Interspecific Competition in Plants: How Well Do Current Methods Answer Fundamental Questions?

Transcription

1 vol. 157, no. 2 the american naturalist february 2001 Interspecific Competition in Plants: How Well Do Current Methods Answer Fundamental Questions? John Connolly, * Peter Wayne, and Fakhri A. Bazzaz Department of Organismic and Evolutionary Biology, Harvard University, Biological Laboratories, Cambridge, Massachusetts Submitted May 25, 1999; Accepted September 14, 2000 abstract: Accurately quantifying and interpreting the processes and outcomes of competition among plants is essential for evaluating theories of plant community organization and evolution. We argue that many current experimental approaches to quantifying competitive interactions introduce size bias, which may significantly impact the quantitative and qualitative conclusions drawn from studies. Size bias generally arises when estimates of competitive ability are erroneously influenced by the initial size of competing individuals. We employ a series of quantitative thought experiments to demonstrate the potential for size bias in analysis of four traditional experimental designs (pairwise, replacement series, additive series, and response surfaces) either when only final measurements are available or when both initial and final measurements are collected. We distinguish three questions relevant to describing competitive interactions: Which species dominates? Which species gains? and How do species affect each other? The choice of experimental design and measurements greatly influences the scope of inference permitted. Conditions under which the latter two questions can give biased information are tabulated. We outline a new approach to characterizing competition that avoids size bias and that improves the concordance between research question and experimental design. The implications of the choice of size metrics used to quantify both the initial state and the responses of elements in interspecific mixtures are discussed. The relevance of size bias in competition studies with organisms other than plants is also discussed. Keywords: competition, thought experiments, design of competition experiments, size bias, replacement series, additive series. * Present address: Department of Statistics, National University of Ireland, Dublin, Belfield, Dublin 4, Ireland; john.connolly@ucd.ie. Present address: Department of Research, New England School of Acupuncture, 40 Belmont Street, Watertown, Massachusetts 02472; pwayne@nesa.edu. fbazzaz@oeb.harvard.edu. Am. Nat Vol. 157, pp by The University of Chicago /2001/ All rights reserved. Much research has been devoted toward understanding how individuals of co-occurring plant species both affect and respond to one another and how these interactions influence structure, dynamics, and evolution within plant communities (Harper 1977; Grime 1979; Schoener 1983; Keddy 1989; Grace and Tilman 1990; Bazzaz 1996). Yet, despite its importance, there is still confusion and considerable debate regarding how to assess interspecific competitive phenomena among plant species (e.g., Inouye and Schaffer 1981; Connolly 1986, 1987b, 1988, 1997; Law and Watkinson 1987; Keddy and Shipley 1989; Herben and Krahulec 1990; Silvertown and Dale 1991; Snaydon 1991, 1994; Grace et al. 1992; Cousens and O Neill 1993; Sackville-Hamilton 1994; Shipley and Keddy 1994). Lack of progress in our understanding of competition has been attributed to a number of factors. These include improper experimental designs and statistical analyses (Connolly 1986, 1987b; Cousens 1988; Goldberg and Scheiner 1993; Sackville-Hamilton 1994; Gibson et al. 1999), too much emphasis on controlled environment versus field studies (Goldberg and Barton 1992), the limited duration of many experiments (Keddy 1989), and the lack of a mechanistic understanding of plant competition (Tilman 1987; Schwinning and Weiner 1998; Berntson and Wayne 2000). While these issues certainly have impeded the development of a coherent theory of plant competition, we believe that progress has also been hindered by an even more fundamental problem: namely, a widespread lack of concordance between the intuitive questions ecologists ask about interspecific competition and the experimental procedures that have been employed to address these questions. We see at least two major causes for this lack of concordance. First, the questions that ecologists and agronomists have asked about interspecific plant competition have not always been well articulated. Questions about the eventuai outcome of competition have not been sufficiently differentiated from questions regarding how much neighboring species affect each other and the mechanisms through which this occurs. In extreme cases, the specific questions driving interspecific competition studies are not

2 108 The American Naturalist even apparent. Cousens (1996, p. 7) noted that it is not uncommon for researchers to employ popular experimental designs (e.g., replacement series) in their research whenever they wanted to look at competition, without stating clearly what it was they were trying to achieve. Moreover, superimposed upon the problem of framing clear questions are long-standing semantic debates. Controversy surrounding the meaning and usage of terms such as competition, interference, neighbor effects, and plant-plant interactions have only hindered the ability of ecologists to agree on the most logical questions to ask and the required methodological tools to address them (Birch 1957; Milne 1961; Milthorpe 1961; Trenbath and Harper 1973; Keddy 1989; Connell 1990). The second difficulty relates to the basis for inferences in competition experiments. The inferences that can be drawn from any experiment are limited by the combination of three elements: the experimental design used, the response and explanatory variables (biotic and abiotic) measured, and the statistical analyses employed. We subsequently refer to this triad of elements as the experimental structure. The relationships between biological questions and experimental structure in the study of interspecific plant competition have not been properly appreciated, with methods pushed far beyond what the experimental structure can validly sustain. It is the goal of this article to clarify these relationships. The core of this article is divided into three sections. In the first section, we briefly review some of the more common questions pursued by researchers studying plant interactions in mixed-species communities. Second, we review the most common experimental structures used to quantify interspecific interactions. In the third section, we employ a series