1 Asymmetric competition is an unequal division of resources amongst competing plants. Thus, competition may be asymmetric in the sense that some individuals remove a disproportionately large amount of resource. Alternatively, competition 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 competition 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 competition 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 competition is modelled by assuming that one species occupies all of the upper positions in the competitive hierarchy and hence dominates the resource.
4 When competition 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 competition between wheat and three species of weeds.
In studying competition within monocultures, it has been found useful to distinguish between two forms of competition (Weiner 1988): symmetric competition is regarded as a sharing of resources amongst individuals, whilst asymmetric competition is an unequal sharing of resources as a consequence of larger individuals having a competitive advantage over smaller ones. Asymmetric competition 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 1972). The degree to which the outcome of competition 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 1980; Firbank & Watkinson 1985a).
This form of asymmetric competition 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 competition from individuals lower in the 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 competition from those at higher positions in the hierarchy, and are unaffected by those lower down. This form of competition has been explored in a number of models for competition in single-species populations. (Firbank & Watkinson 1985a; Pacala & Weiner 1991; Hara & Wyszomirski 1994).
In the same way that plants within monocultures form competitive hierarchies (Ross & Harper 1972; Weiner & Solbrig 1984), it would be expected that in mixtures of species, competition is not equal for all members of the interacting populations. Furthermore, species differ in their ability to capture resources. Watkinson (1985), for example, refers to data from Butcher (1983) on competition 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 (1985) found that competition 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 1991; Kropff & Spitters 1992; Connolly & Wayne 1996) 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 competition may interact with the spatial distribution of the interacting species in determining mean performance (Weiner et al. 2001).
In defining asymmetric competition 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 competition between plant species. Most studies that have considered asymmetric competition between species have considered just one-sided competition, where one species is completely dominant over another (Crawley & May 1987; Rees & Long 1992). Alternatively, models have simply considered differences between resource capture at the species level, but ignore individual-level asymmetric competition (Kropff & Spitters 1991; Kropff & Spitters 1992; Reynolds & Pacala 1993; Benjamin & Aikman 1995; Rees & Bergelson 1997). 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 competition based on the simulation of Firbank & Watkinson (1985a).The model is formulated in the following way:
1The model considers two species, x and y, the densities of which are denoted by Nx and Ny individuals of each species, respectively.
2Individuals 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 qx and qy, 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.
3Individuals 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. 1.
4In the first case, resources are shared evenly between individuals (Fig. 1a). Hence if N individuals overlap an area of habitat, a fraction 1/N of the resource contained within this area is allocated to each individual. We term this symmetric competition, as this corresponds to the mechanism of symmetric competition in single-species stands.
5The second case assumes that competition between species is asymmetric, but that neither species is able to pre-empt the resource (Fig. 1b). (i) The individuals of the two species are organized into a linear hierarchy. There are 1 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 Nx and Ny individuals of species x and y, respectively, the probability of a given position being occupied by species x is Nx(Nx + Ny)−1. (iii) Each individual of x or y successively removes a proportion, dx or dy, respectively, 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 competition.
6The third case assumes that one of the species is able to pre-empt the resource (Fig. 1c). In particular, species y is competitively dominant and, if there are Ny individuals of species y, then the top Ny 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, competition is again asymmetric, with each individual of x or y successively removing a proportion, dx or dy, 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 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 Nx and Ny, other individuals of species x and y, respectively.
We compared the model predictions with field data on competition 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. 1999). 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 370 seeds m−2 on 23 October 1990, 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 (240 and 120 kg ha−1) and laid out in a fully randomized design. Each species was sown at a density of 50 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 1990, 1991 and 1992.
The data analysed are the yields of wheat recorded from within small (20 × 20 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 70 °C for 48 h, and the total dry weight of these was recorded. As these data are taken from small neighbourhoods (an average of 10·8 wheat plants per quadrat), they are ideal for comparison with the prediction of the model, which similarly considers competition 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¯ = ymf(N).
The yield of wheat is related to ym, the yield of wheat in the absence of competition, and f(N), a function that predicts the reduction in yield owing to competition from the weeds. The particular forms of f used were generated from the solutions to the competition 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 1997). Data were logarithmically transformed prior to analysis and the likelihood was numerically maximized using a Rosenbrock pattern search (Rosenbrock 1960). As wheat was grown under weed-free conditions in each year of the experiment, we were able to estimate ym 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 competition function.
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[wx] is the expected mean weight of a target individual of species x interacting with Nx and Ny neighbours of species x and y, respectively:
where p(k) is the probability of occupying position k of the 0 to Nx + Ny positions in the combined hierarchy of the two species. wm is the expected weight of an isolated individual in the absence of competition; f(AxNx,AyNy?k) is the reduction in mean performance experienced by an individual at position k in the hierarchy, where Ax and Ay 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. 1. In terms of the implementation of the model as a series of overlapping circular neighbourhoods, the competition function f is a composite of the removal function (whether individuals share resources, or a proportion dx or dy 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 1 predicts the expected performance of a single target individual with given numbers of intra- and 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 1985). In general, however, the behaviour-of-neighbourhood models are essentially direct functions of the model assumed for mean individual-level effects (Pacala & Silander 1985; Pacala & Weiner 1991). Therefore the form of relationship between performance and density predicted at the individual level by equation 1 will be identical to the form of this relationship at the population level.
Symmetric competition between individuals
When competition between individuals is symmetric (Fig. 1a), 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, px(k) = py(k) = (1 + Nx + Ny)−1 and f(k) = (1 + AxNx + AyNy)−1 for all positions in the hierarchy, assuming that individuals are approximately randomly distributed at the scale of local neighbourhoods (Firbank & Watkinson 1985a). Hence the competition function is:
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 1985b).
