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Keywords:

  • additive series;
  • competition;
  • experimental design;
  • interference;
  • niche differentiation;
  • response surface

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. The replacement series structure
  5. Indices of mixture performance
  6. Interpreting trends on replacement diagrams
  7. Experimental biases
  8. Outcome of competition
  9. Yield comparisons
  10. Conclusion
  11. References
  12. Appendix

1 The replacement series has been used widely to assess interference, niche differentiation, resource utilization, and productivity in simple mixtures of species. Correctly used, the approach can lead to some valid interpretations. In order to avoid criticisms, however, researchers should appreciate the assumptions and limitations of this methodology.

2 A replacement series contains confounded species density treatments. Replacement series experiments therefore provide collective results and cannot distinguish separate contributions to interference by the constituents of a mixture.

3 In a replacement diagram, trends in observed yields per species can be due to a multitude of possible levels of intra- and interspecific interference. Similarly, trends in expected yields per species do not represent specific levels of intra- and interspecific interference. A replacement series is therefore unsuitable for the quantitative evaluation of interference or niche differentiation.

4 A standard replacement series, but not a proportional replacement series, may be used to detect an imbalance between intra- and interspecific interference for a component of a mixture. If competition for resources is assumed or known to be the sole cause of interference, then such experiments yield a qualitative evaluation of complementary resource utilization by the mixture components.

5 The possibility of total density dependence impedes generalization from a replacement series result. Other biases, due to initial size and time of observation, may occur with replacement series and related experimental structures.

6 Results from a replacement series are of questionable value in predicting the long-term outcome of an association between species.

7 Replacement series are a valid setting, but not the only possible setting, for some kinds of yield comparisons. These include comparing productivity of monocultures with that of simple species mixtures.

8 In some cases, interpretations obtained from replacement series have not been confirmed using other methods but, on the whole, conclusions from replacement series do not seem to be characteristically different from those obtained using other approaches.


Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. The replacement series structure
  5. Indices of mixture performance
  6. Interpreting trends on replacement diagrams
  7. Experimental biases
  8. Outcome of competition
  9. Yield comparisons
  10. Conclusion
  11. References
  12. Appendix

Many methods are available for the study of interference and its consequences, but replacement series may have been used more widely than any other experimental structure (Trenbath 1974; Cousens 1991; Gibson et al. 1999). A standard replacement series is comprised of a set of pure and mixed populations in which combined density of the components is held constant. De Wit (1960) formalized the theory of replacement series, but although he has sometimes been given credit for the design he was not its originator. Van Dobben, for instance, carried out replacement series experiments at Wageningen in the early 1950s (de Wit 1960; Cousens 1996), and replacement series and related diallel experiments were also performed elsewhere before de Wit’s (1960) monograph (e.g. Aberg et al. 1943; Black 1958). Since de Wit (1960), replacement series have been used to study interference in plants, bacteria, fungi and animals. In ecology, replacement series have been used to explore many issues, including species coexistence, exclusion, coadaptation, niche differentiation, abundance, distribution, productivity and diversity. Replacement series have often been used to test how interference is modified by various factors such as disease, soil qualities and herbivory. In agriculture and forestry, replacement series have regularly been used in studies of weed–crop associations, and they are the common setting for evaluating yield advantages in intercrops. If judged simply on the basis of longevity of use, adoption by many researchers and range of application, the replacement series is a successful and important scientific methodology.

Opinions concerning the utility and suitability of replacement series experiments, however, vary widely. Replacement series have been favourably assessed as being ‘particularly elegant for studying the interactions involving two species’ (Harper 1977), ‘extremely valuable for comparing the outcome of competition between two plant species under different conditions. Its use has led to important insights into the nature of niche differentiation’ (Firbank & Watkinson 1990), an ‘extremely informative, efficient design for generating hypotheses’ (Cousens & O’Neill 1993), and ‘useful to detect the existence and measure the magnitude of competition, as well as to find the combination of two species which maximizes the total yield of a mixture’ (Rodriguez 1997). On the other hand, it has also been suggested that replacement series lead to ‘biased conclusions’ (Connolly 1987), can be ‘misleading’ (Connolly 1988), are ‘inadequate to assess competitive interactions’ (Connolly 1988), ‘often fail to produce general results’ (Silvertown 1989), are ‘difficult to interpret biologically’ (Snaydon 1991, 1994) and are ‘statistically invalid’ (Snaydon 1991, 1994). Recommendations have recently been made that their ‘continued use should be discouraged’ (Gibson et al. 1999), and that ‘the use of a single-density replacement design should be avoided in future competition experiments’ (Li & Hara 1999).

This controversy is more than an incidental or academic difference of opinions. Dogmatic objections may be suppressing the publication of replacement series results (Cousens 1996) and, if replacement series are valid, this would infringe intellectual freedom. On the other hand, if the technique is not sound, a significant portion of our information base on interference is open to question. The debate concerning replacement series has become emotionally charged. This may be distracting us from the business of studying competition and it may be deterring ecologists from following some lines of research (Gibson et al. 1999).

Publications dealing with this controversy are numerous, dispersed in the literature, and are sometimes partisan and narrowly focused. This presentation is not intended to promote or to discredit the replacement series. It is meant to be a broad and critical evaluation of replacement series methodology as it represents and quantifies interference, and the consequences of interference, in mixtures. It is written mainly for those who are considering the use of replacement series in their research, and for those who are evaluating the extant literature. Those researchers should be aware of the capabilities, limitations and underlying assumptions of replacement series. Given the controversy in this subject area, I make liberal use of quotations in order to give the literature its voice. One theme that will not be considered here is the statistical analysis of replacement series data (see Thomas 1970; Machin & Sanderson 1977; Yates & Dutton 1988; Finney 1990; Federer 1993; Cousens 1996).

