We develop a theory of competition based on two mechanisms that we call the cost of rarity (Mechanism R) and the cost of commonness (Mechanism C).
Evolutionary Ecology, 1993, 7, 142-154
Competition theory and the structure of ecological communities F. A . H O P F t Optical Sciences Center, University of Arizona, Tucson, AZ 85721, USA T H O M A S J. V A L O N E and J A M E S H . B R O W N * Department of Biology, University of New Mexico, Albuquerque, NM 87131, USA
Summary We develop a theory of competition based on two mechanisms that we call the cost of rarity (Mechanism R) and the cost of commonness (Mechanism C). These reduce the rate of population increase only at high densities (asexual organisms) or both at high and low densities (sexual species). The theory predicts that, in certain circumstances, the number of coexisting species in any assemblage will be finite and that these species will differ in their utilization of resources (and in associated morphological traits) more than expected by chance. Specifically, it predicts such nonrandom assortment in assemblages of three or more (i.e. multispecies) sexual species, especially in those communities where a few species are numerically dominant, but not in two-species associations and not in asexual forms. Unlike other theories, however, ours does not predict any specific value of morphological ratio or limiting similarity. We develop procedures to assess the degree of numerical dominance in an assemblage, and then test the predictions of the theory using data on morphological size ratios. The tests yield results that are consistent with the theory. Our analysis of multispecies assemblages of granivorous rodents, bird-eating hawks, Anolis and sexual Cnemidophoruslizards, show that in these assemblages very small ratios are observed less often than expected by chance. Coexisting asexual Cnemidophoruslizards tend to be extremely similar in size. Nesting sparrow and flycatcher assemblages exhibit low numerical dominance which we predict will inhibit detection of regular assortment, and we find no regular pattern in these assemblages. Finally, we fail to detect regular assortment in most two-species associations as expected. We examine several alternative mechanisms that might account for morphological segregation among coexisting species but find little evidence that they have been important.
Keywords: competition; body size ratio; statistical analysis; coexistence; community morphological assortment; limiting similarity; cost of rarity; cost of commonness; stochastic extinction; dominance; standardization Introduction Since Lack (1947) and Hutchinson (1959) called attention to the apparently n o n r a n d o m ratios of morphological traits of ecologically similar, sympatric species, ecologists have speculated about the effects of interspecific competition on community structure and coexistence. Since Gause's (1934) principle of competitive exclusion had claimed that two species with identical requirements could not persist indefinitely in the same environment, ecologists began to ask how similar t A rough draft of the manuscript was completed before F. A. Hopf's untimely death. * To whom correspondence should be addressed. 0269-7653
© 1993 Chapman & Hall
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two species could be and still coexist stably. MacArthur and Levins' (1967) theory of limiting similarity attempted to provide a mathematical solution to this problem. Like many of the early theories of pairwise species interaction, this one was soon shown to be too sensitive to its mathematical assumptions to provide a robust answer to the problem (e.g. May and MacArthur, 1972; May, 1973, 1978; Stewart and Levin, 1973; Abrams, 1975, 1984; Armstrong and McGehee, 1980; Chesson and Warner, 1981; Turelli, 1981; Chesson, 1986). While theoretical studies have explored the mathematics of competitive interactions without providing a general solution (Abrams, 1983), empirical studies have continued to supply evidence that assemblages of ecologically similar, coexisting species are often comprised of species that are more different from each other in their body sizes or dimensions of trophic appendages than would be expected by chance (e.g. Schoener, 1970, 1984; Brown, 1973; Grant and Abbott, 1980; Bowers and Brown, 1982; Case, 1983; Brown and Bowers, 1985; Losos et al., 1987; Taper and Case, 1993a,b; but see Simberloff and Boecklen, 1981; Simberloff, 1984). Thus, while the inductive proposition that morphological differences reduce competition and promote coexistence has been supported, we still lack a theoretical basis for predicting a priori the effects of competition on the morphology of coexisting species. In this paper, we develop a theory of competition that predicts when coexisting species should differ from each other more than is expected by chance. We build on ideas about the commonness and rarity of species introduced by MacArthur (1960), Preston (1962), and Williams (1964). Our treatment is an extension of a theory developed by Hopf and Hopf (1985; see also Hopf, 1990). These authors showed how a 'cost of rarity', that is a necessary consequence of sexual reproduction, results in the clustering of individual organisms into discrete species that are evenly dispersed along a resource axis, even when the resources themselves are distributed continuously along this axis. Here we show how an additional 'cost of commonness' may influence our ability to detect structure in ecological assemblages. We are concerned primarily with the role of interspecific competition in producing larger than expected differences among species in resource utilization or in related morphological traits, which we shall refer to as a regular assortment pattern. We develop the concept of numerical dominance to provide an operational measure of the magnitude of the cost of commonness within a community. This allows us to predict the circumstances in which regular assortment patterns are most likely to be observed. Unlike earlier theories (e.g. MacArthur and Levins, 1967; May, 1973), our theory does not predict any specific value of limiting similarity. Instead, we predict the situations in which the differences between species in resource utilization or in related morphological traits will be larger than expected by chance. We present a preliminary evaluation of the predictions by analysing the ratios of morphological traits in assemblages of (i) asexual organisms, (ii) sexual organisms in which a few species dominate the community numerically, and (iii) sexual organisms in which abundances are relatively evenly distributed among species.
