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Testing performance rank reversals among coexisting species: crossover point irradiance analysis

2003; Wiley; Volume: 17; Issue: 2 Linguagem: Inglês

10.1046/j.1365-2435.2003.07101.x

1365-2435

Kaoru Kitajima, Benjamin M. Bolker,

Remote Sensing in Agriculture

Two potentially competing species can coexist when their relative fitness ranks reverse at different points in a spatially or temporally heterogeneous landscape (Chesson 1985). Trade-offs between components of fitness, such as between survival and growth rates, or between fecundity and survival, may lead to rank reversals in relative fitness even when individual fitness components do not exhibit rank reversals. Alternatively, fitness rank reversals can be driven by rank reversals of a key performance trait, such as growth rate, between contrasting environments. Detecting such rank reversals in any competitive community is thus of interest. In particular, comparative studies of tree seedlings along a light gradient provide a rich arena for testing the relative importance of the rank-reversal hypothesis in community organization, and for exploring the physiological bases of rank reversals. Givnish (1988) and Latham (1992) proposed a hypothesis of rank reversal in relative growth rates (RGR) as the mechanism of niche differentiation and species coexistence along light gradients, and presented supporting evidence from comparative studies using a small number of species. In terms of whole-plant growth rates, more recent comparative studies of a larger number of species have failed to detect substantial rank reversals between gap and shade irradiance (Kitajima 1994; Veneklaas & Poorter 1998; Poorter 1999; Walters & Reich 1999). Sack & Grubb (2001; hereafter SG) introduced a new analytical approach to this problem – the crossover-point irradiance (CPI) method summarized in the next section – and analysed seven published studies to evaluate patterns of rank reversals in tree seedling RGR. Below, we compare statistical properties of the CPI method and two other statistical tests for evaluating the significance of rank reversals, and discuss SG's interpretation of the patterns they found. Sack & Grubb (2001) begin with the idea that mass-based photosynthetic rates of leaves of shade species exceed those of sun species in shade (e.g. Björkman & Holmgren 1963, comparing quantum yields for ecotypes of a herbaceous species; Givnish 1988, comparing sun- and shade-phenotypes of a single genotype; but see Osmond et al. 1999 and Walters & Reich 1999). Regardless of the generality of leaf-level physiological trade-offs, the critical question is how to detect the signature of physiological trade-offs at the levels of whole-plant function and community structure. These two higher levels of inquiry – the organismal and the community – suggest two different null hypotheses. If one wants to test whether lower-level trade-offs scale up to the whole-plant level, a sensible null model is that species maintain the same performance rank in all environments. One does need to ensure that apparent rank reversals are not simply the effect of measurement error, but even a single rank reversal that can be distinguished from measurement error will falsify the model. That SG consider even a small number of rank reversals to be significant suggests that this performance-maintenance model underlies their argument. An alternative null model, useful for testing whether leaf- and organism-level trade-offs scale up to affect community structure, states that the performance rank of a species in two different environments is uncorrelated. Under this model, half the species pairs in a community are expected to show rank reversals by chance; showing that the proportion of rank reversals is significantly different from half falsifies the null hypothesis. Finding fewer rank reversals than expected by chance (positive correlation) suggests that all species are physiologically or phylogenetically constrained to show similar (parallel) reaction norms. Finding more rank reversals than expected by chance (negative correlation) suggests that physiological trade-offs are important in structuring the community; this is the pattern one would see if communities are initially assembled at random, but thereafter stronger species outcompete weaker species in non-reversing pairs. If rank reversals are not more common than expected by chance, one should seek alternative mechanisms for species coexistence. (As with all statistical tests, one can fail to reject the null hypothesis either because the hypothesis is true – in this case, that physiological trade-offs in a single trait do not drive community structure – or because of bias or lack of power.) A standard parametric test for rank reversals quantifies the direction and strength of correlation of performance measured in two different environments by plotting each species' performance score in one environment against that in another environment (Kitajima 1994). A negative correlation implies more rank reversals than expected by chance, while a positive correlation implies significant rank retention among species. Less positive or more negative correlation in one data set than in another indicates more frequent rank reversals in the former. Rank reversal frequency can also be tested non-parametrically with Kendall's test for rank concordance (Sokal & Rohlf 1981). Sack & Grubb (2001) developed a new parametric approach of calculating CPI, the irradiance (percentage of full daylight) at which the reaction norms of RGR of a species pair would intersect along a light gradient. The frequency of rank reversals is evaluated by the proportion of CPI that falls within an arbitrary selected