of quantitative thought experiments to illustrate the potentials and limitations of these experimental structures for addressing particular questions. We emphasize the establishment of appropriate null hypotheses as assisting in clear thinking (Underwood 1991). The results of our analysis are summarized in the form of a table as a guide to experimenters. While our thought experiments emphasize two-species plant mixtures, the conclusions are relevant for multispecies experiments and to some studies with other organisms. Finally, we introduce the essential elements for an alternative approach to analyzing interspecific interactions (J. Connolly and P. M. Wayne, unpublished manuscript). Questions about Interspecific Competition Because the nature and consequences of interspecific plant competition are so broad and touch on so many subdisciplines in ecology, the number of questions that have been formulated, from community- and ecosystem-level consequences of competition to its physiological and genetic basis, is very large. It is not the intention of this article to review these in detail. For a more comprehensive analysis of the classes of questions asked in interspecific plant competition research, see other recent reviews by Connell (1983), Schoener (1983), Keddy (1989), Goldberg and Barton (1992), Goldberg and Scheiner (1993), and Cousens (1996). We limit our scope of inquiry to what we consider to be three broad classes of questions related to interspecific plant interactions: Which species dominates at a particular point in time? Which species gains (relative to others) over a period of time? and What is the effect of one species presence on a neighboring species performance? How these questions differ from one another and why we focus on them is developed below. Which Species Dominates a Mixture/Community at One Point in Time? By dominance, we simply mean that at any given time, in a defined mixture, one species (or any mixture component such as a genotype or functional group) is more abundant (e.g., biomass, seed number, leaf area, or population size) than another. As such, it is clear right from the start that this question has more to do with issues of community composition than with the process of competition, per se. Nevertheless, we include it for two reasons. First, in most interspecific plant competition studies, the primary response variable measured (and used to derive indices) is a single, end of experiment measure of species abundance that is, dominance. Second, the dominance question serves as an important contrast with question 2 (Which species gains over an interval of time?), with which it is commonly confounded. The dominance question frequently arises in studies that aim to assess the effects of environmental factors on the composition of mixed-species stands. Controlled environment studies of this nature are typically designed with mixtures of identical composition in terms of species densities repeated across a range of abiotic (e.g., soil fertility, light, CO 2 level) or biotic (e.g., herbivores, mycorrhizae, pathogens) factors. The response variable is some measure of species relative size in the stand at harvest time. While studies with an experimental structure designed to address the dominance question are useful in assessing how abiotic or biotic factors influence a given species relative abundance at one point in time, the answer to this question tells us very little about the demographic or physiological processes underlying compositional shifts, that is, how the present composition arose and in which direction it is likely to develop. Single, static measures of species abundance cannot be used to assess whether, in a given mixture or environmental regime, one species is gaining

3 Competition: Theory and Questions 109 (or performing better ) relative to another (i.e., question 2). For example, if two species ended up the same size in a mixture, our interpretation of their relative performances would be quite different if we were told that in one case both started at the same seed or seedling size or that, in another case, one species started 20 times smaller than the other. Moreover, single static measures of dominance offer very little insight into whether the presence of one species is suppressing, or possibly even enhancing, the performance of another (question 3). For example, biomass estimates of the components of forest communities will always reveal that canopy trees are much larger than understory herbs, but this does not necessarily mean that the trees suppress or outcompete the herbs. In many cases, understory herbs benefit from being subordinate to dominant canopy species. Yet despite the intuitive logic of these examples, many studies rely on one static, single harvest to infer how species are performing relative to one another and how one species impacts the performance of another. These misconceptions, and their implications, are articulated more fully in the thought experiments we present below. Which Species Gains ( Wins ) in a Mixture? Most interspecific competition studies aim to characterize more than a static snapshot of species relative dominance. One very fundamental question is, Over an interval of time, is one species gaining (e.g., in biomass, population size, seed number, or access to resources) over the other? In its simplest sense, the which species gains question can be assessed by comparing the proportionality between species sizes or abundances at both the beginning and the end of a defined period of time of a given stand s development. If a species contributes proportionally more at the end than at the beginning, it has gained more, and it is fair to say that over that time interval, it outperformed the other species in some sense. Performance defined in this way is very close to the traditional concept of growth efficiency, that is, output per unit input (e.g., gg 1 ), as defined by Blackman (1919), and is analogous to average relative growth rate (Evans 1972). Asking which species gains may provide more insight into the longer-term dynamics of a given mixture. For example, if, over an interval of time, the smaller of two species triples its size while the larger species only doubles its size, one might infer that, given enough time and assuming no changes in growth trajectories, the smaller species might catch up to or exceed the size of its neighbors. Examining the outcome of this question over a range of different conditions may help suggest the outcome of competition in the long term (Connell and Sousa 1983). All that is required to address the which species gains question for a given mixture during a particular stage of