Recently, the frequency-dependency of the per-capita growth rate has been studied theoretically (Chesson 2000; Adler, HilleRisLambers & Levine 2007) and experimentally (Harpole & Suding 2007) in order to make ecological predictions about possible stable coexistence among species. It is argued that if several species have a sufficiently higher per-capita growth rate when the species are rare compared to when the species are common then stable coexistence of the species is predicted (Adler et al. 2007). Furthermore, it was recommended that in order to experimentally test the hypothesis of coexistence it is appropriate to investigate the possible frequency-dependency of the per-capita growth rate in competition experiments where the frequencies of the different plants are manipulated but where the combined density of the two plants is kept constant (Adler et al. 2007; Harpole & Suding 2007). The design of such competition experiments are in the plant ecological literature usually called a ‘substitution design’.
However, it is not generally true that species will display stable coexistence if they have a relative high per-capita growth rate when rare, as acknowledged in the above-mentioned theoretical studies (Chesson 2000; Adler et al. 2007). The use of per-capita growth rates as a criterion for species coexistence is only generally valid in the case when per-capita growth rate declines linearly with both intraspecific and interspecific densities (Chesson 2000), and if the per-capita growth rates decline in a nonlinear way with density, then the criterion for species coexistence is only valid in the specific case when the rare plant is at zero density and the other species are at equilibrium densities (Chesson 2000).
Since it has been demonstrated in literally thousands of ecological and agronomical studies (e.g. Shinozaki & Kira 1956; Law & Watkinson 1987; Cousens 1991) that reproduction and mortality, the two factors that are the basis for population growth rates, are nonlinear functions of plant density, the suggested method for predicting coexistence requires that the substitution experiment is made at the equilibrium densities of the two species (Adler et al. 2007). However, even in the unlikely event that such exact and stable equilibrium densities should exist, then an experimental plant ecologist typically will not be aware of them when planning a competition experiment. Furthermore, such equilibrium densities will generally depend on the environment in which the experiment is carried out and the coexistence proportions of the two species at a possible stable equilibrium, that is, the design of the substitution experiment depends critically on the obtained results.
Hence, it is unjust to make predictions about plant community dynamics based on the results of a substitution experiment because it is not generally known whether the density used in the substitution experiment is relatively low or high compared to the density where the species may coexist. Accordingly, the substitution design has been severely criticized in a number of papers, most notably by Inouye (2001): ‘The use of substitution (and additive) experimental designs has largely precluded generating quantitative estimates of the effects of interspecific competition on population dynamics or coexistence, beyond the inference that species does or does not compete.’ In general, only response surface competition experiments, where both the density and proportion of the investigated species are varied, allow predictions of the long-term ecological outcome of competition (Damgaard 1998, 2004b; Inouye 2001).
That coexistence cannot be predicted from observations at a single density may be illustrated by an examination of the condition for coexistence in a simple nonlinear competition model for annual plants (Damgaard 2004a), where it is found that coexistence depends on fecundity at both low and high density (see Appendix S1 in Supplementary Material). A numerical example using the same model (Fig. 1), in which intraspecific competition is higher than interspecific competition (c12 = c21 = 0·7), shows that the per-capita growth rates display negative frequency-dependency in a simulated substitution experiment with a combined density of 100, although species 1 outcompetes species 2 due to superior fecundity at high densities (β1 = 0·0002, β2 = 0·0005) (Fig. 1).
The study of the frequency-dependency of the per-capita growth rate in a substitution design instead of studying mortality and fecundity in a competition experiment, where both the density and frequency of the investigated species are varied, is comparable to the difference between measuring relative and absolute fitness. Or in the words of thermodynamics, the per-capita growth rate is an intensive variable (since it is normalized by density), whereas mortality and reproduction are extensive variables, and the lesson from thermodynamics is clear that: one should not mix-up intensive variables (e.g. temperature) with extensive variables (e.g. pressure).
In order to circumvent the problems of the substitution design, it has been suggested by Adler et al. (2007) that the ‘manipulations need to be maintained long enough for the composition of the background community to adjust in response to the density of the focal species’. However, this suggestion is in my opinion not a realistic option. Not even in the simple case of annual plants will it be feasible to wait for the stochastic processes of reproduction, dispersal, and recruitment to reach an ecological equilibrium in manipulated experiments, even without considering the complicating factors of a changing environment, spatial dynamics, and a variable pressure from herbivores and pathogens.
In this paper I am not criticising the use of simple heuristic models or the endeavour of linearizing complex nonlinear ecological phenomenon for the purpose of understanding general aspects of, for example, different mechanisms for coexistence (Chesson 2000). However, models that are fitted to ecological data need to include the most important dynamics of the system to make reliable predictions. Furthermore, when making ecological predictions it is becoming increasingly important to quantify the uncertainty of the predictions (Clark 2005), and it is problematic to quantify the uncertainty of a prediction if the ecological data are analysed in a model that is too simple to cover the dynamics of the system. For example, if the above-mentioned competition model (Appendix S1) is fitted to fecundity data from a response surface competition experiment of annual plants, then there is a sizeable degree of statistical uncertainty in the estimates of the parameters due to variable plant growth, and this statistical uncertainty can be interpreted as uncertainty on the predicted densities of the plant species at equilibrium (Damgaard 1998, 2004b). If the same data was analysed in a simpler model using per-capita growth rates then the statistical uncertainty due to variable plant growth will be confounded with structural uncertainty due to the use of a model that is too simple.