The theory The theory developed here is an extension of the non-mathematical model of Bernstein et al. (1986) and the formalization of Hopf and Hopf (1985). The latter is based on a generalized form of MacArthur's (1972) model for multiple consumer species with overlapping requirements for heterogeneous resources. Population growth of species i is described by Equation 1
dNi
T
= biai (Ni)fi(Ri)Ni - diNi
(1)
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where Ni is adult density and Ri is the set of resources used by competitor i for reproduction and survival through the function 1], which is unity in the limit of abundant resources, ai(Ni) regulates reproduction and mortality in a manner independent of competitors and is unity when no such regulation occurs, bi is a coefficient such that the quantity bi ai(Ni) )~(Ri) is the rate of production of new breeding individuals. We loosely refer to this as reproduction, but the process includes the mortality of individuals that have not yet joined the adult breeding population, d i is the coefficient describing the mortality of breeding individuals. Equation 1 departs from most conventional models (e.g. Roughgarden, 1979), because of the inclusion of parameter ai. We allow a i to be a function of Ni, so that population growth can be limited at both high and low densities. Our model is a more general formulation of MacArthur (1972), which uses a special case of Equation 1 in which ai(Ni) = 1. When ai(Ni) = 1, and all species have the same b's, d's andf()'s, but different Ris, the equilibrium number of species in the community equals the number of resources (e.g. MacArthur, 1972; Levins, 1979). If we assume that the niche of each species represents requirements for a unique combination of biotic and abiotic variables, so that the niche can be defined in terms of location along axes of continuous variables, then the number of niches potentially available can approach infinity. In such a scenario, Hopf and Hopf (1985) show that as the total number of individuals of all species (Y~Ni) approaches infinity, the number of species (S) in the community will also approach infinity. Hopf (1990) has verified that the specific value of the limiting similarity derived by MacArthur and Levins (1967) is an artifact of the special assumptions of their model (see also Abrams, 1983). In the kind of Malthusian system represented in Equation 1, no specific values of niche separation can be derived for communities where S = 3 or for any other value of S. While such a result seems biologically unrealistic, Hopf and Hopf (1986) demonstrate it in a non-living competitive system.
The cost of rarity Hopf and Hopf's (1985) solution to the problem of limitless proliferation in biological systems is to assume that ai (Ni) is of the form sketched in Fig. 1. This function is roughly constant, but it decreases at both low and high densities by the action of Mechanisms that we denote as R and C, respectively. Mechanism R, the cost of rarity, and Mechanism C, the cost of commonness, have the consequence of imposing lower and upper bounds, respectively, on the density, Ni, at equilibrium. Hopf and Hopf (1985) show that Mechanism R robustly yields a finite number of species, even when the number of potential n~ches approaches infinity. What are costs of rarity? Sexual reproduction has many consequences that act, directly or
R
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Figure 1. The relationship between regulation of reproduction and mortality, parameterized as ai (Ni), and adult population density, Ni. The function decreases at low and high densities (shaded regions) due to Mechanisms R (costs of rarity) and C (costs of commonness), respectively.
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indirectly, as a cost of rarity (Bernstein et al., 1986). Indeed, all other Allee effects (Dennis, 1989), including population-specific mutualisms or obligate social interactions (e.g., flocking in birds), impose a cost of rarity. Thus, the theory predicts that communities of organisms subject to a cost of rarity (including all sexual organisms) should be comprised of distinct species with differences in resource use that are greater than expected by chance; i.e. these assemblages should exhibit an assortment pattern. Hopf (1990) has shown, however, that in communities composed of only two species (species pairs) the detectability of the assortment pattern depends strongly on species density. The assortment pattern will be strong only when population densities of both species are low. Detectability of an assortment pattern is not a problem when communities contain greater than two species (Hopf, 1990). Asexual organisms, on the other hand, have no cost of rarity. Thus assemblages of parthenogens should behave like the non-living system in Hopf and Hopf (1986) and show no robust pattern of assortment.
