the mean number shared, so that any fall in the number that one pair of species share, due to their excluding each other, must imply a rise in the number shared by some other species pair-i.e., an aggregation.Turning to the second moment of the numbers shared, it is shown that its value in the Vanuatu field data exceeds the largest value to be found in a sample of 1000 matrices, these latter being constructed so that they obey the Connor and Simberloff constraints but are otherwise random. This indicates that exclusion and/or aggregation effects are present in the actual distribution of species, which are not catered for by the null hypothesis.The observed distribution thus emerges as much more exceptional than found by Connor and Simberloff (1979), and even by Diamond and Gilpin (1982), when examining the same ensemble. The reason for this disagreement are sought, and some cautions are offered, supported by numerical evidence, concerning the use of the chi-square test when the data points involved are mutually dependent.

Ecosystem-like behavior of a random-interaction model I

The behavior of large systems of randomly-interacting variables is examined using an intentionally simplified model. The stable positive solutions are found to exhibit to a significant degree some well-known properties of ecological systems. This resemblance (including for example the predominance of “predator-prey” interactions) is all the more striking in view of the lack of biological “data” at the input end. The findings suggest it advisable to distinguish two kinds of properties in ecosystems. One kind would depend on specifically biological mechanisms; the other would characterize a wide class of persistent systems, and arise from the need for a dynamic balance between positive and negative feedback.

Can there be a war of all against all

How important is competition, whether within or between species? This question has been and remains the subject of intense debate (Connell, 1983; Schoener, 1983; Strong et al., 1984; Diamond and Case, 1986). How often is the structure of an ecosystem or the fate of a species decided by competition? How does competition from other species compare in significance with, for instance, the availability of a suitable habitat? If we have the figures for the way that species are distributed-over the islands of an archipelago, for example-can we use them to prove that species are avoiding each other, and that competition is therefore a crucial factor in colonization (Strong et al., 1984; Diamond & Case, 1986; Harvey & May 1985; Roberts & Stone, 1990)? These questions are just a few of the aspects of competition that remain controversial. Here we pose a broader question by examining the concept itself of a competition system-that is, one in which the interactions between species are mutually antagonistic and harmful. We ask: can one model such a system without much difficulty, or do inconsistencies arise? If the latter, then how must the model be repaired so as to remove these inconsistencies, what real-world behaviors do these adjustments imply, and how plausible is the resulting model? We report here work on these questions which suggests answers that may be found surprising. The type of model we have examined (Stone & Roberts, 1991) is a very general one. It describes any assemblage of species in which the rate at which a species increases its population, at a given time, depends only on its own population, and those of other species, at that time. The results obtained did not depend on assuming any specific form for these interactions. There is a well-known and apparently simple way to make this type of model into one of competition (May, 1973) First, consider the rate at which a species-let us call it A-grows in population: the difference between the number of births and the number of deaths in some time period, a day, for example. For brevity, call this its birthrate. Assume that each birthrate has a maximum value typical of the species, which it will enjoy only when there is a plentiful supply of all the resources it needs (food, sun, space, etc.). Then reduce this “intrinsic” birthrate by a term that expresses how other members of A will cut down available resources, or otherwise interfere with each other’s livelihood. Next, describe a second species B similarly, and regard their joint behavior as a system of two “competing”