Functional Ecology 2 Asymmetric between plant species Blackwell Science, Ltd R. P. FRECKLETON* and A. R. WATKINSON *Department of Zoology, University of Oxford, Oxford OX 3PS, UK, and Schools of Environmental and Biological Sciences, University of East Anglia, Norwich NR4 7TJ, UK Summary. Asymmetric is an unequal division of resources amongst competing plants. Thus, may be asymmetric in the sense that some individuals remove a disproportionately large amount of resource. Alternatively, may be asymmetric in that one species removes a disproportionately large amount of resource. The mechanisms determining the two forms of asymmetry may be similar, for example through initial size advantage or over-topping. 2. We explore the consequences of these two forms of asymmetry for models that predict mean performance as a function of the density of interacting species. We do so using neighbourhood models that explicitly consider the allocation of resources to individuals within an interacting mixture. 3. Asymmetric individual is modelled by assuming that individuals are formed into a competitive hierarchy such that individuals at the top of the hierarchy are able to remove more resources than those at the bottom. Mean performance declines exponentially, moving from top to bottom of the hierarchy. Asymmetric species-level is modelled by assuming that one species occupies all of the upper positions in the competitive hierarchy and hence dominates the resource. 4. When is asymmetric at the species level, yield density responses follow an exponential decline. Otherwise, arithmetic mean performance follows a classic hyperbolic response. 5. Using this approach, we explore the asymmetry of between wheat and three species of weeds. Key-words: Contest, non-linear model, maximum likelihood, resource, scramble Functional Ecology (2) Ecological Society Introduction The outcome of in mixtures of plant species within a community will be determined by a variety of processes, including the spatial distribution of individuals; the resources being competed for; and the ability of the species to compete for these resources. In single-species stands, a variety of studies have emphasized the importance of variation in competitive ability and resource acquisition at the level of the individual (Hara 984a; Hara 984b; Firbank & Watkinson 985a; Weiner 986; Firbank & Watkinson 987; Weiner & Thomas 986; Pacala & Weiner 99; Hara & Wyszomirski 994; Nagashima et al. 995). In particular, these studies have concentrated on how variability in individual growth rates affects size hierarchy formation and the response of mean performance to changing density. Author to whom correspondence should be addressed. E-mail: robert.freckleton@zoology.oxford.ac.uk In studying within monocultures, it has been found useful to distinguish between two forms of (Weiner 988): symmetric is regarded as a sharing of resources amongst individuals, whilst asymmetric is an unequal sharing of resources as a consequence of larger individuals having a competitive advantage over smaller ones. Asymmetric may arise, for example, as a consequence of variation in emergence times within a population, with those plants emerging first gaining an advantage over later-emerging ones (Ross & Harper 972). The degree to which the outcome of is either symmetric or asymmetric plays a fundamental role in determining the strength of the effects of increasing population density and shape of the response curve (Watkinson 98; Firbank & Watkinson 985a). This form of asymmetric may be viewed as a competitive hierarchy. Individuals at the top of the hierarchy (for example, those plants that emerge first) obtain the most resources, are affected little by from individuals lower in the 65
66 R. P. Freckleton & A. R. Watkinson hierarchy, and hence grow largest. Individuals lower in the hierarchy grow smaller as they have access to fewer resources than those at the top of the hierarchy. In the extreme case of asymmetric hierarchy formation, individuals are affected only by from those at higher positions in the hierarchy, and are unaffected by those lower down. This form of has been explored in a number of models for in single-species populations. (Firbank & Watkinson 985a; Pacala & Weiner 99; Hara & Wyszomirski 994). In the same way that plants within monocultures form competitive hierarchies (Ross & Harper 972; Weiner & Solbrig 984), it would be expected that in mixtures of species, is not equal for all members of the interacting populations. Furthermore, species differ in their ability to capture resources. Watkinson (985), for example, refers to data from Butcher (983) on between varieties of peas and Avena fatua. In that study, it was found that the form of the frequency