The replacement series structure

  1. Top of page
  2. Summary
  3. Introduction
  4. The replacement series structure
  5. Indices of mixture performance
  6. Interpreting trends on replacement diagrams
  7. Experimental biases
  8. Outcome of competition
  9. Yield comparisons
  10. Conclusion
  11. References
  12. Appendix

A standard replacement series involves at least two different components, usually different species. The components are mixed and some measure of yield of each component per unit area, such as dry biomass or seed number, is assessed at constant total density while the proportions of individual components in the mixture vary from 0 to 100%. Results from replacement series are often presented as replacement diagrams (Fig. 1). The components can also be different genotypes or phenotypes within species. Only two-component systems will be discussed here; but multicomponent arrangements are possible. Another variation is the proportional replacement series (Connolly 1986; Snaydon 1991), in which the proportions of species are varied but the monoculture densities differ between the two species, i.e. in a proportional replacement series the combined density of the mixture changes as species proportions change. It is the adequacy of single density (standard), or single level (proportional), replacement series that is the focus of the controversy.

image

Figure 1. Replacement diagram showing trends in total and component species dry mass (solid lines) of Lolium perenne (YL) and Trifolium repens (YT) grown in 13-cm diameter pots. Trends were generated using yield–density response models for 16-year-old pastures (Turkington & Jolliffe 1996). Models were YL = XL/(− 0.0478 + 0.210XL + 0.0847XT), R2 = 0.88, and YT = XT/(0.668 + 0.119XT + 0.0626XL), R2 = 0.99, where XL and XT are the number of Lolium and Trifolium plants per pot (XL + XT = 24), respectively. Expected yield trends (broken lines) are obtained when intra- and interspecific interference are equal, e.g. YTexp=XT/[0.668 + 0.119(XT + XL)]. When XL + XT = 24, identical expected yield trends result from the alternative model described in Fig. 3, e.g. YTexp=XT/[0.0968 + 0.143(XT + XL)].

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Replacement series are often contrasted with several related experimental structures (Fig. 2). Terminology in this subject area has not been consistent (Austin et al. 1988; Rejmanek et al. 1989; Cousens 1991, 1996; Snaydon 1991; Gibson et al. 1999) and I will try to use terms that have chronological priority, largely matching the usage of Cousens (1991, 1996). The replacement series is a substitutive design because, along a series, the presence of one species is regularly substituted for that of another (Fig. 2a). Simple additive series (Fig. 2b), which have been used since the 1920s (Harper 1977), involve a set of treatments in which a target species is held at constant density while a companion species is varied in density from zero to some higher value. Simple additive series have also been called partial additive series by other researchers (Rejmanek et al. 1989; Gibson et al. 1999). Another form of additive series, involving a set of balanced mixtures, was described by Austin et al. (1988) and Snaydon (1991), but will not be considered here. A group of replacement series at different total densities (Fig. 2c) is an addition series (Spitters 1983). Addition series have also been called additive series (Rejmanek et al. 1989; Gibson et al. 1999), although like Snaydon (1991) I disagree with that nomenclature on the basis of chronological priority and because it leads to confusion with simple additive series. Mixture diallels are a set of all binary combinations (Gibson et al. 1999), and diallel experiments have often included species monocultures. This corresponds to a binary factorial structure (Fig. 2d) in which the densities of the constituents are varied independently of one another (Mead 1979; Snaydon 1991). Pair-wise comparisons can be made between any two cells in the factorial matrix, often with the species being mixed in a fixed ratio at different total densities. Hence, replacement series and additive series are different one-dimensional trajectories on the larger yield–density response surfaces that can be constructed from data obtained using factorial or addition series approaches (Fig. 3). Still other experimental structures, such as spatially explicit designs, are available. The focus of this presentation, however, is the replacement series. Alternative experimental structures will only be referred to in order to illustrate certain points. Broader comparisons of the different structures have been made elsewhere (Harper 1977; Mead 1979; Mead & Riley 1981; Radosevich 1987; Rejmanek et al. 1989; Roush et al. 1989; Cousens 1991; Snaydon 1991; Gibson et al. 1999).

image

Figure 2. Several commonly used experimental structures combine species at different levels of population density. The number of density levels used per species could be fewer or greater than shown. (a) Single-density replacement series. Other single density replacement series could also be formed along the SW–NE diagonals. If the corresponding monoculture density levels for the two species are unequal, a proportional replacement series is obtained. (b) Simple additive series of varying densities of species j added to fixed density of species i (▪), and varying densities of species i added to fixed density of species j (□). Other simple additive series could be formed along the vertical or horizontal transects, as long as both species are always present. (c) Addition series, consisting of a set of three replacement series (SW–NE diagonals). (d) Complete binary factorial structure. Monoculture yield–density series are found in the top row and left hand column of the factorial structure.

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image

Figure 3. Inverse yield–density response surface (solid lines) for Trifolium formed from the yield–density response model used in Fig. 1. The inverse of yield per plant equals XT/YT. The response surface is displayed at densities down to zero plants per pot, but data were not obtained where XT < 1 or XL < 3. Transect AB across the response surface, at a constant total density of 24 plants per pot (bold line), equals the replacement diagram line for Trifolium in Fig. 1. Slopes of the response surface in the XT and XL dimensions measure effects of intra- and interspecific interference, respectively. Broken lines illustrate an alternative model that shares the same replacement line AB: XT/YT = XT/(0.0968 + 0.143XT + 0.0864XL). For an additive series, for example BC, the shape and position of the larger response surface are also not fixed – it is an assumption of the model that trend BC is the result of variation in Lolium density rather than species proportions. A constant proportion transect of the response surface, such as a line joining the origin to C, connects a set of simple pair-wise combinations.

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Due to their formal organization, the simplicity of the associations they evaluate and, often, the short duration of study, there is some artificiality to the use of all of these experimental structures. They do, however, lend themselves to much research that is directly relevant to agricultural cropping systems. They are appropriate settings for attempts to explore the basic rules, events and consequences of interference. Presumably, some findings obtained from these simple experimental structures will be applicable to complex associations that are more variable in space and time. This expectation, however, has seldom been formally tested. Indeed, attempts to make such tests should be carried out with caution. As Gibson et al. (1999) state: ‘a clearer realization of the limitations of short-term experiments in providing anything but simple indicators in respect to the outcome of long-term competition is desirable’.

An attraction of replacement series, simple additive series and simple pair-wise experiments is that they require fewer sampling units than addition series or factorial arrangements. Hence, provided they can satisfy the aims of an experiment, replacement and additive series are more efficient than the larger structures. Replacement series can be inefficient, however, in their allocation of sampling units to mixtures. For example, a replacement series involving one mixture and two monocultures diverts two-thirds of the effort to non-mixture units. There are other practical difficulties, especially in field circumstances. Non-uniform establishment and mortality can move component and total densities away from their intended levels, particularly in long–lived associations.

The structure of a replacement series requires that the densities of component species be confounded with their proportions (DeBenedictis 1977; Jolliffe et al. 1984). As a result, one cannot immediately determine whether the changes in yield that occur along a replacement series are due to the decline in density of one species or to the increase in density of the other species. It does not help to re-express this matter in terms of species proportions because, as the proportion of one species declines, that of the other increases. A somewhat comparable situation applies to simple additive series. Here, there are exact correlations between the experimental factors, and one cannot immediately determine whether changes in yield are due to changes in companion species density, companion species proportion or total density.