range of irradiance, for which SG selected 2–10% to represent commonly encountered light availability in forest understories. Typical forms of the RGR reaction norm show an asymptotic increase, often with a drop at the highest irradiance (Poorter 1999; Valladares et al. 2000). Thus, up to 15–20% irradiance several functions, including logarithmic, Michaelis–Menten and non-rectangular hyperbola, may fit equally well to observed reaction norms that have multiple data points in this range (Poorter 1999). Sack & Grubb (2001) chose a semilog plot to linearize the reaction norm: RGR = R ln(irradiance + 1) + L Any two non-parallel lines in a two-dimensional space must intersect; the irradiance at which the reaction norms for species A and B are predicted to cross (CPI) may be calculated according to equation 2 of SG. The CPI method has several advantages. Crossover-point irradiance directly infers the irradiance at which two species switch their performance ranks, and can be compared across experiments that have used different light treatments. It can also be calculated when performance data are collected in only two light environments; this feature is convenient because having more than two light treatments may be logistically difficult in a comparative study of many species. Crossover-point irradiance has the potential to extrapolate results for experiments that measure performance in too narrow a range of light environments, which would otherwise bias results against detecting rank reversals in a single performance trait. This is perhaps the most attractive feature of the CPI which, as shown by SG's analysis, often occurs at irradiances below the minimum used in typical experiments. However, the CPI method has several drawbacks. First, SG apparently aim to test the null hypothesis of RGR rank maintenance rather than of random assembly; we do not know what fraction of CPI is expected to fall within a given range of irradiance by chance. Second, the calculated CPI values may be biased by structural error – mis-specification of the underlying function for the true reaction norm. Third, the CPI method as implemented by SG does not take into account the potential effects of measurement error. If measurement errors are large, the estimated fraction of CPI cannot contain much true information. In order to compute appropriate confidence intervals to falsify the null hypothesis of rank maintenance, one would require additional statistical techniques that incorporate measurement error. Below, we use simulations to explore the sensitivity of CPI and other methods to measurement and structural error. All simulation analyses were done using the R language (Ihaka & Gentleman 1996). We estimated reaction norms for 15 species from the data of Poorter (1999) at irradiances of 3, 6, 12 and 25% of daylight (excluding the highest light treatment of 50%, as done by SG), using either logarithmic functions (as did SG) or vertically offset Michaelis–Menten curves, which can have a positive or negative y intercept (non-zero RGR at zero irradiance). We also did the analysis by fitting non-rectangular hyperbolas that have been widely used to fit the light response of instantaneous photosynthetic rates (Lieth & Reynolds 1987); the results are similar to those for the Michaelis–Menten curves and are not discussed further. The Michaelis–Menten curves fit the RGR reaction norms of Poorter (1999) better than SG's logarithmic curves, as expected since Michaelis–Menten curves have one more parameter than logarithmic curves. Median r2 for Michaelis–Menten fits was 0·83 (range 0–0·99), while the median r2 for logarithmic fits was 0·67 (range 0·004–0·96). We then took either function to be the 'true' reaction norm, and explored the effects of measurement and structural errors on the estimated fraction of CPI that occurs between 2 and 10% daylight (= range used by SG). We evaluated measurement error by adding truncated normal deviates to RGR estimates. We evaluated structural error by mis-specifying the parametric function for calculation of CPI – for example, using a logarithmic function when the true function for the simulation was Michaelis–Menten, and vice versa. The CPI values were calculated for logarithmic curves using SG's equations 1 and 2; standard numerical methods (Brent's method; Press et al. 1994) were used to find the CPI for Michaelis–Menten curves. The true CPI fraction is always correctly estimated if no noise is added to the reaction norm and if CPI is estimated with a parametric form that matches the true form. If the true reaction norms are Michaelis–Menten (Fig. 1a), the estimates are relatively unaffected by noise if the correct function (Michaelis–Menten) is specified. However, incorrectly using a logarithmic function to fit the reaction norms predicts a significantly smaller fraction of CPI in the 2–10% daylight range (0·15) than the true estimate (0·23 with Michaelis–Menten) with little noise, and estimates increase with noise. In contrast, if the true reaction norms are logarithmic (Fig. 1b), CPI estimates using the true function increase with increasing noise, while CPI estimates using Michaelis–Menten are biased downward from the true fraction when measurement error is low, and increase to approach the true estimate (0·16) at the higher noise levels. Estimated fraction of crossover points falling between 2 and 10% irradiance as a function of estimated phenotypic reaction norm shape and measurement error. Simulations use reaction norms fitted to RGR data for the 15 tree species of Poorter (1999); measurement error is added from a truncated normal distribution with mean parameter zero and standard deviation (in RGR units) as shown on the horizontal axis. True reaction norms are (a) Michaelis–Menten; (b) logarithmic. Dashed and dotted lines show 95% confidence limits from 10 simulations at each noise level. We conclude that the estimation of CPI can be sensitive to both measurement and structural error. The assumption of a logarithmic relationship adopted by SG tends to bias the estimated fraction of CPI upward with increasing measurement error (Fig. 1). Fortunately, if that assumption is wrong (if the true reaction norms are Michaelis–Menten), the estimated fraction of CPI is biased in a conservative direction (underestimates the fraction of CPI in the specified range of irradiance; Fig. 1a). We do not have a good mechanistic argument to derive correct functions for RGR reaction norms, so that any function is necessarily an approximation. The asymptotic shape of reaction norms of RGR as a function of percentage daylight may resemble that of instantaneous photosynthetic light response curves for a leaf, but this coincidence does not imply common mechanisms. Sack & Grubb (2001) argue that the slope (R) of the RGR reaction norm is analogous to photosynthetic light-use efficiency, whereas the intercept (L) is a function of dark respiration. However, both parameters (L and R) reflect morphological as well as physiological traits of the species. The slope R is not analogous to the 'quantum yield' determined as the initial slope of the photosynthetic light-response curve, but rather is an indicator of morphological and physiological plasticity (Valladares et al. 2000). The intercept, L, of the reaction norm is a statistical extrapolation, unlike the dark respiration in the photosynthetic light-response curve. Estimated intercepts and slopes are unlikely to be independent of each other for a simple reason, without a need to invoke biological mechanisms proposed by SG; under a null model of a common light compensation for all species, intercepts and slope should be negatively correlated with each other. We used Poorter's data of RGR at 3 and 12% daylight (corresponding roughly to understorey vs gap irradiance) to simulate how correlation analysis may be affected by measurement error. Parametric correlation analysis does not assume a particular parametric function for the shape of reaction norms. It is also easy to calculate significance levels for correlation, and adding noise simply dilutes the significance of the results, rather than biasing them (Fig. 2a). Correlation analysis does have the disadvantage that it can use only actual data at two different irradiances, which must be assumed to represent the irradiance extremes experienced by plants in the field (a potentially dangerous assumption; cf. Montgomery & Chazdon 2002). Pearson correlations (a) and number of crossovers (b) between RGR at low (3%) and high (12%) irradiances as a function of noise added to RGR data for the 15 tree species of Poorter (1999). Dashed lines show 95% confidence intervals on correlations from 10 simulations at each noise level; horizontal dotted lines show 95% significance levels for non-zero correlations. Next we examined a non-parametric test for the number of observed rank reversals against a null model of random rank shuffling, which is mathematically equivalent to Kendall's τ statistic for rank retention (Sokal & Rohlf 1981). Kendall's τ gives the significance level of an observed number of crossovers as the probability that the same number or more (or fewer) crossovers occur for a random permutation of the species performances in the second environment. This non-parametric crossover analysis (Fig. 2b) gives results similar to those from parametric correlation analysis. A small number of crossovers corresponds to a positive correlation, and the estimated number of crossovers increases with increasing noise amplitude (Fig. 2b), corresponding with the decreasing positive correlation in Fig. 2(a). The estimated number of crossovers becomes non-significant at a lower noise level than does the estimated correlation, because non-parametric analysis trades off statistical power for robustness to outliers. In summary, the three statistical methods have pros and cons that stem from different assumptions (Table 1). The main disadvantages of the CPI method are that we do not know how many crossovers characterize confidence limits for the null case, nor can we distinguish the effects of high noise levels from a true community signal of rank reversals. One may argue that even a few rank reversals contribute significantly to species coexistence, but unless rank reversals are observed more frequently than expected by random permutation, rank reversals in a given performance trait cannot be the main reason for observed microsite preferences. The true value of the CPI method will emerge when there are sufficient data (accurate measurements at sufficiently different irradiances) to make statistically strong conclusions about the true shape of the reaction norms. The title of Sack & Grubb (2001), 'Why do species of woody seedlings change rank in relative growth rate between low and high irradiance?', does not match the results of their analysis. Only one of the seven studies examined by SG showed a negative correlation – and even this correlation was only suggestive and not statistically significant (P = 0·083). Accordingly, we suggest that future searches for mechanisms of species coexistence should be directed toward testing alternatives to the rank-reversal hypothesis, such as allocation-based trade-offs in life-history traits (e.g. negative correlation between growth rates and survival; Brokaw 1987; Kitajima 1994; Dalling & Hubbell 2002; Kitajima 2002; Wright 2002) and ontogenetic crossovers (e.g. species that are shade tolerant as juveniles becoming shade intolerant with age and vice versa; Grubb 1996; Svenning 