distributions of individual biomass between and within species depended on which variety of pea competed with the Avena: in one case, the frequency distributions of the two species were very similar, whilst in another case the largest plants were all Avena, indicating that the two species were organized in a competitive hierarchy. Similarly, Weiner (985) found that in mixtures simultaneously determined the distribution of sizes of individuals within species, as well as the distribution of biomass across species. Work that has manipulated the relative emergence times of competing species (Kropff & Spitters 99; Kropff & Spitters 992; Connolly & Wayne 996) shows that the degree to which one species may be able to overtop another, and thus dominate light resources, may be influenced by relative emergence time, and that this affects the relative amounts of resource captured by each species. Furthermore, asymmetric may interact with the spatial distribution of the interacting species in determining mean performance (Weiner et al. 2). In defining asymmetric between species, it will be useful to distinguish two components of asymmetry of resource capture. In addition to the division of resource amongst individuals, in species mixtures it is also necessary to consider the division of resources between the species, and the degree to which one species or the other is able to pre-empt resources. To date, these processes have not been separated in studying asymmetric between plant species. Most studies that have considered asymmetric between species have considered just onesided, where one species is completely dominant over another (Crawley & May 987; Rees & Long 992). Alternatively, models have simply considered differences between resource capture at the species level, but ignore individual-level asymmetric (Kropff & Spitters 99; Kropff & Spitters 992; Reynolds & Pacala 993; Benjamin & Aikman 995; Rees & Bergelson 997). The aim of this paper is to present and test a simple model that incorporates and contrasts these two components of asymmetry, and to consider the implications of interspecific competitive asymmetry for yield density responses in two-species mixtures. Materials and methods NEIGHBOURHOOD MODELS INCLUDING ASYMMETRY The model we analyse is a simple model of resource based on the simulation of Firbank & Watkinson (985a).The model is formulated in the following way:. The model considers two species, x and y, the densities of which are denoted by N x and N y individuals of each species, respectively. 2. Individuals are located at discrete points in space and remove resources from spatially restricted neighbourhoods. The neighbourhoods of individual plants of species x and y are of area q x and q y, respectively. For simplicity we assume that the neighbourhoods are circular, although we could assume that the neighbourhoods are of any shape; the key assumption is that the neighbourhoods are spatially restricted. 3. Individuals remove resources from their neighbourhood. The amount of resource removed determines the size of the plant. Adjacent plants compete for resources when their neighbourhoods overlap. We incorporate three rules for determining how resources are allocated between individuals that imply different degrees of asymmetry of resource capture. These are illustrated schematically in Fig.. 4. In the first case, resources are shared evenly between individuals (Fig. a). Hence if N individuals overlap an area of habitat, a fraction /N of the resource contained within this area is allocated to each individual. We term this symmetric, as this corresponds to the mechanism of symmetric in single-species stands. 5. The second case assumes that between species is asymmetric, but that neither species is able to pre-empt the resource (Fig. b). (i) The individuals of the two species are organized into a linear hierarchy. There are to N positions in the hierarchy, where N is the total number of individuals of the two species. Individuals are assigned to positions in the hierarchy with individuals removing resources in the order in which they are assigned to the hierarchy. (ii) The hierarchy is randomly assembled such that if there are N x and N y individuals of species x and y, respectively, the probability of a given position being occupied by species x is N x (N x + N y ). (iii) Each individual of x or y successively removes a proportion, d x or d y, respectively,