If no other information or assumptions can be brought to bear, then neither replacement nor simple additive series can serve as a basis for establishing how the component species contribute to interference. Some studies have simply accepted this limitation and have not attempted to attribute species’ behaviour as being due to any one of the interrelated experimental factors. It is possible to analyse replacement series results for the effects of species proportions (Cousens 1996), but another tactic has been to assert that species proportions and total density are not relevant biological factors but rather that it is component-species densities that count. This is not an unorthodox view: similar assumptions are implicit in some neighbourhood models, as well as in the structure of some yield–density response functions. Such response functions have only occasionally required density-interaction terms, implying that species proportions are usually not consequential. If we use these arguments to remove total density and species proportions as contributing factors, a proposition which is not accepted by all (e.g. Sackville Hamilton 1994), then yield responses in simple additive series can be interpreted as being due to changing density of the companion species (Snaydon 1991). For the replacement series, however, further interpretation remains blocked because component species’ densities remain confounded. In this sense, a replacement series experiment produces a collective result; it does not allow us to disentangle the contributions to interference due to each of the constituents of the mixture (Jolliffe et al. 1984; Snaydon 1991; Gibson et al. 1999).

Indices of mixture performance

  1. Top of page
  2. Summary
  3. Introduction
  4. The replacement series structure
  5. Indices of mixture performance
  6. Interpreting trends on replacement diagrams
  7. Experimental biases
  8. Outcome of competition
  9. Yield comparisons
  10. Conclusion
  11. References
  12. Appendix

‘The interpretation of the outcome of competition can depend critically on the way competition is measured’ (Freckleton & Watkinson 1999). Quantitative analysis of the results from replacement series experiments has regularly employed indices derived from observations of yield, although this is not essential. Indices can be helpful for a variety of reasons. They can both appropriately express a relevant characteristic and facilitate the condensation and presentation of experimental results. Indices may help us to compare results from different studies, particularly when they are some relative or standardized measure. For several reasons, however, indices must be used with some caution. The melding of several variables into an index inevitably results in some loss of detail, compared to the original data (Gilliver & Pearce 1983; Mead 1986). The melding obscures knowledge of relationships that may exist between the variables, and can result in large confidence limits (Jasienski & Bazzaz 1999). The loss of information, the obscuring of relationships and the decrease in statistical precision all tend to worsen with increasing numbers of variables (and additional mathematical operations) being involved in an index. Hence, ‘the statistical behaviour of the indices is difficult to comprehend’ (Firbank & Watkinson 1990) and interpreting their biological meaning can be difficult (Snaydon 1991). Some researchers have deliberately chosen to use simple indices (e.g. Austin et al. 1988; Keddy & Shipley 1989), such as ratio of species yield in mixture to its yield in monoculture (i.e. relative yield per species). Simplicity is helpful, but it is not the only important property. Size bias (Grace et al. 1992) and the form of the species-density response curve (Li & Hara 1999) may still prevent the unambiguous interpretation of a replacement series result. Finally, indices of mixture performance usually involve arbitrarily chosen monoculture points (Connolly 1988, 1997).

More than 25 indices (see Appendix) have been used in conjunction with replacement series, although some of these are also applicable to other experimental structures. New indices have periodically been introduced as researchers have attempted to refine the measurement and expression of particular issues or as they have attempted to address new issues. Some formal comparisons of different indices have been made (e.g. Mead & Riley 1981; Connolly 1986; Wilson 1988a; Snaydon 1991; Cousens & O’Neill 1993; Loreau 1998; Freckleton & Watkinson 1999; Jolliffe & Wanjau 1999). Different indices can be correlated, often because they share the same input variables. In some cases, an index has had more than one type of use. For example, relative yield total was introduced in the context of the variation in population structure over a series of generations (de Wit & van den Bergh 1965). It has also been used to express the relative productive capabilities of mixtures and monocultures, and to indicate species complementarity and niche differentiation. In this context, it is important to note that in replacement series experiments it is interference (sensuHarper 1961) that is expressed in the observed yield responses. Interference can occur by various processes, including competition for resources, allelopathy and facilitation. Separating the contributions of these processes is notoriously difficult (Harper 1977; Weidenhamer 1996), and impossible in many experiments. Despite this, the literature contains many examples of replacement series experiments that have been interpreted narrowly, generally attributing the effects simply to competition for resources. Where evidence concerning possible mechanisms of interference is absent, the use of replacement series to assess complementary use of resources, for example, requires an assumption that non-competitive processes, such as allelopathy and facilitation, are inconsequential.

Estimator bias (Connolly 1986; Grace et al. 1992), i.e. effects of scale on indices of interference, is a further complication. Where species differ in their monoculture yields, and when species proportions in mixture are very unequal, bias can be introduced via the mathematical structure of an index, in concert with the statistical properties of the data. For example, relative yield total is the sum of two ratios. If it is derived for a mixture of 99% species A and 1% species B, then the errors of estimation of the two ratios may be very different, leading to estimator bias compared to a 50 : 50 mixture. Other commonly used indices subject to the same bias are relative crowding coefficient and aggressivity.

When using indices, it is still necessary to appreciate the properties of the replacement series structure. Following de Wit (1960), it is common to see statements of the type: ‘the relative crowding coefficient of species A against species B’. If component species influences are inseparable in replacement series, then such a statement is unjustified. Similarly, relative yield per species (the ratio of species yields in mixture and monoculture) should not be interpreted simply as an indicator of the effect of the companion species, as it reflects not only possible contributions from increased density of a companion species but also lower density of the target species.

Interpreting trends on replacement diagrams

  1. Top of page
  2. Summary
  3. Introduction
  4. The replacement series structure
  5. Indices of mixture performance
  6. Interpreting trends on replacement diagrams
  7. Experimental biases
  8. Outcome of competition
  9. Yield comparisons
  10. Conclusion
  11. References
  12. Appendix

Visual inspection of replacement diagrams has often been the method used to interpret replacement series results, and the convenience of this has been highlighted as one reason for the popularity of replacement series (Firbank & Watkinson 1985). If usage is an indication, many researchers have agreed that by inspecting replacement diagrams ‘it is possible to determine whether competition is occurring, and if so which is the more successful component in the mixture’ (Hill & Shiamoto 1973). A characteristic terminology has arisen, describing the position and patterns of component and total yield trends on replacement diagrams (e.g. Hill & Shiamoto 1973; Trenbath 1976; Harper 1977; Mead & Riley 1981). Our current understanding of the origins of replacement lines, however, suggests that such terminology (e.g. the ‘mutual cooperation’, or ‘stimulation’ of associated species) is incorrect (Snaydon 1991).