2000). Because median CPIs for the seven studies cited by SG increase with duration of growth period, SG suggest simply that future studies should use longer durations. However, higher CPIs do not necessarily mean an increase in the frequency of rank reversals. If all species exhibit parallel shifts in RGR reaction norms to higher irradiance (perhaps caused by a growth-associated increase in respiratory tissue relative to photosynthetic leaf area), then median CPI will increase without an increase in the number of rank reversals. Median CPIs are not correlated with either cross-species correlation of RGRs, or fraction of CPI in the 2–10% range for the seven studies examined by SG (P = 0·47 and 0·38, respectively). Their suggestion that frequency of rank reversals increases with study duration is not supported statistically. The observed increase in scatter of CPI, on the other hand, may suggest an increase in measurement errors with experimental duration. Further, a proper test of the relationship between experimental duration and CPI distribution patterns should be done within the same study, rather than using means from studies that differ greatly in many other aspects of experimental design. A better title for SG's paper would be: 'Why don't young woody seedlings change their RGR ranks between high and low irradiance in short-term studies?', which better represents the physiological hypotheses that they propose: (1) seed mass is the main initial determinant of specific leaf area (SLA); (2) SLA determines RGR initially in both high and low irradiance; (3) SLA becomes independent of seed mass as seedlings grow; (4) SLA remains a main determinant of species differences in RGR in shade, but unit leaf rate (ULR) becomes the main determinant of species differences in RGR in sun; and (5) cross-species correlation between RGR at high and low irradiance is initially high, but weakens with seedling age. We agree these are reasonable hypotheses that deserve more explicit tests, but we would add some qualifiers. Cross-species correlation between SLA and seed mass is not necessarily caused by physical constraints, but may be largely caused by life-history correlates and phylogenetic inertia. High SLA for the first seedling leaf reflects not merely 'seed size-linked differences' (SG), but is better viewed as a physiological strategy to maximize initial RGR. Sack & Grubb (2001) emphasize that 'very short studies do not adequately represent the processes of long-term natural establishment'. We agree in principle. If some crossovers are undetectable because species differ little in their absolute growth in shade, longer studies may help detect species differences via increased signal-to-noise ratio. However, whether a short- or long-term study is appropriate depends on the research objective. If the main objective is to understand the physiological and morphological reasons why species differ in their survivorship during the first few months after establishment (Kitajima 1994), then a short-term study is adequate. Short-term growth analysis at standardized ontogenetic stages yields meaningful physiological information because the theoretical relationship RGR = LAR × ULR assumes short harvest intervals (Hunt 1982). Longer-term greenhouse studies also have many potential pitfalls. Longer-term studies may compare a fast-growing species at an advanced ontogenetic stage with a slower-growing species at an early ontogenetic stage. The roots of fast-growing species are also likely to experience pot limitation earlier than those of slower-growing species. Moreover, long-term greenhouse studies may not adequately predict field performance. Growth and survival in the field are influenced by numerous factors, including herbivory, pathogens, drought, etc. (Kitajima 1996; Dalling & Hubbell 2002; Montgomery & Chazdon 2002). We therefore propose that short- and long-term studies in greenhouse and field, respectively, should be viewed as complementary. Effects of pretreatment light conditions deserve even more attention than asserted by SG. Typically, plants are raised in moderately high light before transfer to shade. Acclimation from sun to shade often involves abscission of sun-phenotype leaves, which occurs rapidly for species with inherently faster growth rates and tissue turnover rates. These acclimation responses tend to penalize inherently fast-growing species and favour detection of crossovers, especially if the study is relatively short. Unless the intention of the experiment is to examine the responses of gap-established seedlings to canopy closure, use of different pretreatment light conditions should be avoided. Determining whether or not species switch performance ranks between contrasting environments is important in developing a mechanistic understanding of species coexistence. Although CPI is a biologically meaningful concept, it uses a null hypothesis that does not address the importance of rank reversals in maintaining coexistence. Furthermore, its application is questionable because of structural and measurement errors unless the shape of the reaction norm can be estimated reliably. Hence the design shortfall of measuring performance in an inadequate range of environments cannot be overcome by the CPI method applied to two-point reaction norms. It is important that experiments aiming to examine patterns of reaction norms should employ appropriate light treatments, and avoid artefacts from pretreatment conditions and ontogenetic heterogeneity. We thank Lourens Poorter for use of his data and discussion, Ethan Bolker and Ira Gessel for information on the probability distributions of rank reversals, and Craig Osenberg, Ed Tanner, Robert Dudley, Jim Dalling and Egbert Leigh for constructive comments.