stand development is two sequential measures of abundance. Surprisingly, the importance of making measurements at several successive times for assessing interspecific competitive phenomena is commonly neglected (Milthorpe 1961; Connolly et al. 1990; e.g., Underwood 1992 for species other than plants). How Does One Species in a Mixture Affect the Performance of Another? While the answer to the which species gains question provides information about species relative performances in a given mixture, it does not necessarily provide information about the causes underlying species performances. More specifically, it tells us very little about the extent to which the relative success of one plant species was caused by any direct (e.g., abrasion) or indirect (e.g., resource competition, pathogen spread, facilitation) interactions with another species. While it seems logical that the which species gains question should be inextricably linked to the extent to which co-occurring species affect one another, this may not always be the case. Returning to the hypothetical forest example used above, it is quite conceivable that, even if understory herbs are outperforming or gaining on canopy trees by growing 10 times faster per unit size, they may not be influencing the growth of trees. Canopy trees may be getting their soil resources (e.g., water and nutrients) from different strata of the rhizosphere than herbs, and their shoots would be largely uninfluenced by the herbs. To assess how plant species affect one another, we generally need to measure more than relative performance in one mixture; we need to compare the performance of a target species across a range of mixtures. A very simple additive design, for example, in which a set number of individuals of a target species is grown with and without individuals of a second species, would tell us whether the presence of the second species influenced the performance of the target species. The analysis of this design might be as simple as a one-way ANOVA; however, its conclusion regarding the effects of neighbors on targets would also be simple and limited to a binary answer yes or no. More commonly, ecologists want answers to more complex questions, such as, How much of a target species performance is attributable to a neighboring species presence? What is the effect of a heterospecific neighbor relative to a conspecific neighbor? Is the effect of a heterospecific neighbor contingent on its relative abundance (density/frequency) and/or on other environmental factors? Can we (hierarchically) rank species within a community with respect to their effects on a target species performance? While fundamental and intuitive, these questions are far more com-

4 110 The American Naturalist plex than the simple yes/no question, Are species influencing one another? These questions have spurred the development of a number of experimental designs and associated indices, including substitutive, or replacement series, designs (e.g., dewit 1960; Harper 1977), various additive designs (e.g., Harper 1977; Miller and Werner 1987; Snaydon 1991), and response surface approaches (e.g., Connolly and Nolan 1976; Suehiro and Ogawa 1980; Firbank and Watkinson 1990). To date, each one of these approaches has received criticism, yet few attempts have been made to compare comprehensively their advantages and limitations with respect to their ability to answer fundamental questions. Below we attempt to carry out such an analysis through the use of thought experiments. Overview of Experimental Structures Employed in Plant Competition Experiments A key element of any experimental structure is the design. The simplest design is the pairwise (PW) experiment (fig. 1A), which consists of a single mixture repeated across a range of levels of a treatment factor (e.g., fertility). However, historically, many laboratory and field experiments were designed either as additive series (AS; e.g., Harper 1977; Miller and Werner 1987; Snaydon 1991), replacement series (RS; e.g., Harper 1977; Trenbath 1978; Keddy and Shipley 1989), or response surface (RE) designs (e.g., Suehiro and Ogawa 1980; Firbank and Watkinson 1985). A review of laboratory plant competition experiments (Gibson et al. 1999) showed that, of 107 competition experiments reported in 10 leading journals in ecology and weed science over the period , the frequencies of PW, AS, and RS experimental designs were 24, 28, and 42, respectively. There were only a few RE designs used. In the field, manipulation experiments, in which one or more species is added to, or eliminated from, a community, are also common (e.g., Silander and Antonovics 1982; Aarsen and Epp 1990; Goldberg and Barton 1992) but will not be discussed here. An AS (fig. 1B) design consists of a number of mixtures in which the density of one species, the target species, is the same in all mixtures and that of the other species, the associate species, varies. The AS design frequently includes the target species in monoculture, often at several densities. Apart from their use in agriculture for determining the reduction in crop yield (i.e., target species) by weed species (i.e., associate species), the AS method is frequently used to establish competitive hierarchies among plant species in natural communites (e.g., Harper 1977; Miller and Werner 1987; Goldberg and Landa 1991; Keddy et al. 1994). The RS design, first introduced by de Wit (1960), consists of a number of mixtures and monocultures in each of which the total density of individuals is constant, but species relative frequencies vary (fig. IC). A large number of indices have been developed around the RS method and employed in plant competition studies (e.g., de Wit and van den Bergh 1965; McGilchrist and Trenbath 1971; Mead 1979; Willey and Rao 1980; Snaydon and Satorre 1989; Sackville-Hamilton 1994), purporting to measure various facets of species interactions, such as species aggressivity, enhancement and suppression, competitiveness, and resource use. Replacement series experiments frequently seem to be set up to address the which species gains and the how does one species affect the other s performance questions, although the two questions are rarely distinguished. The use of RS designs became less popular following sustained criticism in the 1980s (Inouye and Schaffer 1981; Connolly 1986; Law and Watkinson 1987) but seem to be achieving a resurgence in recent years (Keddy and Shipley 1989; Cousens and O Neill 1993; Sackville-Hamilton 1994; Shipley and Keddy 1994; Gibson et al. 1999). Partly in response to dissatisfaction with RS and AS designs, RE designs (fig. 1D) have been advocated (e.g., dewit 1960; Connolly and