Although not necessarily required, ‘expected yield’ trends (Fig. 1) have often served as the frame of reference for interpreting replacement series, and this is implicit in several indices, such as relative crowding coefficient and relative yield total. For a standard replacement series the expected yield trends are straight lines that result from equal intra- and interspecific interference (Fig. 1), an interpretation that is supported by exercises using yield–density functions (Li & Hara 1999). An expected yield trend, however, does not reveal whether interference is zero, weak or severe, just that the intra- and interspecific components of interference are in balance.

For a proportional replacement series, curvilinear trends in expected yields are required to represent balanced intra- and interspecific interference, a point that does not seem to have been noted previously. The amount of curvature required to represent balanced interference will not be known from a single proportional replacement series, and this prevents a conventional interpretation of its results. This point may relate to work by Connolly (1986, 1988) who compared the interpretation obtained from standard and proportional replacement series. For the same mixture, the arbitrary choice of different monocultures changed the expected yield trends and this affected the interpretation of the results. As Connolly (1988) stated: ‘it is not acceptable that the conclusions of an experiment be dependent on its design’.

For a standard replacement series, if the observed yield of a component species departs from its expected yield, intra- and interspecific interference are not in balance. For the combined species yields, however, the occurrence of similar observed and expected values does not prove that there is a state of balanced interference (as offsetting unbalanced responses may be occurring in the component species). If a simple (i.e. not dramatically irregular in slope) yield–density response surface exists then, when intraspecific interference exceeds interspecific interference the observed yield trend for a component species in mixture rises above its expected yield. If interspecific interference exceeds intraspecific interference, the observed yield trend moves in the other direction. The size of the departure from the expected yields, however, is difficult to interpret. Exercises with yield–density response functions suggest that the observed yield trends are controlled in complex and sometimes non-intuitive ways (Jolliffe & Wanjau 1999). It is not only the relative values of intra- and interspecific interference that determine a result, but also the species densities and the potential size of individuals.

There are further difficulties in attempting to interpret interference using results from a replacement series. An observed replacement series trend forms part of some larger yield–density response surface, but it does not specify a particular surface. For example, in the inverse yield–density frame of reference (Fig. 3) transect AB across the response surface is the convex replacement curve for Trifolium shown in Fig. 1. An alternative response surface, one of many possibilities that share the identical transect, is also shown in Fig. 3. The slope of the response surface is a measure of interference, and the alternative model in Fig. 3 represents 20% larger intraspecific interference (0.143/0.119) and 38% greater interspecific interference (0.0864/0.0626) than in the original model. Hence, an observed replacement line is not an indicator of unique levels of intra- or interspecific interference. This difficulty extends to other assessments. The substitution rates, i.e. the ratio of interspecific interference to intraspecific interference, also differ between the two models: substitution rates are 0.52 (= 0.0626/0.119) in the original model and 0.60 (= 0.0864/0.143) in the alternative model. A compound ratio, related to the substitution rates, was used by Spitters (1983) to measure niche differentiation. That ratio also differs between the different potential response surfaces.

This discussion therefore leads to the view that a replacement diagram may reveal (in)equality of intra- and interspecific interference, and it may indicate the directions of the imbalances. A replacement diagram is inadequate, however, for the quantitative assessment of interference. The effects of component species are confounded along the series, and at any density combination neither the observed nor the expected yields are attributable to unique levels of interference. This conclusion applies to the graphical interpretation of interference in replacement series, as well as to interpretations based on indices. It should be noted that this argument does not require a presumption of the simple, inverse yield–density response surface used in Fig. 3. More complex, curved surfaces could be used to make the same point.

Other researchers have reached similar conclusions about the limited capability of replacement diagrams and their attendant indices to serve as means for interpreting interference (Connolly 1988; Snaydon 1991, 1994; Sackville Hamilton 1994; Cousens 1996; Gibson et al. 1999; Li & Hara 1999). Connolly (1986) anticipated the present line of argument, and stated: ‘the fundamental difficulty is that a replacement series is one-dimensional, being points on a straight line in the two-dimensional plane. This raises the possibility of two-dimensional density effects being distorted or confounded on reduction to one dimension.’

The use of expected yield trends as a basis for interpreting interference stems from the analogy made by de Wit (1960) between Raoult’s law and organismal competition. For an ideal mixture of solvents, the equilibrium vapour pressure of any constituent (analogous to component species yield per unit land area) is determined by its saturation vapour pressure and mole fraction. A structure directly comparable to the organismal replacement series is formed when total moles are held constant and the constituents are mixed in different proportions. This is approximated by pure and mixed solutions of benzene and toluene. Because total moles are constant, the vapour pressure of each constituent is directly proportional to the number of moles of that constituent (analogous to component species population density). Hence, an assumption of the analogy is that of a linear relationship between density and yield, and this assumption has been criticized (Austin & Austin 1980; Jolliffe et al. 1984). As de Wit (1960) stated, the situation described by Raoult’s law corresponds to cases ‘in which there is no competition … There are many mixtures for which Raoult’s law does not hold. They are treated with the introduction of activity coefficients’. In the subsequent derivation, the relative crowding coefficient is described as being ‘analogous to the activity coefficients of liquids’ (de Wit 1960). Jolliffe et al. (1984) indicated that there is a discontinuity in the logical development of de Wit’s theory. The analogy with Raoult’s law predicts trends in species yield (‘projected yields’) that follow the initial slope of the monoculture yield–density curve. The analogy does not lead to the expected yield trends, which lie below the projected yield trends when interference occurs. Although de Wit’s theory can be criticized, his contributions to ecology should not be disparaged. Whether de Wit’s approaches were right or wrong, he was an important pioneer, and his attempt to formulate a quantitative theory of competition, based on mechanistic concepts, was admirable and unusual for his time.