67 Asymmetric between plant species (a) (b) case of light extinction within a canopy. We term this asymmetric hierarchy formation. The model is solved analytically to predict the expected mean weight of an individual of species x interacting with N x and N y, other individuals of species x and y, respectively. Fig.. Schematic diagram illustrating the three modes of employed in the modelling. For the sake of illustration it is assumed that the two species (differentiated by shading) are competing for light, such that taller individuals are able remove more resources and are unaffected by the presence of smaller plants. (a) Symmetric : all individuals are able to remove the same amount of resource and no individual achieves competitive dominance. The effects of are thus more-or-less equal for all individuals, and resources are simply shared amongst competing individuals. (b) Asymmetric individual-level : individuals vary in competitive ability, with the result that some individuals are able to remove more resources than others. No one species is on average competitively superior to the other. (c) Asymmetric between species: one species is able to dominate the resource, and all individuals of this species are able to pre-empt resources and make them unavailable to the second species. of the resource from its neighbourhood. Competition is asymmetric as the performance of an individual at a given position in the hierarchy is affected only by those individuals above in the hierarchy, and not by those below. There is an exponential decline in expected performance moving from the top of the hierarchy to the bottom, and the relative variance in performance increases with increasing density. We term this asymmetric individual-level. 6. The third case assumes that one of the species is able to pre-empt the resource (Fig. c). In particular, species y is competitively dominant and, if there are N y individuals of species y, then the top N y positions in the hierarchy are occupied by species y, and lower positions in the interspecific hierarchy are occupied by species x. At the level of the individual, is again asymmetric, with each individual of x or y successively removing a proportion, d x or d y, respectively, of the resource from its neighbourhood. This value is set at a constant for each species irrespective of position in the hierarchy. This implies that although the proportion of resources removed from an individual s neighbourhood does not vary with rank, the net amount of resources removed will decline as an exponential function of the position of an individual, for example, as in the (c) FIELD DATA We compared the model predictions with field data on between winter wheat and three species of arable weed: Galium aparine, Anisantha (= Bromus) sterilis and Papaver rhoeas. A detailed description of the experiment is given elsewhere (Lintell Smith et al. 999). The experiment consisted of 48 3 3 m plots marked out in an area of field (36 48 m) that had been ploughed and rolled prior to the start of the experiment. Plots were separated by a 3 m discard area. The field was drilled with wheat at a depth of 4 cm at a density of 37 seeds m 2 on 23 October 99, following roterra cultivation to 6 cm depth. Three replicates of each of eight weed treatments (all three species, all pairwise combinations, each species alone and weed-free) were sown at two nitrogen levels (24 and 2 kg ha ) and laid out in a fully randomized design. Each species was sown at a density of 5 seeds m 2. Weeds were allowed to set seed at the end of each season. These germinated to form the weed population in the next season. The experiment was repeated using the same protocol in 99, 99 and 992. The data analysed are the yields of wheat recorded from within small (2 2 cm) quadrats taken within the main experimental plots. All above-ground biomass of plants was removed from these areas, and the number of weeds recorded. The wheat plants were dried at 7 C for 48 h, and the total dry weight of these was recorded. As these data are taken from small neighbourhoods (an average of 8 wheat plants per quadrat), they are ideal for comparison with the prediction of the model, which similarly considers between plants within small neighbourhoods. We used non-linear regression analysis to fit models that predict the yield of wheat as a function of the combined density of surviving weeds. We fitted a model of the form = y m f(n). The yield of wheat is related to y m, the yield of wheat in the absence of, and f(n), a function that predicts the reduction in yield owing to from the weeds. The particular forms of f used were generated from the solutions to the model (below). We used a maximum-likelihood approach to fit the models. Exploratory analysis indicated that model fits were extremely sensitive to a small proportion of residual values. We therefore fit the model using a maximum-likelihood approach assuming a Cauchy distribution of error, which is less sensitive to outliers and changes in residual variance than other distributions (Hilborn & Mangel 997).