Nolan 1976; Suehiro and Ogawa 1980; Wright 1981; Spitters 1983; Joliffe et al. 1984; Firbank and Watkinson 1985; Connolly 1987a; Hakansson 1988; Connolly et al. 1990; Menchaca and Connolly 1990; Connolly and Wayne 1996). In typical two-species plant mixture experiments, the relationship between a species response and the explanatory variables (usually the densities of the two species and sometimes an abiotic factor) is estimated for each species. A primary index derived from response surface approaches is the substitution rate (Wright 1981; Spitters 1983; Connolly 1987a; Connolly et al. 1990; Menchaca and Connolly 1990), also called competition coefficients (Firbank and Watkinson 1985), which measures the effect on a species of a unit change in the density of the associate species relative to a unit change in its own density. For all methods, responses and biotic explanatory variables (e.g., the yields of associate species), other than density, generally appear to have been measured solely at time of harvest. As will be seen below, this imposes major limitations on the inferences that can be drawn from any experimental structure. Thought Experiments Thought Experiments as a Tool The use of thought experiments has a rich history in many scientific traditions (Brown 1991) but has been surprisingly underutilized in ecology. Traditionally, they have been employed to clarify concepts and to test the claims of various proposals without the need to carry out experiments. At

5 Competition: Theory and Questions 111 Figure 1: Four experimental designs commonly used in competition experiments. The densities of the two species are denoted by d 1 and d 2. The designs are (A) pairwise (PW), (B) additive series (AS), (C) replacement series (RS), and (D) response model (RE). For the response model any selection of mixtures/monocultures that will allow the estimation of the response functions is a valid design. the heart of thought experiments are the creation of simple, unambiguous, compelling examples that serve as standards against which proposed theories must hold true. Thought experiments can be very useful in demonstrating that certain approaches will not work because of internal inconsistency or logical flaws, and they can also obviate the need for additional research. We believe that thought experiments can contribute significantly to resolving some of the confusion surrounding the area of interspecific plant competition. The thought experiments below have been designed to elucidate the inferential range and limitations of a number of currently used experimental structures in answering fundamental questions regarding interspecific competitive phenomena among plant species. The Scope and Limitations of Our Thought Experiments For practical purposes, we have limited the scope of our three thought experiments in a number of ways. First, we emphasize shorter-term studies on plants. With modification, however, many of the issues and ideas presented below are equally relevant to longer timescales, that is, competitive processes that span multiple seasons or generations. Second, our thought experiments address phenomena at the stand level (i.e., considering the average individual performance of a species). Finally, our thought experiments focus on the phenomenological level of competition. While mechanistic aspects of interspecific plant interactions, such as relative foraging efficiencies (Hutchings 1988; Bazzaz 1991), the physiology underlying resource uptake and usage (Caldwell et al. 1987; Schwinning 1996; Wayne and Bazzaz 1997), and nature of indirect plant-plant interaction mediated through microbes (Sanders et al. 1995), are receiving increasing attention, we feel that sorting out confusion on the phenomenological level first will create a clearer arena within which to explore how these and other mechanisms relate to net species interactions.

6 112 The American Naturalist Thought Experiment 1: Without Sequential Measures over Time, the Which Species Gains Question Cannot Be Answered In introducing the which species gains question above, we suggested that our interpretation of species relative success at any given point in time in a mixture should take into account information about the relative sizes of individuals of the two species at the beginning of their growth phase. In this thought experiment, we develop this intuitive, yet generally overlooked, idea more formally. Competitive outcomes for two hypothetical species, Sp1 and Sp2, are illustrated in figure 2 for three cases in which the density of each species is the same, but the initial biomass per individual of Sp2 is twice that of Sp1. In figure 2A, the final biomass ratio (biomass of a species relative to the other species) is the same (2 : 1) as the initial biomass ratio between the two species. This indicates that, over this particular growth interval, the species are competitively balanced in some sense; neither has gained or lost. In figure 2B, the final biomass ratio for Sp2 exceeds its input biomass ratio, and in figure 2C, the output ratio for Sp2 is less than its input biomass ratio. If final biomass data were the only data available, as is the case in the majority of competition experiments, it is likely that Sp2 would be judged as more competitive than Sp1 for all three cases it is the dominant component in all mixtures. However, when initial size data are included, it becomes clear that Sp2 outperforms Sp1 in only one case (fig. 2B); in both other cases its relative abundance in mixture either remains the same or declines (fig. 2A, 2C). From the perspective of understanding competitive interactions, ecologists should be interested in explaining changes in species proportions over time, instead of, or perhaps as well as, ratios at one point in time. Stated more formally, the null hypothesis for the which species gains question should be that the final biomass ratio for species in a given mixture equals the initial biomass ratio, and not simply that the final biomass ratio equals unity. Thus, Figure 2: We intuitively discount for initial size differences in answering the which species gains question. This figure compares the interpretation of three possible outcomes of a pairwise experiment when the different initial sizes of individuals of the two species are known with the interpretation when they are not known. The three outcomes are (A) harvest output size ratio equals initial input size ratio, (B) harvest output ratio shows that the species with initially larger individuals has gained proportionately more, and (C) harvest output ratio shows that the species with initially smaller individuals has gained proportionately more.