Expected yield trends are not the only possible frame of reference for interpreting replacement series. For example, designs where additional plots were included allow mixture yields to be compared against monoculture yields at constant component species density, as was proposed by Jolliffe et al. (1984). That procedure has been used in several studies (e.g. DiTommaso et al. 1996) and was found to provide a more detailed interpretation than obtained using a conventional replacement series (Roush et al. 1989). The procedure does not, however, test for total density dependence and assumes that no interactions occur between component species densities. While interactions seem to be uncommon, they cannot be ruled out a priori, and the procedure does not include a test for interactions. Austin et al. (1988) also noted the value of additional target species monocultures in relation to pair-wise experiments. Both these suggestions (Jolliffe et al. 1984; Austin et al. 1988), and the modifications by Rodriguez (1997) that will be mentioned later, constitute a move to expand the dimensionality of replacement or additive series structures.

Experimental biases

  1. Top of page
  2. Summary
  3. Introduction
  4. The replacement series structure
  5. Indices of mixture performance
  6. Interpreting trends on replacement diagrams
  7. Experimental biases
  8. Outcome of competition
  9. Yield comparisons
  10. Conclusion
  11. References
  12. Appendix

Several factors may introduce bias into replacement series experiments, including total density, initial size, and time of observation. If there is density dependence in the behaviour of mixed species, or in the indices of mixture performance, then the choice of a single total density for a standard replacement series may lead to an idiosyncratic result (Rejmanek et al. 1989; Silvertown 1989). Early replacement series studies did include some cases where series were performed at more than one total density (de Wit 1960; Baeumer & de Wit 1968), but evidence for systematic density dependence was not obtained. Later results (Marshall & Jain 1969; Khan et al. 1975; Wu & Jain 1979; Weiner 1980; Aarssen 1985; Firbank & Watkinson 1985), however, suggested that the magnitudes of species responses, and perhaps species dominance, may vary with total density. Connolly & Nolan (1976) indicated that total density should be considered when using replacement series to identify the most productive mixtures, and a similar recommendation was made by others in the context of the outcome of competition (Inouye & Schaffer 1981). DeBenedictis (1977) concluded that results obtained using the de Wit approach could simply reflect total density dependence.

A number of replacement series indices have been reported to be density-dependent, at least when density is insufficient to produce the maximum possible final yield. For example, de Wit (1961) noted that total density may affect relative reproductive rate. Other indices reported to exhibit density dependence are relative crowding coefficient (e.g. Firbank & Watkinson 1985; Connolly 1986, 1987; Law & Watkinson 1987; Rejmanek et al. 1989; Sackville Hamilton 1994), relative yield total (Firbank & Watkinson 1985; Taylor & Aarssen 1989; Sackville Hamilton 1994; Jolliffe & Wanjau 1999), aggressivity (Law & Watkinson 1987) and yield suppression ratio (Taylor & Aarssen 1989). The degree of density dependence has been found to vary between the different indices compared (Connolly 1986; Taylor & Aarssen 1989). As noted earlier, different indices can share common elements and therefore, if density dependence occurs in some indices, it may also exist between other replacement series indices. Some indices may be innately density-dependent because of their mathematical structure (e.g. Sackville Hamilton 1994; Li & Hara 1999), although this interpretation may reflect the assumption of a particular type of yield–density model.

The possibility of density dependence is therefore an important difficulty confronting researchers hoping to make generalizations from replacement series research. This issue, however, is not just restricted to replacement series. Yield–density response functions, for example, have sometimes needed modification to account for density dependence (Law & Watkinson 1987; Connolly et al. 1990; Menchaca & Connolly 1990; Blackshaw & Schaalje 1993). Whilst density dependence of replacement series results might well be expected from our knowledge of yield–density relationships (Cousens & O’Neill 1993), it has not always been found (Cousens & O’Neill 1993; Watkinson & Freckleton 1997). In particular, Cousens & O’Neill (1993) noted that evidence for density dependence came from a limited number of studies and lacked a strong statistical basis.

In agricultural studies, the choice of replacement series density has often been based on the optimum density for monoculture yield (Trenbath 1976; Rejmanek et al. 1989). In ecological contexts it has frequently been suggested that the total density should be sufficiently high to allow asymptotic or maximum yields to be approached in the monocultures. This may assist interpretation because, above certain densities, some replacement series indices may become independent of total density (Rejmanek et al. 1989; Taylor & Aarssen 1989). Taylor & Aarssen (1989) state that if ‘the relative yield suppression of mixture components is to be interpreted strictly as relative competitive ability, and mixture productivity is to be interpreted strictly as niche overlap, then they must be estimated when the demands on resources equal the supply. i.e. at densities which achieve constant final yield.’ Limiting the use of replacement series to a high total density, however, is more restrictive than the original requirement simply to use one particular total density, and this limitation has other consequences. It excludes the use of replacement series for exploring lower densities, where interference may still occur, and the phenomenon of density dependence may in itself be an interesting subject of study. Combined with the requirement for constant total density, a requirement for high density would make the use of a standard replacement series impractical (Snaydon 1994) for certain combinations of species (e.g. tree/grass mixtures).

The standard replacement series, with its use of a single density, would be particularly disadvantageous if total density alters species dominance. Cousens & O’Neill (1993) explored this question using yield–density response functions and concluded ‘there is little positive evidence of qualitative changes in behaviour from replacement series with increasing density. Dominance seldom changes with density in these data’. Others have also found that total density seems to have more of a quantitative than qualitative effect on dominance (Marshall & Jain 1969; Firbank & Watkinson 1985; Rejmanek et al. 1989; Hetrick et al. 1994). It is difficult, however, to be conclusive on this matter because resolution may depend on the possible shapes of yield–density response surfaces, which are, as yet, imprecisely known. As Li & Hara (1999) recently stated: ‘when the shapes of the yield–density relationships in monocultures of study species are unknown, we may misinterpret the results of replacement series experiments no matter whether the replacement series were conducted at low or high densities’.

Possible bias due to initial size has sometimes been discussed specifically in relation to replacement series (Grace et al. 1992), but it also applies to other experimental structures. Gurevitch et al. (1990) pointed out that replacement series and many other analyses of competition ‘fail to disentangle fully competitive effects and responses from differences in potential growth (i.e. inherent size differences among species)’. Problems of bias due to size and time are interrelated, and can be introduced through arbitrary decisions made by researchers. There has been an extended discussion on how size biases may affect the detection of size hierarchies in plant communities (Herben & Krahulec 1990; Grace et al. 1992; Shipley & Keddy 1994; Connolly 1997). The issue of size bias has also been considered in the specific context of replacement series (Connolly 1986; Herben & Krahulec 1990; Grace et al. 1992; Connolly 1997). There are several concerns. An assumption of the replacement series is that there is initial equivalence in size of species (Harper 1977; Aarssen 1985). Validation of this assumption has seldom been reported in replacement series studies. Another concern is that, because growth tends to be self-compounding, differences in initial size may tend to determine the end result of a competing association. This could involve a feedback in which size differences magnify as larger individuals increasingly suppress smaller individuals (Wilson 1988b). Also, some estimators may be biased in favour of the more productive species (‘sampling effect’, Aarssen 1997; Hector 1998). Initial size differences can determine the early interactions between associated individuals, affecting the subsequent development of size asymmetries (e.g. Weiner et al. 1997). In prolonged associations, however, problems of bias related to initial size may not always be significant (Grace et al. 1992). Firbank & Watkinson (1989) pointed out that if initial size was ‘the only factor determining the outcome of competition between species over several generations then communities would be dominated by those plants with the highest relative growth rates. This is clearly not the case.’