68 R. P. Freckleton & A. R. Watkinson Data were logarithmically transformed prior to analysis and the likelihood was numerically maximized using a Rosenbrock pattern search (Rosenbrock 96). As wheat was grown under weed-free conditions in each year of the experiment, we were able to estimate y m independently through calculating the average yield of the weed free plots. These estimates were used as hard parameter estimates, and the non-linear fitting procedure was used to estimate the parameters of the function. Results MODEL ANALYSIS In general terms, the model may be solved by expressing mean performance in terms of the probability of an individual occupying a given position of the combined hierarchy and its expected performance at that position. The model may be expressed in the following form, where E[w x ] is the expected mean weight of a target individual of species x interacting with N x and N y neighbours of species x and y, respectively: Silander 985; Pacala & Weiner 99). Therefore the form of relationship between performance and density predicted at the individual level by equation will be identical to the form of this relationship at the population level. SYMMETRIC COMPETITION BETWEEN INDIVIDUALS When between individuals is symmetric (Fig. a), expected performance is the same irrespective of which position in the combined hierarchy the individual occupies, and each individual has the same probability of occupying each position in the hierarchy. Thus, p x (k) = p y (k) = ( + N x + N y ) and f(k) = ( + A x N x + A y N y ) for all positions in the hierarchy, assuming that individuals are approximately randomly distributed at the scale of local neighbourhoods (Firbank & Watkinson 985a). Hence the function is: N x + N y Ew [ x ] = ( -------------------------------- ------------------------------------------- + N x + N y ) ( + A x N x + A y N y ) eqn 2 N x + N y Ew [ x ] = pk ( )fa ( x N x,a y N y k), eqn = ----------------------------------------. + A x N x + A y N y eqn 3 where p(k) is the probability of occupying position k of the to N x + N y positions in the combined hierarchy of the two species. is the expected weight of an isolated individual in the absence of ; f(a x N x,a y N y k) is the reduction in mean performance experienced by an individual at position k in the hierarchy, where A x and A y are the average proportions of the neighbourhood of the target plant overlapped by any neighbour of x and y, respectively. The form of f will depend on the degree to which neighbours of both species remove resources and hence make them available to other individuals, as well the organization of the competitive hierarchy, both intra- and interspecifically, as shown schematically in Fig.. In terms of the implementation of the model as a series of overlapping circular neighbourhoods, the function f is a composite of the removal function (whether individuals share resources, or a proportion d x or d y of resources is removed by each individual from its neighbourhood) and the formation of the hierarchy (the number of dominant individuals overlapping the neighbourhood of a target individual and hence removing resources). Equation predicts the expected performance of a single target individual with given numbers of intraand interspecific competitors. To develop a full neighbourhood model to predict the mean weight of an individual within a stand, it would be necessary to model the probability distribution of the numbers of intra- and interspecific (Pacala & Silander 985). In general, however, the behaviour-of-neighbourhood models are essentially direct functions of the model assumed for mean individual-level effects (Pacala & Mean weight thus declines as a hyperbolic function of the density of both species. This is the familiar form of response of mean size to density (Firbank & Watkinson 985b). ASYMMETRIC COMPETITION BETWEEN INDIVIDUALS When between individuals is asymmetric, the function f() has to consider both the probability that a neighbour occupying a given position is of one species or the other, and the amount of resources removed by neighbours at each position. Thus at position i in the hierarchy, the amount of resource removed by species x is the probability that the position is occupied by species x (N x p x, where p x is the probability that the position is occupied by a given individual of species x, as above) multiplied by the resource removed by an individual, ( A x d x ). Hence f (k) has the following form, where p x and p y are, as above, the probabilities that a given individual of x or y, respectively, will occupy a given position in the hierarchy: k fk ( ) = [ N x p x () i ( A x d x ) + N y p y () i ( A y d y )]. eqn 4 i= In equation 4, the competitive effect is calculated across the k dominant competitors in the hierarchy above the target individual. The values of p x and p y determine whether resource pre-emption by one or other of the species occurs through determining whether one species is more likely to occupy the higher positions in the hierarchy.