The cost of commonness Mechanism C, the cost of commonness, imposes an upper bound on the density of a particular species in a manner independent of interspecific competition. Examples of such a mechanism include many kinds of density-dependent predator-prey, plant-herbivore, and host-pathogen interactions (e.g. Janzen, 1970; Ricklefs and O'Rourke, 1975; Holt, 1984). If Mechanism C acts strongly on an association, it can prevent the detection of an assortment pattern. To see this, consider first an association at equilibrium in which a Mechanism R, but no Mechanism C, is operating. Next, turn on a Mechanism C, and make it act strongly enough that population density for each species is substantially reduced. As a result, fewer resources are consumed by the existing species. The increased resource level allows another set of species, previously excluded from the assemblage, to use the same range of resources and hence to invade the community. The species that will survive and coexist will depend largely on the history of the sets. Whilst the species within any one set may exhibit assortment rules relative to other members of the same set, their properties of resource utilization will tend to be distributed at random with respect to members of the other set. Hopf (1990) shows that if Mechanism C is strong enough to increase the number of species by a factor of 1.5 - 2, then the resulting asssortment pattern will be difficult to detect. The only mechanism left to generate patterns in associations where C is strong is local (i.e. within community) character adjustment. Our use of average morphologies to characterize entire species over large regions largely prohibits testing for character adjustments. Thus, patterns we can observe are restricted to assortment effects, and these are predicted to occur only if Mechanism C is weak or absent. Thus, while the theory does not require that we identify the specific factor C that may act on a particular community, it will be important to determine the strength of Mechanism C in the associations studied. All Cs should have the effect of reducing the density of the commoner species relative to the density of the rarer ones (e.g. Glasser, 1979). To assess the strength of C on a community, we define an association as having high numerical dominance if it contains a few species that are very abundant and consistently present (core species of Hanski, 1982) and many species that are rare and ephemeral (satellite species). Since C reduces numerical dominance, we can assess the strength of C by measuring the relative abundances of species in the community. Communities experiencing a strong C will exhibit low numerical dominance while those experiencing a weak C will exhibit high numerical dominance.
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Preliminary evaluation of the theory Measuring the cost of commonness We measure the strength of a Mechanism C on a community by examining the relative (fractional) abundances of each species. Assemblages that contain a few abundant and many rare species are said to exhibit high numerical dominance. Assemblages in which most species are relatively equally abundant are said to exhibit low dominance. Numerical measures of dominance depend on the total number of species in the assemblage (S). Most such measures can be said to be biased estimators, because the number of species counted empirically is often less than the number, S, actually present (Pielou, 1977; Lloyd and Ghelardi, 1965; Hopf and Brown, 1986). Sampling of few individuals at a site (small sample size) will tend to underestimate S (Preston, 1962), and the magnitude of this bias will increase with increased numerical dominance. Therefore, when a sample of an assemblage appears to exhibit low dominance, we need to be able to determine whether this is due to undersampling an assemblage with high dominance or to true reduced dominance. To do this we follow the method of Hopf and Brown (1986) for obtaining a quantitative measure of the relative abundances of species that is independent of the total number of species in an assemblage. Their method can be used to calculate an Sindependent measure of dominance from the value of Simpson's (1949) diversity index. Numerical dominance (D) is expressed as a score, that can range from 1 - 20, for each community. Communities of low dominance (even distribution of abundances) will yield low scores, communities of high dominance will give high scores, and communities with a random distribution of abundances among species will give a uniform random distribution of Ds. We illustrate the application of this method and the bias that can come from undersampling using extensive census data that are available for breeding and wintering assemblages of North American ducks (see Appendix 1). These are suitable for this purpose because there are censuses of numerous sites during both nesting and wintering seasons, and some of these include very large numbers of individuals so they should be relatively free of undersampling bias. In Fig. 2 we show that when the samples include large numbers of individuals, scores for the winter assemblages cluster tightly around 20, but those for the nesting communities are much more evenly distributed. Samples of both nesting and wintering assemblages that contain few individuals gave more evenly distributed scores. Note that the winter assemblages exhibited substantially higher dominance than the nesting ones, and that the smaller samples, especially of the winter assemblages, gave a more even distribution of D's than the larger samples. The winter assemblages show that bias from undersampling can cause communities with high dominance to appear to exhibit a much lower degree of dominance. Hopf and Brown (1986) demonstrated a similar effect of undersampling bias by computer simulation, drawing at random small subsamples from a community exhibiting high dominance. Tests Here we provide a brief, preliminary evaluation of the theory by comparing three different kinds of assemblages; (i) asexual species, (ii) sexual species exhibiting low numerical dominance, and (iii) sexual species exhibiting high numerical dominance. We chose these three categories because in the first there is no cost of rarity (no Mechanism R); in the second there is a Mechanism R, but a high cost of commonness (a strong Mechanism C);