Several methods of correcting for size bias are available (Connolly & Wayne 1996; Gibson et al. 1999). Where observations are made at more than one sampling time, size-independent estimators such as relative efficiency index (Connolly 1987 Grace et al. 1992) can be used. However, attempts to correct for initial size can raise other problems, such as how to deal with within-species size variation, and the arbitrary choices of size measure and initial time (e.g. mass vs. area, or time of emergence vs. time of initial interactions between neighbours). Some experiments may be more concerned with determining the winners and losers of competition than with establishing fair play (equal initial size) among the competitors. In such cases the need to correct for differences in initial size may not exist.

Replacement series experiments have often determined yield at only one time of observation. As with the choice of a single total density this may lead to idiosyncratic results. Replacement series have been found to provide different results at different harvest times (Baeumer & de Wit 1968; van den Bergh 1968; Aarssen 1985; Akey et al. 1991). Ontogenetic differences occur between species, so results from different times of observation should provide different perspectives, even if initial sizes are equal. The desirability of following the progressive time course of interference has been repeatedly stated in the literature (Hutchings & Budd 1981; Aarssen 1985; Roush & Radosevich 1985; Connolly et al. 1990; Tilman 1990; Goldberg & Barton 1992; Grace et al. 1992; Gibson et al. 1999). Final yield provides no knowledge of the dynamics of an association (Connolly et al. 1990; Turkington & Jolliffe 1996) and this shortcoming ‘is possibly the single most neglected and important issue in current practice’ (Gibson et al. 1999). If we are to understand better the processes, mechanisms and rules of interference, we will need to know the sequence of events that occur as organisms coexist in close association. The end result of an experiment is inadequate for this purpose, whether it involves a replacement series or not.

Outcome of competition

  1. Top of page
  2. Summary
  3. Introduction
  4. The replacement series structure
  5. Indices of mixture performance
  6. Interpreting trends on replacement diagrams
  7. Experimental biases
  8. Outcome of competition
  9. Yield comparisons
  10. Conclusion
  11. References
  12. Appendix

Since de Wit (1960), replacement series have sometimes been considered for use in predicting the long-term fate of competing populations, such as eventual community composition and the possible exclusion of species. Indeed, relative reproductive rate was the first index derived in de Wit’s (1960) monograph, and in that work he introduced ratio diagrams as a graphical method for assessing mixture performance across generations. It was this aspect of de Wit’s theory, however, that first came under question. In early work, de de Wit et al. (1966) indicated that relative replacement rate should not be over-interpreted in cases where frequencies change over time. Experimental results of Marshall & Jain (1969) suggested that density dependence also applies to ratio diagrams. Harper (1977) noted that, while standard replacement series require constant total density, natural populations undergo changes in both proportions and total density. Many researchers (Inouye & Schaffer 1981; Goldberg & Werner 1983; Law & Watkinson 1987; Silvertown 1989; Herben & Krahulec 1990; Silvertown & Dale 1991; Cousens & O’Neill 1993; Cousens 1996; Connolly 1997; Rodriguez 1997) have since commented on the inadvisability of attempting to interpret or predict long-term population dynamics using standard replacement series. Perhaps – on this issue – there is no controversy. As stated by Herben & Krahulec (1990), ‘the de Wit designs tend to produce artificially deterministic interactions, since the standardization of the design excludes many of the processes which produce variability in the competitive outcome.’Rodriguez (1997) proposed modifications that might allow the use of replacement series to predict outcomes, through comparisons of mixtures across several generations at equivalent stages in life cycles. In order to do this, the dynamics and density dependence of the system need to be known, and assumptions must be made concerning survivorship and fecundity between times of observation.

Yield comparisons

  1. Top of page
  2. Summary
  3. Introduction
  4. The replacement series structure
  5. Indices of mixture performance
  6. Interpreting trends on replacement diagrams
  7. Experimental biases
  8. Outcome of competition
  9. Yield comparisons
  10. Conclusion
  11. References
  12. Appendix

The interpretation of interference and the prediction of mixture outcome are important ecological interests, but they are not the only interests addressed by replacement series studies. There is a considerable literature pertaining to studies where replacement series and other experimental structures have essentially been used as a setting for yield comparisons. Common themes involve comparisons of mixtures and monocultures, or rank ordering of different mixture combinations. Here, there may be little interest in interpretation and, when this is the case, many of the limitations to using replacement series that were discussed earlier may not apply. Experimental biases due to density, size and time may still contribute to the outcome, but controlling these factors may not be detrimental to the experimental aims.

For example, and in a different context, yields obtained in additive series experiments were used to rank competitive effect and response of species in different environments (Keddy et al. 1994). The additive series is also held to be an appropriate experimental structure for evaluating crop losses at different levels of weed infestation (Radosevich 1987; Cousens & O’Neill 1993; Cousens 1996; Gibson et al. 1999). In such cases, the detection of an economic threshold for applying weed control measures may be the experimental goal; interpreting crop behaviour in terms of weed density, species proportions or total density may be of no concern. It is interesting that replacement series have also been regularly used in studies of crop/weed associations (Rejmanek et al. 1989; Gibson et al. 1999). In one sense, however, replacement series are inefficient for this purpose as the weed monoculture required in a replacement series may be of little agricultural interest. A partial replacement series, involving a crop monoculture and a crop/weed mixture at the same total density, might be useful for comparing the relative suppression of crop production by different weeds. However, a full replacement series may be advantageous in that it requires the evaluation of both the crop and the weed, while in an additive series only the crop is commonly assessed. Interpretation can also be a goal of crop/weed studies, and the use of larger experimental structures has been recommended for attempts at gaining a better understanding of crop/weed associations (Pantone & Baker 1991; O’Donovan 1996).