69 Asymmetric between plant species SYMMETRIC COMPETITION BETWEEN SPECIES; ASYMMETRIC COMPETITION BETWEEN INDIVIDUALS When interspecific hierarchies are formed randomly, that is, there is no asymmetry for resource access between species, a target plant has an equal probability of occupying any position in the hierarchy. Hence p x (k) = p y (k) = ( + N x + N y ) for all k positions in the combined hierarchy. Equation 4 therefore becomes: Mean weight (log scale) Exponential model Hyperbolic model k N x ( A x d x ) + N y ( A y d y ) fk ( ) = ------------------------------------------------------------------ + N x + N y i= ( = N x( A x d x ) + N y ( A y d y )) k --------------------------------------------------------------------------. ( + N x + N y ) k eqn 5 Density (log scale) Fig. 2. Contrasting density responses: the exponential function (equations and ) and hyperbolic model (equations 3 and 7) predicting the mean performance of species x as a function of species y, plotted on a doublelogarithmic scale. Substituting this into equation then yields: Ew [ x ] = ------------------------- + N x + N y N x + N y ( N x ( A x d x ) + N y ( A y d y )) k ---------------------------------------------------------------------- ( + N x + N y ) k ( A y d y ) N y k N y i = ( A x d x ) fk ( ) = -------------------------------------------------------------. + N x eqn 8 = --------------------------- + N x + N y + + ( N x ( A x d x ) + N y ( A y d y )) N x N y ( + N x + N y ) N x N y [ ] ---------------------------------------------------------------------------------------------------------------------------------. ( N x ( A x d x ) + N y ( A y d y ))( + N x + N y ) eqn 6 By evaluating this equation in the limits that N x and N y, it is possible to approximate equation 6 by: Ew [ ] = -----------------------------------------------, ( + α xx N x + α xy N y ) eqn 7 where α xx = A x d x and α xy = A y d y. In this case, therefore, the form of is of the same form as predicted by equation 3. The difference between the model for symmetric and this model for asymmetric, however, is that there is considerably more variance in performance in the model for asymmetric, as equation 5 predicts an exponential decline in performance moving down from the top of the competitive hierarchy. This variance forms the basis for diagnosing the form of interactions at the individual level from field data (below). ASYMMETRIC COMPETITION BETWEEN SPECIES ( + + ) When species y is able to pre-empt resources through domination of the interspecific hierarchy, then for the first k =... N y positions in the combined hierarchy, p y (k) = N y and p x (k) =. Then for the k = N y +... + N x + N y, p y (k) = and p x (k) = ( + N x ). Hence, for the first k =... N y positions in the combined hierarchy, f(k) = ( A y d y ) k, and for the k = N y +... + N x + N y other positions, Substituting these expressions into equation then yields: ( A y d y ) N y ( Ew [ x ] ----------------------------------- A xd x ) ( A x d x ) + N x ( ) = -------------------------------------------------------------------------. + N x A x d x This then may be approximated by the model: exp( βn y ) Ew [ x ] = ------------------------------, + α xx N x eqn 9 eqn where β = ln( A y d y ). The important difference between this equation and the forms of yield density responses predicted by equations 3 and 7 is that in equation the mean weight of species x declines exponentially as the density of species y increases. The yield density response predicted by equation is therefore very different from that predicted under the other forms of (Fig. 2). Under the hyperbolic model, log mean weight declines linearly with increasing log density at high densities. In contrast, under the exponential model, the rate of decline in log weight with increasing density is proportional to density. Since the sensitivity of the model (the rate of change in log weight with log density) to changing the density of species y is very much higher than that for changing the density of species x, mean performance according to equation is much more sensitive to changing the fraction of species y in the community. Note that when hierarchies are formed asymmetrically such that the combined hierarchy is always dominated by species y, the yield density relationship is always of the form of equation, irrespective of the form of symmetry of between individuals of species x.
