On the relative yield of plants in two-species mixture

Two contrasting experimental designs have been used to study inter-specific competition in binary mixture. One is the additive design, in which two species are grown together, and the density of one species is maintained constant while that of the other is allowed to vary; this design is possible in nature. In Harper's (1977) words, this design allows the experimenter to be free to vary both density and proportion; thus it is easy to design experiments, but difficult or even almost impossible to interpret the results. The other is the replacement (or substitutive) design, which is artificial in nature. In a standard replacement design, the total density of the mixture is held constant, and the same density is also used for both component species in their respective monocultures. The most common replacement design is the one in which two monocultures of component species and their 0.5:0.5 mixture are used. The 'style-free' additive design has received a lot of criticism for its varying both total density and proportion. Harper (1977) even stated that most problems of additive designs are eliminated in replacement designs and the replacement experiments are particularly elegant for the study of interactions between two plant species. Rejmdnek et al. (1989) have also criticised the additive designs: they result in simultaneous changes of proportion and total density, and make the interpretation of results difficult, seriously limiting their application. Similar criticisms have also come from other researchers (see review by Snaydon 1991 and references therein). While the criticisms of the additive designs persist or the advantages of the additive designs are recognised again, criticisms of the replacement designs have also become common (Snaydon 1991, 1994). Snaydon (1991) concluded that measures of competitive abilities (e.g. relative crowding coefficient) derived from the replacement designs are difficult to interpret biologically, because they are affected by a number of factors such as the density of each component in its monoculture, the shape of the yield-density relationship curves of the components and the proportions of the components in mixture, and that additive designs should be used in preference. However, his conclusions were also subject to debate (Sackville Hamilton 1994). Many authors (e.g. Firbank and Watkinson 1985, Connolly 1986, Law and Watkinson 1987, Austin et al. 1988, Taylor and Aarssen 1989, Snaydon 1991) have noticed that the results of the replacement design may be density-dependent: the density dependence may be restricted solely to densities below the value at which constant final yield is reached, or to the conditions where monoculture yield of component species is always dependent on density (i.e. yield-density relationship is not asymptotic, but parabolic) (Taylor and Aarssen 1989). The parabolic yield-density relationship has been widely observed in plant populations, in particular when yield components are considered (Bleasdale and Nelder 1960, Farazdaghi and Harris 1968, Willey and Heath 1969, Health et al. 1991, HashemiDezfouli and Herbert 1992, Van-Averbeke and Marais 1992). It has also been demonstrated for total yield of crops (e.g. Sparks 1988, Li 1995, Li et al. 1996, B. Li, T. Hara and J.-I. Suzuki unpubl.). Moreover, the parabolic yield-density relationship can also be theoretically derived (Pacala and Weiner 1991; see also Appendix). In the Appendix, we theoretically show under what conditions the parabolic yield-density relationship may emerge. Several authors (e.g. Austin et al. 1988, Taylor and Aarssen 1989, Cousens and O'Neill 1993) have demonstrated how the results of replacement series experiments depend on density in relation to the reciprocal yield-density equation (i.e. asymptotic yielddensity relationship). However, the effects of the parabolic yield-density relationship on the results of the replacement designs have never been considered theo-