Another interest in agriculture and forestry has been to determine whether intercrops are more productive than their corresponding monocultures (Trenbath 1974; Willey 1979; Jolliffe & Wanjau 1999). This connects to the more complex ecological issue of the relationship between species diversity and community productivity. The prevalent way of visualizing intercrop performance has been through the use of replacement diagrams, sometimes with the intent of finding the mixture having species proportions that result in optimum combined yields. Such a use of replacement series does not require an attempt to account causally for the yield trends (Cousens 1996). Many studies with intercrops have been restricted to a single total density, which sometimes reflects an assumption that the most productive mixture will be obtained at the density at which the monocultures are ordinarily raised for agronomic purposes. Such an assumption has little foundation, and its adoption limits the flexibility of this approach to intercropping research. As with interference, increasing the dimensionality of the experimental structure can be helpful in studying mixture productivity. This was recognized by Mead (1979) who stated, ‘in some form it will be necessary to use a two-dimensional representation of yield advantage’.

As with interference, mixture productivity can be assessed using a number of measures (Willey 1985; Garnier et al. 1997; Jolliffe & Wanjau 1999). Several indices express the relative productive performance of mixtures and monocultures. These include relative yield total (RYT; de Wit & van den Bergh 1965), land equivalent ratio (LER; Mead & Willey 1980) and relative land output (RLO; Jolliffe 1997) and details concerning the calculation of these three indices are given in the Appendix. In these three indices, a value of 1.0 indicates similar productivity in mixture and monocultures. Values above 1.0 indicate better production from the mixture, and values below 1.0 indicate better production from the monocultures. The use of RYT in this context is in addition to its use in interpreting the nature of interference, mentioned earlier, whereas LER and RLO were not developed to interpret interference, but simply to quantify mixture productivity.

There is evidence that the index used to assess mixture productivity can affect the conclusions drawn from a study (Garnier et al. 1997). Although RYT and LER had long been used to compare the productivity of mixtures and monocultures, I chose not to use these indices in a recent survey (Jolliffe 1997) whose aim was to analyse whether binary species mixtures differ in productivity from their monoculture counterparts. Productivity is a function of land area as well as numbers of individuals, but these differ between the monocultures and mixtures used to calculate RYT and LER. For example, consider a standard replacement series containing two monocultures and a 50 : 50 mixture. The monocultures occupy twice the land area and contain twice the numbers of individuals per species compared to the mixture, and in that sense the monocultures and mixtures differ in key inputs pertaining to productivity. In the case of RLO, these inputs are the same. It should be noted, however, that RLO is positively correlated with RYT and a similar correlation presumably exists with LER (Jolliffe & Wanjau 1999). For example, if RYT is calculated for the same data sets as in Jolliffe (1997; Table 2), a similar conclusion is reached that relative yields are significantly higher in mixtures than in monocultures (P. Jolliffe, unpublished data).

Since de Wit (1960), many studies have used absolute rather than relative yields. This requires that the measures of yield be qualitatively the same for the different species. It has often been found that the combined absolute yields of species mixed in replacement series fall near the line connecting the two monocultures (e.g. van den Bergh 1968; Garnier et al. 1997), i.e. the mixture is often less productive than the better monoculture, and this can be true when RYT or LER exceed 1.0. Hence, these indices of relative productivity are of questionable effectiveness in attempts to identify the mixture that has the highest absolute yield (Garnier et al. 1997). Indeed, RYT and RLO were found to be uncorrelated with absolute mixture yields in a range of studies (Jolliffe & Wanjau 1999). Where absolute productivity is the issue, researchers can simply evaluate combined mixture production (Jolliffe & Wanjau 1999), or take the ratio of combined yield in mixture to that of the more productive monoculture (Trenbath 1974; Finney 1990 Garnier et al. 1997). Hence, different methods can be used to quantify mixture productivity. None of these approaches is valid or invalid per se. It is up to researchers to decide on the most appropriate measure(s) to use in relation to their particular experimental aims.

Conclusion

  1. Top of page
  2. Summary
  3. Introduction
  4. The replacement series structure
  5. Indices of mixture performance
  6. Interpreting trends on replacement diagrams
  7. Experimental biases
  8. Outcome of competition
  9. Yield comparisons
  10. Conclusion
  11. References
  12. Appendix

The replacement series is a valid experimental structure, forming one dimension within a larger yield–density response matrix. As with other tools, however, the use of replacement series for a particular purpose may be inefficient, ineffective or invalid. The decision to use a replacement series should be preceded by the consideration of a number of factors, including experimental design, practicality, capability for interpretation, and the experimental goals (Cousens 1996). Further considerations would involve the merits and limitations of alternative procedures, which have not been considered here.

In making a critical evaluation of replacement series, this discussion has naturally focused on possible areas of concern. This may give an impression that there are simply so many difficulties that the use of replacement series should be abolished. However, none of the difficulties so far identified is fatal to all potential uses of the replacement series. Perhaps some of the present limitations of replacement series might be overcome (Rodriguez 1997), and this prospect would disappear if the technique was extinguished. A decision to keep the replacement series in our repertoire is in keeping with calls for pluralism in ecological methodology (Weiner 1995; Aarssen 1997), and does not deny researchers the opportunity of using other procedures.

Nevertheless, researchers who are considering the use of replacement series, or who are reading the literature, need to appreciate the limitations and assumptions of this methodology. There are problems in obtaining detailed interpretations from the trends on replacement diagrams. Experimental biases due to choice of density, initial size and observation time can exist. Some of the recent suggestions to discourage the use of replacement series may have arisen in part from the expectation that other procedures may offer greater possibility for rapidly advancing our understanding of interference and this may well be true. Many replacement series experiments have been done, so perhaps the chance of using replacement series to make further significant ecological discoveries is small. This discussion has not ruled out the use of replacement series to detect inequalities in interference. There also seems to be no reason to exclude the use of replacement series in making simple yield comparisons, although biases due to density, initial size and time may contribute to the observed yield variations.