62 R. P. Freckleton & A. R. Watkinson DIAGNOSIS The aim is to test data in order to distinguish between that is asymmetric at the individual level, that is, in terms of the removal of resources by individual plants, and asymmetry in terms of resource pre-emption by one species or another. The distinction between asymmetric in terms of hierarchy domination by one of the species, and symmetric where interspecific hierarchies are randomly formed, is straightforward as the yield density responses predicted by the models for these two forms of are very different. Specifically, as shown in Fig. 2, a plot of log mean plant weight versus log density should reveal clear differences in response. Distinguishing between symmetric and asymmetric at the level of individual plants, assuming that hierarchies are randomly formed, requires further analysis of the model. Specifically, when between individuals is asymmetric, the relative variance (a) in mean performance should increase dramatically as density increases. This is characteristic of asymmetric in single-species populations. A simple way to analyse this behaviour is to look at geometric mean performance. The geometric mean is always smaller than the arithmetic mean, the difference between the two being a function of the relative variability of the data. Specifically, if the variance in size changes systematically with density, then the geometric mean will respond differently from the arithmetic mean to changing density. Under symmetric, the difference between the predictions of the two means will be minimal, as there is no assumed mechanism for generating variance as a function of competitive intensity. By contrast, for the asymmetric model without resource pre-emption by one of the species, the geometric mean should differ considerably from arithmetic mean performance. Specifically, geometric mean performance may be predicted by modifying the general form of model (equation ) to consider log performance. In the case of asymmetric without resource pre-emption, the particular form (equation 6) is modified to: N x + N y E[ logw x ] = -------------------------- k log[n x ( A x d x ) + N x + N y + N y ( A y d y )] k log[ + N x + N y ] = ----------------------- + N x + N y (log[ N x ( A x d x ) + N y ( A y d y )] N x + N y log[ + N x + N y ]) k Yield of wheat (g m 2 ) (b) Density of weeds (m 2 ) Fig. 2 3. Yield density British responses in mixtures of winter wheat and three species of weeds Ecological (see text for Society, details). The curves show the best-fit exponential and hyperbolic models Functional (parameters Ecology, in Table ). (a) Raw data; (b) smoothed response, calculated from a 5, running 65 623 geometric mean of the ordered data. = ----- ( N 2 x + N y )(log[ N x ( A x d x ) + N y ( A y d y )] log[ + N x + N y ]). Hence geometric mean performance is given by: N x ( A x d x ) + N y ( A y d y ) GM( w x ) = w m ------------------------------------------------------------------ + N x + N y eqn This again is an exponential yield density relationship that contrasts with the hyperbolic relationship between arithmetic mean performance and density. The difference between the response of log AM and log GM performance to increasing density should therefore measure the degree of asymmetry of performance at the individual level. ANALYSIS OF FIELD DATA -- ( N 2 x + N y ) Figure 3a shows the relationship between yield and total weed density. There were no clear differences between the yield density responses with the different.