Competition and community structure
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S Figure 2. Distribution of scores from the evenness test applied to censuses of nesting and wintering ducks containing different numbers of individuals per site: a) >300, b) >1000, c) < 100, d) 50200. The effect of undersampling is illustrated by the much wider distribution of scores (and correspondingly reduced ability to reject the null hypothesis) for the sites where few individuals were recorded, especially in winter. and in the last there is a Mechanism R, but a weak Mechanism C. Therefore the theory predicts that we should observe assortment patterns only in the last case. Before comparing the theoretical predictions with data, it is necessary to determine the strength of Mechanism C necessary to obscure the effects of competition on assortment patterns. Our method of measuring numerical dominance provides a quantitative assessment of the strength of Mechanism C, but the theory is not yet developed to the stage where we can predict the level of numerical dominance required to observe regular assortment. We can, however, make two qualitative predictions. First, regular assortment patterns should be observed only in assemblages of ecologically similar sexual species (i.e. in the same guild) exhibiting high values of D. Second, there should be a threshold relationship between the level of numerical dominance and assortment. That is, all assemblages above some level of D should show an assortment pattern while those below such level should not. The theory predicts larger than expected differences in resource utilization among coexisting, ecologically similar species. Resource utilization is rarely measured directly, but can be inferred from body size or dimensions of other morphological traits (e.g. Lack, 1947; Hutchinson, 1959). It is the ratios (or the differences in the logarithms) of morphological traits that most accurately reflect differences in resource utilization, because the requirements for resources and the morphological traits used in acquiring and processing resources are allometric functions of body size (Peters, 1983; Calder, 1984). The assessment of regular patterns of assortment is complex. There are many biological and statistical issues involved in the selection of assemblages that are appropriate for analysis and in the statistical methods that are appropriate for detecting nonrandom patterns of assortment. For example, regular assortment patterns can be expressed and tested either as ratios that are more even than expected by chance, or as the tendency of very small ratios to be observed less
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frequently than expected by chance. We address these issues elsewhere (Hopf et al., in preparation). Here, for simplicity, we use the most rigorous of the methods that we have developed in this preliminary assessment of the theory. We have compiled from the literature data sets on the species composition of local communities and on the relative abundances and morphologies of the species in seven guilds of terrestrial vertebrates (for references and additional details see Appendix 2). These assemblages include granivorous desert rodents, nesting sparrows, nesting flycatchers, bird-eating hawks, Anolis lizards, sexual Cnemidophorus lizards, and assemblages of exclusively asexual (i.e. parthenogenetic) Cnemidophorus lizards. Our theory makes three predictions that can be evaluated using these data: (i) the parthenogenetic assemblages should not show regular assortment patterns; (ii) the two-species assemblages will often not exhibit detectable assortment; (iii) when all sexual assemblages are ranked according to dominance, there should be some value of dominance above which all assemblages exhibit regular assortment and below which all assemblages do not exhibit detectable assortment. Table 1 presents data on the average degree of dominance in these assemblages and the results of the minimum ratio test, subdivided into two categories: two-species assemblages and communities containing three or more species. The data are consistent with all of these predictions. First, all assemblages of asexual Cnemidophorus tended to have significantly smaller ratios than expected by chance. Second, of the other guilds, we failed to detect regular assortment in all two species assemblages except in the hawks. And third, there was a significant correlation between numerical dominance and tendency to avoid small ratios (Spearman rank correlation, Rs = -0.77, p < 0.05). That is, significantly fewer than expected small ratios were observed in multi-species assemblages of all guilds of sexual forms that had as high or higher dominance than the rodents, while we failed to observe regular assortment patterns in the two guilds of sexual animals that exhibited lower dominance than the rodents (Fig. 3).
Table 1. Numerical dominance (D) and the results from the test for the tendency to avoid small ratios in the assemblages. Each assemblage is subdivided into two-species associations (S = 2) and multispecies associations (S > 2). g is a quantitative measure of assortment pattern (Hopf et al., in preparation). Negative g values indicate a tendency to avoid small ratios S=2 Assemblage
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