7 Competition: Theory and Questions 113 to evaluate competition in a pairwise experiment, it is important, at a minimum, to know the composition of the mixture at both the start and end of a given growth interval. Because the nature of species interactions can vary considerably throughout a given mixture s development, numerous sequential measurements of species relative proportion might be desirable. A feature of this example is that the naive interpretation of final yield leads to the conclusion that the initially larger species is more competitive, that is, a size bias in favor of the species with larger individuals. Thought Experiment 2: Estimates of Species Effects on One Another Will Exhibit Size Biases When Competition Indices Rely Solely on Plant Response at a Single Harvest In thought experiment 1, we emphasized the role of initial size and time in assessing the which species gains question. Below, we focus on how ignoring initial size and relying solely on static measures of final yield can lead to consistent size biases in assessing how one species affects the performance of another. We do this by analyzing the results of a simple thought experiment employing commonly used indices associated with three experimental designs: replacement series (RS), additive series (AS), and response surfaces (RE). This thought experiment begins by assuming that in a monoculture of equal-sized individuals of a hypothetical species (Sp1), we can arbitrarily merge or link two individuals to form a larger individual and label it as a distinct individual of a second hypothetical species (Sp2). Merged (Sp2) and unmerged (Sp1) individuals can then be imagined to grow together and can be used to simulate a mixture of two hypothetical species (fig. 3). With more formal notation, what was once considered a monoculture (at density d) could now be considered a two-species mixture of pseudospecies Sp1 and Sp2 (at densities d 1 and d 2,respectively, where d p d 1 2d 2). This partitioning of pop- ulations into pseudospecies of different sizes can be envisaged for a range of original monoculture densities, and different partitions can be selected to simulate different relative frequencies of pseudospecies across any range of densities. For the purpose of this thought experiment we note that the arbitrary merging and relabeling of individuals (into pseudospecies) within monocultures in no way changes the biology of the plants. Thus, the larger pseudospecies have identical growth, allocation, and physiological characteristics per unit size as do the original, smaller individuals from which they were aggregated. An appropriate analysis should lead to the conclusion that these pseudospecies only differ Figure 3: Illustration of idea behind thought experiments 2 and 3. Total number of individuals is d; d 2 pairs are arbitrarily linked to form individuals of pseudospecies Sp2, and the remaining d 1 individuals are nominated as being of pseudospecies Sp1. In this case d p d 2d. 1 2 in size, and that neither is competitively enhanced or suppressed by the other. To generate predicted final sizes for various mixtures of pseudospecies so that parameters associated with RS, AS, and RE designs can be assessed, we assume an inverse linear model for the relationship between size and density (e.g., Shinozaki and Kira 1956; Holliday 1960). The relationship between mean yield per individual at harvest (w) for the true species and its initial density (d) isdescribed by 1 w p, (1) a bd where the coefficient b (assumed positive) partly determines the rate at which yield per individual declines with increasing density (the rate also depends on a). Suppose, without affecting the generality of the argument, that all individuals at a particular density are the same size, defined by equation (1). Now suppose, as defined by the assumptions of our thought experiment, that in a particular plot at density d, d 2 pairs of individuals are selected and each such pair of individuals is considered as an individual of a pseudospecies called Sp2 (fig. 3). Let the remaining d 1 individuals be labeled as pseudospecies Sp1. It then follows that d p d 2d. (2) 1 2 While the density in terms of total numbers of individuals of the pseudospecies is now not d but d 1 d 2 (with d 1

8 114 The American Naturalist Table 1: Values of a range of indices for RS designs at a range of total densities Total density RYT components Relative crowding coefficient RY1 RY2 Sp1 Sp2 Coefficient of aggressivity Competitive ratio a p.1; b p.02: Index value indicating equal competitiveness a p 0; b p.02: Index value indicating equal competitiveness Note: Indices are calculated assuming, first, that a p 0.1 and b p 0.02 and, second, that a p 0 and b p 0.02 in equation (3). Included are relative yield components (RYT), the relative crowding coefficients, the coefficient of aggressivity, and the competitive ratio. The index values that indicate equal competitiveness are also shown. Index values are calculated from the performance of individuals of the two species in the 50 : 50 density mixtures (predicted using eq. [3]) compared with predicted performance at the appropriate monoculture densities. The indices are computed as follows. The overall density is d, and the binary mixtures are at densities d/2 for each species. The yields per individual in mixture are w 1 and w 2 for Sp1 and Sp2, respectively, and yields per individual in monoculture are w 10 and w 20 for Sp1 and Sp2, respectively. Then, formulae for index values are RYT p RY1 RY2, where RY1 p dw 1/2dw 10, RY2 p dw 2/2dw20. The RYT is 1 for all cases considered here, but this is not so in general. RCCs for Sp1 and Sp2, respectively, are 1/(2w 10 /w 1 1) and 1/(2w 20 /w 2 1). Coefficient of aggressivity is (w 1 /w 10 ) (w 2 /w 20 ) and competitive ratio is (w 1 /w 10 )/(w 2 /w 20 ). standard sized and d 2 double-sized individuals in the stand), the yields per individual of Sp1 and Sp2 (w 1 p w and w2 p 2w, respectively) can be written, by substi- tution of equation (2) in equation (1), as 1 1 w1 p w p p, a bd a bd 1 2bd w2 p 2w p p. (3) a bd a b d 1 bd These simulate response equations for a two-species