62 Table. Fits of models to the data presented in Fig. 3a. The functions Asymmetric fitted were either the exponential or hyperbolic models, n = 2 in both cases. In both cases the estimate of y m was obtained from data on plants grown in the absence of weeds (n = 2). The model-fitting procedure is described in the text between plant species Model Parameter estimate (± SE) Log likelihood Exponential y = y m exp( an) 2885 335 25 6 Hyperbolic y = y m /( + an) 25 64 25 95 y m 697 5 42 4 Biomass of species x Density of species y Fig. 4. Simulated yield density responses for comparison with Fig. 2. The simulation results were derived from a spatially explicit realization of the model described in the text. Plants were allocated to random positions within a habitat area of 2 2 units. Neighbourhoods of both species were 3 square units in area. Species x was sown at a constant density of plants; the density of species y varied from to plants, with replicates at each density. The competitive hierarchy was formed randomly such that individuals could occupy any position within the combined hierarchy of the two species. The arithmetic mean yield density response for this model is predicted to follow a hyperbolic yield density model (equation 7), whereas, the geometric mean or smoothed responses are predicted to follow an exponential response (equation 7). (a) Raw results; (b) smoothed response based on a running mean of the ordered data. (a) (b) weed combinations or clear effects of nitrogen (Lintell Smith et al. 999). Hence we do not consider these differences further, but analyse the data as a function of total weed density. Wheat yield is not a simple function of weed density (Fig. 3a), and visually does not appear to correspond well to either of the forms shown in Fig. 2. Whilst at low densities there is little response of wheat yield to weed density, at high weed densities the response is highly variable. Notably, however, the variance in performance increases as density increases. One consequence of this is that at intermediate weed densities some plots yield mean plant sizes as high as those at lower densities, whereas other plots yield plants less than % of the size of those at lower densities. Table summarizes the analysis of the raw data. The exponential model fit the data better, as indicated by the value of the log-likelihood function. This improvement of fit is minimal, however, and it is clear from Fig. 3a that neither model describes the data entirely satisfactorily, and there are clear systematic deviations in both cases. In order to look at trends in geometric mean yield density responses we generated smoothed responses. Smoothing is often used, for example, in time-series analysis in order to damp local stochastic variation with the aim of discerning long-term trends. By analogy, we employed smoothing in order to remove some of the variation about the mean response for the data in Fig. 3a, and hence to determine which function underlies the yield density response. We calculated running means for successive observations of the ordered data (Fig. 3b). Presented in this way, it is clear that the yield density response is essentially intermediate between the two forms of model. Except at the highest densities, the correspondence between the exponential model and the smoothed data is considerably better than that of the hyperbolic model. Our interpretation of the yield density response is therefore that is asymmetric at the level of individual plants, but that neither one species nor the other completely dominates the combined competitive hierarchy of the two species. This interpretation is reinforced by the results of an explicit simulation of the model where is asymmetric, but without pre-emption of resources by either of the species (Fig. 4). Although the arithmetic mean response follows the hyperbolic model (equation 7), the geometric mean response follows the exponential model (equation ). Both the raw simulation results (Fig. 4a) and the smoothed response (Fig. 4b) show remarkable qualitative similarity to the original data (Fig. 3). At a high densities in both the data and simulation, mean weight may vary by two orders of magnitude or more. The main difference is that there appear to be fewer points in the very top right of the response observed in the data (Fig. 3a) than the simulation (Fig. 4a). This difference probably relates to the incomplete dominance postulated above. Presumably in reality the hierarchy is not completely dominated by the weed species, but contains an intermediate area of overlap where either species may occur. This could be modelled, for example, by using a continuous switching function to model asymmetric species (Freckleton 997).
622 R. P. Freckleton & A. R. Watkinson Discussion The notion of competitive asymmetry in plants has generally been applied to within singlespecies stands, and has only rarely been extended to explore between species (Weiner 985; Schwinning & Fox 995; Connolly & Wayne 996; Weiner et al. 2). In contrast, animal ecologists have tended to use the term asymmetric specifically in relation to between pairs of species, without reference to the individuals that are competing (Lawton & Hassell 98; Calow 998). Here we have highlighted how asymmetric between species should be considered as having two components: the asymmetry of between individual plants; and the asymmetry of at the level of the species. The models and data we present demonstrate that these effects can have dramatic impacts on yield density responses, and are readily detected under field conditions. The asymmetry of between individuals in mixed-species stands is basically the same in nature as asymmetric between individuals in single-species stands. This