Asymmetric competition between individuals
When competition between individuals is asymmetric, the competition 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 (Nxpx, where px 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, (1 − Axdx). Hence f(k) has the following form, where px and py are, as above, the probabilities that a given individual of x or y, respectively, will occupy a given position in the hierarchy:
In equation 4, the competitive effect is calculated across the k dominant competitors in the hierarchy above the target individual. The values of px and py 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.
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 px(k) = py(k) = (1 + Nx + Ny)−1 for all k positions in the combined hierarchy. Equation 4 therefore becomes:
Substituting this into equation 1 then yields:
By evaluating this equation in the limits that Nx → ∞ and Ny → ∞, it is possible to approximate equation 6 by:
where αxx = Axdx and αxy = Aydy. In this case, therefore, the form of competition is of the same form as predicted by equation 3. The difference between the model for symmetric competition and this model for asymmetric competition, however, is that there is considerably more variance in performance in the model for asymmetric competition, 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 = 1 … Ny positions in the combined hierarchy, py(k) = Ny−1 and px(k) = 0. Then for the k = Ny + 1 … 1 + Nx + Ny, py(k) = 0 and px(k) = (1 + Nx)−1. Hence, for the first k = 1 …Ny positions in the combined hierarchy, f(k) = (1 −Aydy)k, and for the k = Ny + 1 … 1 + Nx + Ny other positions,
Substituting these expressions into equation 1 then yields:
This then may be approximated by the model:
where β = ln(1 − Aydy). The important difference between this equation and the forms of yield–density responses predicted by equations 3 and 7 is that in equation 10 the mean weight of species x declines exponentially as the density of species y increases. The yield–density response predicted by equation 10 is therefore very different from that predicted under the other forms of competition (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 10 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 10, irrespective of the form of symmetry of competition between individuals of species x.
The aim is to test data in order to distinguish between competition 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 competition in terms of hierarchy domination by one of the species, and symmetric competition where interspecific hierarchies are randomly formed, is straightforward as the yield–density responses predicted by the models for these two forms of competition 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 competition at the level of individual plants, assuming that hierarchies are randomly formed, requires further analysis of the model. Specifically, when competition between individuals is asymmetric, the relative variance in mean performance should increase dramatically as density increases. This is characteristic of asymmetric competition 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 competition, 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 1) to consider log performance. In the case of asymmetric competition without resource pre-emption, the particular form (equation 6) is modified to:
Hence geometric mean performance is given by:
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
Figure 3a shows the relationship between yield and total weed density. There were no clear differences between the yield–density responses with the different weed combinations or clear effects of nitrogen (Lintell Smith et al. 1999). 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 1% of the size of those at lower densities.
Table 1 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.
Table 1. Fits of competition models to the data presented in Fig. 3a. The functions fitted were either the exponential or hyperbolic models, n = 112 in both cases. In both cases the estimate of ym was obtained from data on plants grown in the absence of weeds (n = 20). The model-fitting procedure is described in the text
Parameter estimate (± SE)
y = ym exp(−aN)
y = ym/(1 + aN)
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 competition 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 competition 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 11). 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 competition (Freckleton 1997).
The notion of competitive asymmetry in plants has generally been applied to competition within single-species stands, and has only rarely been extended to explore competition between species (Weiner 1985; Schwinning & Fox 1995; Connolly & Wayne 1996; Weiner et al. 2001). In contrast, animal ecologists have tended to use the term asymmetric competition specifically in relation to competition between pairs of species, without reference to the individuals that are competing (Lawton & Hassell 1981; Calow 1998). Here we have highlighted how asymmetric competition between species should be considered as having two components: the asymmetry of competition between individual plants; and the asymmetry of competition 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 competition between individuals in mixed-species stands is basically the same in nature as asymmetric competition 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 1988). More generally, this asymmetry results from individual-level variations in resource capture resulting from, for example, variations in initial emergence.
Asymmetric competition 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 competition between species results from mean differences between species.
The approach we have taken to explore the consequences of asymmetric competition 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. 2001), as well as allowing the population- and community-level impacts of asymmetric competition on performance to be modelled (Schwinning & Fox 1995). Several studies have explored how the parameters of single-species yield–density models are affected by varying the symmetry of competition (Firbank & Watkinson 1985a; Pacala & Weiner 1991; Hara & Wyszomirski 1994; Freckleton 1997). 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 1985a). Under asymmetric competition, the yield–density response always follows the hyperbolic form defined above. When competition is symmetric, however, under- or over-compensating yield–density responses may be predicted to occur, depending on the efficiency with which individuals convert resources into biomass (Firbank & Watkinson 1985a; Freckleton 1997). The impacts of symmetric competition on yield–density responses in plants are thus fundamentally different from those predicted for animal populations (Royama 1992) where ‘collapsing’ competition–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 1989), and hence the consequences of symmetric competition 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 competition 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 11). This distinction is important if we are interested in predicting the effects of competition 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 1998). 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 50% of individuals will be smaller than average, and 50% will be larger. On the arithmetic scale most individuals will be smaller than the average. The predictions of the geometric mean model (equation 11) may thus be more relevant to understanding the impacts of asymmetric competition 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 competition at high densities under the exponential model. Although founder effects have been postulated to arise through asymmetric competition between species (Reynolds & Pacala 1993; Rees & Bergelson 1997), the mechanism postulated here is very different, resulting from variance in performance at the individual level.
In conclusion, we present models and data that demonstrate important impacts of the asymmetry of competition between individuals and species on the form of competition density response. Particularly if we look at geometric mean performance, the consequences of asymmetric competition 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 competition to be characteristic of competitive hierarchies in single species, asymmetric competition between species may be an important – but largely overlooked – phenomenon.
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.
Received 12 January 2001; revised 28 April 2001; accepted 30 April 2001