mixture where individuals of the species differ only in size: individuals of Sp2 are twice the size of Sp1, capture twice as much resource as Sp1, and give twice the yield per individual, and take up twice the space but have the same asymptotic yield per unit area in monoculture (1/b). These equations, and in particular the characteristic that the coefficient of d2 is twice as large as that of d1, bring out the issue that the responses of both species, while still being density dependent, also reflect the size difference between individuals of the two species. This is what would naturally be expected given the construction of the two pseudospecies, since a unit increase in the density of Sp2 is equivalent to an increase of 2 in the density of Sp1 and so should have a greater effect. This suggests that, in real stands, if species differ in size at the start of an experiment, the assessment of a species impact on associates should allow for both the initial densities and sizes of species. What conclusions would be reached if an experiment was carried out on this pseudomixture system using three experimental designs, an RS, an AS, and a response surface design? A numerical example using the coefficients a p 0.1 and b p 0.02 in the inverse linear model (eq. [3]) is used to illustrate the argument. For RS a second case with a p 0 is used as this simulates constant yield per unit area for each species, where it has been claimed (Taylor and Aarssen 1989; Cousens and O Neill 1993) that RS works well. Replacement Series. Five RS increasing in total density from two to 100 are considered. Each RS comprises the two monocultures and the mixture consisting of half the monoculture density of each species. The values of a range of commonly used indices associated with RS designs (Mead 1979; Connolly 1986) are calculated (table 1). The relative yield components (RY1 and RY2; de Wit and Van den Bergh 1965) are, for a species, the ratio of its total

9 Competition: Theory and Questions 115 yield in mixture to its total yield in monoculture. The relative crowding coefficient for each species is the ratio of the yield per individual of a species in mixture to its yield per individual in monoculture (de Wit 1960). The coefficient of aggressivity (McGilchrist and Trenbath 1971) between two species is half the difference between these two ratios, and the competitive ratio (Willey and Rao 1980) is their ratio. In the RS methodology, when competition is equal between species, these indices take values of 0.5, 1, 0, and 1 for the four indices, respectively, and these values would naturally form the basis of a null hypothesis for whatever the index purports to measure. In the classical interpretation of the results in table 1, none of the indices returns a value indicating equal competitiveness. Rather, they all take values suggesting that Sp2 (i.e., merged individuals) outcompetes Sp1. Thus, all the indices are size biased, and so it is impossible to frame the appropriate null hypothesis unambiguously since it would confound bias with the feature of interspecific interaction being investigated. Size bias is even evident in case B where the RS results are independent of density and its magnitude varies with density when a ( 0. Additive Series. Two additive series are considered with a density of 5 of Sp1 (the target or crop pseudospecies) and increasing densities of associate (or weed) pseudospecies Sp2 or Sp3 (Sp3 being formed in the same way as Sp2 except merging three individuals to form an individual of Sp3). Equations (3), and their analog for mixtures of Sp1 with Sp3, are used to predict the crop yield per individual at a range of weed densities (fig. 4). Figure 4 shows that the yield of Sp1 is decreased more by an individual of Sp3 than of Sp2 and thus Sp3 would be considered as more competitive than Sp2 but the diagram shows no simple 3 : 2 ratio as might be imagined. The aggressiveness (sensu Harper 1977; the reduction of the target species by a given density of the associate) displayed by Sp3 is greater than that of Sp2, but as can be seen from the artificial nature of the construction of the example, this aggressiveness should not be overinterpreted; it simply reflects an initial size difference. In the AS, Sp3 would reduce the yield of Sp1 by more (at an equivalent density) than would Sp2 because its individuals were initially larger and grew at the same rate per unit initial biomass as individuals of Sp2 and hence captured more resources. However, to use this result as the basis of a competitive hierarchy in which Sp3 would be judged as more competitive than Sp2, with any connotation of Sp3 gaining in the mixture, would be quite spurious. This is clearly seen if one considers a mixture consisting of one individual of each of the three pseudospecies. In biomass terms, both the input and output ratios between species would be 1:2:3, and so no species has gained or lost Figure 4: Results from two AS showing the yield per individual of Sp1 at a density of 5 as affected by increasing density of Sp2 (continuous line) or Sp3 (hatched line). Sp2 and Sp3 are pseudospecies whose individuals are aggregates of two and three individuals of Sp1, respectively. Response models for species 1 are derived from equation (3) as w1 p 1/(a 2bd 2) and w1 p 1/(a 3bd 3) for Sp1 in mixture with Sp2 or Sp3, respectively. The response of Sp1 individuals in monoculture at density five is 1/a. Values are calculated assuming a p 0.1 and b p over the course of the experiment. One must be careful in concluding from an experiment that a species is competitively better than another if, perhaps, all that is being said is that its individuals were initially larger. Without information on initial sizes, it is impossible to test null hypotheses as to which species gains. Not knowing initial size differences is also likely to result in confounding estimates of suppressive/enhancing effects. Response Surface. Equation (3) gives the response functions for the relationship between final yield per individual and initial density for this mixture system. When the competition coefficients (sensu Firbank and Watkinson 1985) or substitution rates (Wright 1981; Spitters 1983; Connolly 1987a; being the ratio of inter- to intraspecific coefficients of density) are calculated for this example, they give values of 2 and 0.5 for Sp1 and Sp2, respectively. This would traditionally be interpreted as Sp2 being more competitive than species 1, but again, these values merely reflect initial size differences. Including information about species relative sizes in estimates of substitution rates is critical for testing hypotheses about species effects on one another without bias. In all three methods, there is a danger of size-biased interpretation when only final yield is measured, favoring