asymmetry is modelled phenomenologically by generating an asymmetric division of resources amongst individuals. This is because those individuals at the top of a competitive hierarchy, such as those that emerge first, are able to remove a disproportionately large amount of resource, for example through size advantage (Weiner 988). More generally, this asymmetry results from individual-level variations in resource capture resulting from, for example, variations in initial emergence. Asymmetric between species results from the differential ability of the species to be able to occupy higher positions in the competitive hierarchy. This may result, for example, from height differences between species with one species being able to completely over-top another and hence pre-empt access to light. The determinants of this competitive asymmetry may be similar to those that determine competitive asymmetry at the level of individuals. The difference is that whereas competitive asymmetry at the level of individuals results from variance among individuals, asymmetry in between species results from mean differences between species. The approach we have taken to explore the consequences of asymmetric is set within the framework of model yield density responses. The advantage of this approach is that such models allow predictions of yields ( Weiner et al. 2), as well as allowing the population- and community-level impacts of asymmetric on performance to be modelled (Schwinning & Fox 995). Several studies have explored how the parameters of single-species yield density models are affected by varying the symmetry of (Firbank & Watkinson 985a; Pacala & Weiner 99; Hara & Wyszomirski 994; Freckleton 997). In single-species stands the degree of asymmetry determines how a fixed amount of resource is allocated amongst competing individuals. The impact of changing the degree of asymmetry on model parameters basically depends on the nature of resource use at the individual level (Firbank & Watkinson 985a). Under asymmetric, the yield density response always follows the hyperbolic form defined above. When is symmetric, however, underor over-compensating yield density responses may be predicted to occur, depending on the efficiency with which individuals convert resources into biomass (Firbank & Watkinson 985a; Freckleton 997). The impacts of symmetric on yield density responses in plants are thus fundamentally different from those predicted for animal populations (Royama 992) where collapsing density responses result from the inability of individuals to survive below some threshold level of resource acquisition. Plants generally do not show such thresholds for survival or reproduction (Rees & Crawley 989), and hence the consequences of symmetric for yield density responses in plant monocultures are quite different. In contrast to the predictions for single-species stands, our models suggest that in mixtures of species, the form of yield density responses may be quite profoundly changed by altering the degree of competitive asymmetry (Fig. 2). When was asymmetric at the level of individual plants, but the combined hierarchy was formed at random such that neither species asymmetrically dominated, mean performance was predicted to follow the hyperbolic model (equation 7), whereas geometric mean performance followed the exponential model (equation ). This distinction is important if we are interested in predicting the effects of on long-term dynamics, as the model predictions are isotropic only on the logarithmic scale. An isotropic distribution implies that the weight of a randomly chosen individual is likely to be bigger or smaller than the average with equal probability (Lande 998). The predictions of the geometric mean model are isotropic as the distribution of plant sizes (which decline exponentially moving from the top of the hierarchy to the bottom) is linear on the logarithmic scale. Hence, on the logarithmic scale 5% of individuals will be smaller than average, and 5% will be larger. On the arithmetic scale most individuals will be smaller than the average. The predictions of the geometric mean model (equation ) may thus be more relevant to understanding the impacts of asymmetric on interspecific interactions for either species. One consequence of this may be to generate founder effects in community dynamics, resulting from the disproportionately intense impacts of at high densities under the exponential model. Although founder effects have been postulated to arise through asymmetric between species (Reynolds & Pacala 993; Rees & Bergelson 997), the mechanism postulated here is very different, resulting from variance in performance at the individual level.
623 Asymmetric between plant species In conclusion, we present models and data that demonstrate important impacts of the asymmetry of between individuals and species on the form of density response. Particularly if we look at geometric mean performance, the consequences of asymmetric at the level of individuals may be important, irrespective of whether one species or the other tends to dominate access to resources. As a wide body of evidence has shown asymmetric to be characteristic of competitive hierarchies in single species, asymmetric between species may be an important but largely overlooked phenomenon. Acknowledgements We would like to thank Richard Law and Jake Weiner for extensive discussion of this work, and two anonymous referees for helpful comments. Also many thanks to Graham Hopkins for providing the impetus for this work. References Benjamin, L.R. & Aikman, D.P. (995) Predicting growth in stands of mixed species from that in individual species. Annals of Botany 76, 3 42. 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