Conclusions that have been reached using replacement series have often been obtained using other methods, such as response surface analysis. These include the lack of reciprocity in interference between associated species, plus the fact that that interference between species is not constant, but can vary during growth and under different environmental conditions occurs (e.g. van den Bergh 1968; Fowler 1982). On the other hand, there are some examples where response surface techniques have led to conclusions different from those given by replacement series methods (Firbank & Watkinson 1985; Connolly 1986, 1987, 1997; Connolly et al. 1990; Rodriguez 1997). Much of the literature pertaining to the replacement series preceded the criticisms that have been levelled against it. The present discussion suggests that care will be needed in accepting conclusions from published studies in which replacement series have been used to interpret interference, complementarity of resource use, niche differentiation or the outcome of competition. At least, the assumptions made in reaching those interpretations need to be recognized. It should be noted that many studies have used replacement series in concert with other procedures, i.e. the conclusions of such studies are more broadly based than they would be if only replacement series had been used. Also, it does not appear that the published results and interpretations obtained with replacement series are characteristically at odds with the main findings from other procedures. If such discordance occurred regularly it should be obvious by now. For these reasons, the extant replacement series literature should not be ignored or dismissed.

Several limitations of replacement series can be overcome by expanding its dimensionality. This would involve conducting some larger experiments with more treatment combinations. Large experiments are needed in order to deal with the multiple relationships among population composition, density, time of association and other factors (Goldberg & Barton 1992). Also, thorough investigations of resource use by associated species ought to provide more direct evidence concerning matters of competition, complementarity of resource use and niche differentiation. The effort to undertake such studies can be justified by the ecological and economic importance of interference. To some extent, such research is phenomenology (Tilman 1990), but the information obtained from such studies is essential for the refinement, calibration, and verification of mechanistic models of interference.

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  2. Summary
  3. Introduction
  4. The replacement series structure
  5. Indices of mixture performance
  6. Interpreting trends on replacement diagrams
  7. Experimental biases
  8. Outcome of competition
  9. Yield comparisons
  10. Conclusion
  11. References
  12. Appendix
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Received 14 September 1999revisionaccepted 8 February 2000

Appendix

  1. Top of page
  2. Summary
  3. Introduction
  4. The replacement series structure
  5. Indices of mixture performance
  6. Interpreting trends on replacement diagrams
  7. Experimental biases
  8. Outcome of competition
  9. Yield comparisons
  10. Conclusion
  11. References
  12. Appendix

Appendix: Indices and replacement series

Indices that have been used in conjunction with replacement series include: relative reproductive rate, relative crowding coefficient (de Wit 1960), relative replacement rate, relative yield, relative yield total (de Wit & van den Bergh 1965), aggressivity (McGilchrist & Trenbath 1971), land equivalent ratio (International Rice Research Institute 1974; Mead & Willey 1980), relative space (Elberse & de Kruyf 1979), competitive ratio (Willey & Rao 1980), relative monoculture response, relative mixture response, combined relative monoculture response (Jolliffe et al. 1984), yield suppression coefficient, yield suppression ratio (Aarssen 1985), relative efficiency index (Connolly 1987), competitive balance, competitive intensity, interaction in competitive intensity, relative yield of mixtures (Wilson 1988a), net aggressivity (Yates & Dutton 1988), relative suppression coefficient, absolute suppression coefficient, log10 relative crowding coefficient (Snaydon 1991), competitive saprotrophic ability (Wardle & Parkinson 1992), relative competition index, absolute competition index (Grace 1995), relative land output and total land output (Jolliffe 1997). Additional indices for assessing mixtures, such as relative physiological performance (Austin 1982), could be employed with replacement series but have not yet been used for that purpose.

Indices can use similar input variables, and distinctions between indices are sometimes not obvious. For example, consider three indices used to express the relative productivity of mixtures and monocultures: relative yield total (RYT), land equivalent ratio (LER) and relative land output (RLO). RYT was introduced in the context of replacement series experiments (de Wit & van den Bergh 1965). Its early use was to evaluate niche differentiation, but it was later applied in surveys of the relative productive performance of plant species mixtures (Trenbath 1974). As with LER and RLO, data on species yields per unit land area, in mixture and monocultures, are used to calculate RYT. There are, however, important distinctions between these three indices concerning the specific input data used as well as their mathematical formulation. Consider two species, designated by subscripts i and j, respectively, grown in a standard replacement series in monocultures (subscript m) or in mixture (subscript x). If Y represents some measure of species yield per unit land area, RYT is calculated from:

  • RYT  =  (Yix/Yim)  +  (Yjx/Yjm)

That is, RYT is the sum of the relative yields per species grown in a standard replacement series.

Land equivalent ratio (LER) was developed to measure the amount of land required for an intercrop to be as productive as the same crops grown in monocultures (Mead & Willey 1980). The formula for calculating LER is the same as for RYT:

  • LER  =  (Yix/Yim)  +  (Yjx/Yjm)

LER has however had a more flexible meaning than RYT (Willey 1985). It can be identical to RYT, in which case the experimental structure is a standard replacement series and Yim and Yjm are the species monoculture yields per unit land area at the same density as the combined species density in mixture. Alternatively, LER has also been calculated on the basis of maximum or optimum yields of the monocultures attainable (Willey 1985), and on the basis of average monoculture yields (Pilbeam et al. 1994). In these cases the species’ monoculture densities may differ from one another as well as from the combined mixture density. In such cases, values of Yim and Yjm used to calculate LER may differ from those used to calculate RYT.

Relative land output (RLO) is the ratio of combined yields in the mixture compared with those in monocultures (Jolliffe 1997):

  • RLO  =  (Yix  +  Yjx)/(Yim  +  Yjm)

RLO is a generalization of an index developed earlier and independently by Wilson (1988a). For RLO there is a requirement that the land area used to grow the mixture be equal to the total land area on which the monocultures are grown. This provision contrasts with a standard replacement series, for example, where twice as much land area is used for the monocultures as for a 50 : 50 species mixture (for a two-species system). In addition, RLO requires that the same numbers of individuals be represented in mixture and monoculture. This provision also contrasts with a standard replacement series in which the monoculture plots contain more individuals of each species than are found in any mixture. For example, consider a mixture containing 10 individuals of species i and 20 individuals of species j growing on a plot of land having an area of 3 × Α. The corresponding monocultures for the calculation of RLO would be a plot of area A containing 10 plants of species i and a plot of 2 × A containing 20 individuals of species j. As in a standard replacement series, the same total population density is used in the monoculture and mixture plots required for determining RLO. Contrary to Loreau (1998), RLO does not require a specific planting pattern within the plots. The requirement for all three of these indices is that planting patterns (spatial and temporal distributions) should correspond between the mixture and monoculture plots. RLO, and to a lesser extent LER, are not restricted to replacement series, but results from replacement series experiments can be used to calculate any of these indices. Others have suggested that various criteria and indices are valid for the assessment of mixture productivity (Willey 1985; Vandermeer 1989), and I share that point of view.