10 116 The American Naturalist species with initially larger individuals. While size and sizerelated traits have been shown to modify competitive ability (Thomas and Weiner 1989; Connolly and Wayne 1996), this example shows that the contribution of initial size differences to the indices and methods must be discounted in assessing competitive phenomena. How is this discounting to be done? The third thought experiment suggests avenues for resolving some of these difficulties. Thought Experiment 3: The Reliance on Species Densities as Explanatory Variables Is at the Heart of the Size-Biased Estimates of Interspecific Plant Competition Thought experiment 2 demonstrated that for two hypothetical pseudospecies in a mixture, identical in all aspects of biology (e.g., growth rate, physiology, shape) except for initial size, the initially larger species will be consistently misjudged as competitively superior by most competition analyses. This final thought experiment suggests that the traditional use of initial density in AS, RS, and RE, that is, the number of individuals of species and not their initial biomass or size, is at the heart of the difficulties. We begin this experiment by creating pseudospecies as we did in thought experiment 2, that is, arbitrarily selecting d 2 pairs of individuals in a monoculture at density d and calling them pseudospecies Sp2, leaving d 1 unpaired in- dividuals and labeling them as members of a pseudospecies Sp1, with d p d 1 2d 2. In this experiment, we convert the models in equation (3) to models that use initial stand biomass of each species, rather than initial density, as explanatory variables. To do so we let the initial biomass of individuals (unpaired) of Sp1 be w 0, making the initial size of pseudospecies Sp2 individuals 2w 0. A density of d 1 individuals of size w 0 leads to initial stand biomass for Sp1 of y1 p dw 1 0 (from which d1 p y 1/w0). Initial stand bio- mass for Sp2 is y2 p 2d2w0, which gives d2 p y 2/2w0. In a mixture of d 1 and d 2 plants of Sp1 and Sp2, respectively, substituting for densities in equation (3) gives the following equations relating yield at harvest to the initial stand biomass of the two species: 1 1 w1 p p, a b(y /w ) 2b(y /2w ) a fy fy w2 p (4) (a/2) (b/2)(y 1/w 0) b(y 2/2w 0) 1 p, (a/2) (f/2)y 1 (f/2)y 2 where we use the symbol f to represent b/w 0. Changing variables to the initial biomass scale thus gives response models in which the coefficient of initial biomass (y 1 or y 2 ) of either Sp1 or Sp2 is f for Sp1 responses and f/2 for Sp2 responses. As in thought experiment 2, an appropriate analysis should lead to the conclusion that these pseudospecies only differ in size and that neither is competitively enhanced or suppressed by the other. Appropriate null hypotheses are that neither species gains for any mixture (i.e., that the models of output per unit input, w i /w i0, readily derived from eq. [4] are identical for both species) and that neither species suppresses or enhances the other (i.e., the substitution rates are both 1, or the RS indices in table 1 take the values 0.5, 1, 0, and 1, respectively). As before, this is examined for RS, AS, and RE methods. Replacement Series. An initial biomass replacement series (IBRS) is a set of mixtures and monocultures all of which have the same initial total biomass (rather than the same initial total density as in the more traditional RS). Since the explanatory variables in model (4) are expressed in biomass terms, it is easy to use it to simulate the results from a replacement series design where replacement between the two pseudospecies is on an IBRS basis. Suppose total initial biomass of y p 0.8 for an IBRS consisting of three stands, a monoculture of initial stand biomass of 0.8 for each pseudospecies, and a mixture with initial stand biomass of 0.4 for each. Suppose values of 0.01, 0.1, and 0.02 for w 0, a, and b, respectively. This gives a value of 0.02/0.01 p 2 for f. For this (and for any such IBRS), the replacement series indices suggest (table 2) that neither pseudospecies is enhanced/suppressed and thus reflects what is defined to be the truth for this experiment. This result is very different from the size-biased results obtained when using a RS based on density (thought experiment 2). When the indices (tables 1, 2) are unbiased, the index values indicating equal competitiveness (table 1) form the basis of null hypotheses for the particular aspect of competitiveness captured by the respective index. The traditional RS indices do not address the which species gains question, but clearly in this example, for any mixture along any IBRS, neither species gains. Additive Series. An initial biomass additive series (IBAS) for two species consists of a monoculture of one species at a given level of initial stand biomass and a set of mixtures in which the initial stand biomass of that species is held constant at its monoculture level but the initial stand biomass of the other species changes over mixtures. The traditional AS can be readily converted into an IBAS if the initial sizes of individuals of the two species are known. Consider the three pseudospecies Sp1, Sp2, and Sp3, with individuals of Sp2 and Sp3 formed by arbitrarily grouping two or three individuals of Sp1, respectively, as described in thought experiment 2. Suppose an additive

11 Competition: Theory and Questions 117 Table 2: Results of simulated competition experiment based on equations (4) along an initial biomass replacement series (IBRS) Monoculture Sp1 Sp1 Mixture Sp2 Monoculture Sp2 Yield per individual Yield mixed/mono 1 1 Values of competition coefficients Relative yield components.5 Relative crowding coefficients 1 Coefficient of aggressivity 0 Competitive ratio 1 Relative yield total 1 Note: An initial stand biomass of 0.8 for each species in monoculture and 0.4 for each in mixture was used. Parameter values of 0.01, 0.1, and 2 for w 0, a, and f were used for calculations. series, with Sp1 always at a density of 5 and Sp2 at densities 0, 2, 4, 6, and 8 and a second additive series for Sp1 and Sp3 with the same design. We have seen that traditional AS analyses of these examples gives size-biased results. By using equation (4), we can simulate the results of such an experiment conducted on an initial biomass basis. We use the same parameter values (w 0, a, and b equal to 0.01, 0.1, and 0.02, respectively, and hence f p 2) as for the IBRS example above. This leads to response models with parameter values a p 0.1 and f p 2 for Sp1 and parameters values for Sp2 and Sp3, which are one-half and one-third the size of these, respectively. Harvest biomass per individual is used as the response. Response of Sp1 per individual is plotted against the initial biomass of the associate pseudospecies (fig. 5A) and against the density of the associate pseudospecies (fig. 5B). On a density basis (fig. 5B), Sp3 appears more aggressive than Sp2, but on a per unit initial biomass basis (fig. 5A), the effect on Sp1 of either pseudospecies is the same (as would be the effect of Sp1 on itself). This identical effect of both species per unit initial biomass is what the thought experiment indicates should occur. This example suggests that the appropriate null hypothesis to be tested is that the per unit initial biomass effect on the target species is the same for both associate species. Response Surface. Since pseudospecies 1 and 2 are formed by arbitrarily labeling some pairs of identical individuals, it follows that the response of a pseudospecies must be identical for a unit change in initial stand biomass of itself or its associate, which is just a unit change in total initial stand biomass. This is what is indicated in model (4), where the coefficients of initial biomass are the same within either equation. In any mixture, a unit change in the initial biomass of either pseudospecies has an identical effect on its target. In model (3), based on density, the substitution rates are size biased, but in model (4) they are unity ( f/f and [1/2]f/[1/2]f ), indicating that any size bias has been eliminated. The appropriate null hypothesis in this model is that the substitution rates equal 1. The results here have been derived for output per individual as the response. They also apply where output per unit input or RGR is used; correcting for differences in initial biomass would remove size bias. This thought experiment can be simply extended to show that, even where output per unit input is identical for both species in each mixture (as it would be in this thought experiment), estimates of competitive performance are size biased when produced by RS, AS, and RE for equations based on density but not when based on initial biomass of species. A variant of this thought experiment in which initial (N i0 ) and final (N i1 ) densities are the measured variables for the ith species, with the same arbitrary grouping of individuals in pairs, will lead to a model of the form N11 N21 1 p p. (5) N10 N20 g hn10 2hN20 Thus, with respect to which species gains, neither does. However, with respect to the effects of species on each other, the relative sizes of individuals appears in the substitution rate between coefficients. This creates difficulties in establishing a null hypothesis in a real experiment for the substitution rates or for enhancing/suppressive effects, unless initial size differences are discounted in some way, since the observed substitution rates will confound an effect as a result of different initial sizes with suppressive or enhancing effects. If initial sizes differ, it is also clear that RS and AS will have similar interpretative difficulties to those outlined in thought experiment 2 above, even for the simple model in equation (5). Equation (5) is a simple

12 118 The American Naturalist Figure 5: Results of simulated AS competition experiment based on equations (4). Three pseudospecies Sp1, Sp2, and Sp3 are compared where individuals of Sp2 and Sp3 are composed of two and three individuals of Sp1, respectively. Two additive series are constructed, Sp1 as target with Sp2 and Sp1 as target with Sp3. For parameter values for equation (4) given in the text, the diagrams show (A) yield per individual of Sp1 versus initial stand biomass of Sp2 or Sp3 and (B) yield per individual of Sp1 versus density of Sp2 or Sp3. case of the discrete analog to the Lotka-Volterra equations (Leslie 1958), and the substitution rates are the competition coefficients for that model. Thus, these difficulties also attend the interpretation of the competition coefficients of the Lotka-Volterra equations. In summary, the major difficulties we have identified relate to two sources of size bias. The first arises in addressing the which species gains question on the basis of final harvest yield and not adjusting for initial size differences between species (thought experiment 1). The second arises in answering the effects of neighbors question, using approaches based on initial density rather than some measure of initial species contribution to the stand such as initial biomass (thought experiments 2 and 3). Insofar as any experimental structure fails to address these two issues, the estimates of competitive ability and hence tests of null hypotheses, whether for the which species gains question or the effects of neighbors question, will be potentially size biased in favor of the initially larger species. Discussion To synthesize the findings of our analyses and our thought experiments, we summarize (table 3) the utility of various experimental structures with respect to their ability to address specific questions. We then discuss a few key issues