• asymmetry;
  • competition indices;
  • Gompertz equation;
  • monocultures;
  • zone of influence


  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  • 1
    A dynamic competition kernel for a plant growing in the presence of a neighbour is a function describing how the competitive effect experienced by the plant depends on its own size, and the size of and distance to a neighbour. Competition experiments on Arabidopsis thaliana were used to derive dynamic competition kernels for this species.
  • 2
    The experiments entailed growing target plants in isolation, and with single neighbours of different relative sizes, and at different distances. Growth was determined from repeated measures of the area occupied by each plant. Target plants with smaller neighbours, or with neighbours that were further away, grew to a larger size than targets with larger, or closer, neighbours.
  • 3
    Relative growth rate (RGR) of isolated plants was best described by the Gompertz equation, a standard plant growth equation that reduces RGR in proportion to the logarithm of plant size.
  • 4
    Competition kernels were constructed by modifying the Gompertz equation to include the competitive effect of a neighbour. Alternative kernels (including the zone of influence model), with and without a parameter of asymmetry of competition, were constructed from observed growth rates of target plants with neighbours over short time intervals. The kernels were evaluated based on their ability to predict the size of targets and neighbours over a much longer time period.
  • 5
    The kernel that best described growth overall was a simple function proportional to the logarithm of neighbour size, and decreasing with distance to the neighbour. The zone of influence model was best able to describe the effects of distance on competition, but was relatively unsuccessful when the two plants differed substantially in size. Including a parameter for asymmetry did not provide any notable improvements in predictive ability, and in many cases made the predictions worse.


  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

Although even-aged monospecific stands of plants are the simplest plant populations, they exhibit some interesting behaviours. As the plants grow, the population develops a size hierarchy, with a few large and many small individuals (Hara 1988), and this pattern can become reversed, with the plants becoming more similar in size, following the onset of self-thinning (Weiner & Solbrig 1984). Monocultures also exhibit the law of constant final yield, the reciprocal yield law, and the −3/2 power law (Harper 1977, pp. 153–194; although see Weller 1987). Our understanding of the processes underlying these patterns has been greatly increased by considering plant competition and growth at the level of the individual, and the associated use of individual-based models (IBMs), where growth is modelled separately for each plant and the consequences examined at the population level (e.g. Aikman & Watkinson 1980; Weiner & Thomas 1986; Bonan 1988; Aikman 1992; Weiner et al. 2001). The literature contains both static and dynamic IBMs of monocultures, the former predicting size after a set period of time, the latter predicting growth rates continuously through time; both spatial and non-spatial models have been used (Hara 1988).

Dynamic spatial IBMs in particular have led to a better understanding of the factors that determine the formation of size hierarchies, including the degree of asymmetry of competition, the spatial arrangement of individual plants, variation in emergence time of seedlings, environmental heterogeneity and disturbance regime (Bonan 1988, 1991; Miller & Weiner 1989; Yastrebov 1996; Yokozawa et al. 1998; Weiner et al. 2001). However, at the heart of these models there is a problem: one must specify a set of rules that dictates how the growth of a plant depends on its size and on the size of and distance to its neighbours (a so-called competition kernel, Law et al. 2001). The problem is that little is known about what kind of function should be used (the forestry literature below, Benjamin & Sutherland 1992, Benjamin 1993 and Benjamin 1999 are exceptions). Because of this, competition kernels are usually based on assumptions about mechanisms underlying plant competition, or on local rules motivated by global patterns of growth (especially the reciprocal yield law, e.g. Weiner 1984). This has led to a wide variety of kernels (e.g. Ford & Diggle 1981; Weiner 1984; Bonan 1988; Miller & Weiner 1989; Yastrebov 1996; Yokozawa & Hara 1999), with little empirical evidence as to which are most appropriate. To improve understanding of growth in plant populations it would help to have competition kernels based on the empirical properties of the growth of competing plants.

A number of studies have fitted spatially dependent neighbourhood models to observed data, but the models in question have typically been static, using details of the local neighbourhood to predict the final plant size or seed set after an extended period of growth (Mack & Harper 1977; Waller 1981; Weiner 1982; Pacala & Silander 1985, 1990; Silander & Pacala 1985; Vandermeer 1986; Benjamin & Sutherland 1992; Bergelson 1993). Such kernels cannot be used to describe the growth of a plant continuously through time; to do this would require dynamic kernels that change in value as the plant and its neighbours grow. Those studies that have employed dynamic kernels have tended to specify a kernel first, and then to fit it to data, rather than to examine different kernel types (e.g. Weiner 1984; Weiner & Thomas 1986; Benjamin 1999). The forestry literature is an interesting exception to this rule because of the strong economic incentive foresters have in finding dynamic kernels that describe tree growth in the presence of competing neighbours; in this literature there are several studies comparing the predictive ability of different kernels on the same set of growth data from large stands of trees (e.g. Wagner & Radosevich 1991; Biging & Dobbertin 1992; Soares & Tomé 1999). A possible limitation to this approach is that, even if the rules underlying competition are simple, when applied to a large number of individuals the resulting behaviour may be too complex to give insights into the underlying rules (Firbank & Watkinson 1987; Bonan 1993). This may explain the fact that several studies in the forestry literature have found that spatial (distance-dependent) kernels are no better at predicting growth than non-spatial ones (e.g. Lorimer 1983; Martin & Ek 1984).

This paper reduces plant competition to its simplest elements by taking just two individuals at a time. Dynamic kernels are derived for Arabidopsis thaliana (L) Heynh. from experiments that varied the relative sizes, and dis-tance between, two competing plants. Our intention was to produce kernels simple enough to use in models of plant population dynamics, but capable of accurately predicting observed patterns of plant growth. For this reason, we did not consider kernels that require a com-plex algorithm to calculate their value, such as Thiessen polygon models (Hara 1988), although we did include a zone of influence (ZOI) model, in view of its interest as a mechanistic model of competition (Bonan 1988, 1991; Benjamin & Sutherland 1992; Czárán & Bartha 1992; Benjamin 1993, 1999; Weiner et al. 2001; Larocque 2002). Mechanistic models from the physiological literature describing growth at the level of the individual are rather too intricate for the purpose of this paper (e.g. Lemaire & Millard 1999) and our approach was therefore phenomenological at the level of physiology. We tested whether simple competition experiments can give insight into dynamic competition kernels, and whether simple kernels, based on modifications to the Gompertz equation, can provide an accurate description of the growth of two competing plants.

Materials and methods

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

Growth of a target and a neighbour individual of Arabidopsis was investigated in two experiments: (1) a size experiment in which the pairs of plants were set at the same distance apart, but at different relative sizes; and (2) a distance experiment in which the pairs of plants were set to similar initial size, but at different distances apart. Seed was Columbia ecotype, produced from populations grown in conditions similar to those under which the experiments were conducted.


Size experiment

One target plant and one neighbour plant were grown in each of 180 pots. All targets were sown on the same day (9 November 2000); neighbours were started at different times, giving 30 replicates in each of six treatments (9, 6 or 3 days before the target and 4, 7 or 10 days after the target). All pairs of plants were grown 5 mm apart, with the mid-point of the pair at the centre of the pot. Thirty control target plants (without neighbours) were also included. The pots were placed in a glasshouse under high-pressure sodium lamps on a cycle of 16 h light per day. The average daily temperature over the course of the experiment was 19.2 °C. The pots were arranged randomly, and were re-randomised five times during the course of the experiment to reduce the effects of any environmental gradients within the glasshouse. The experiment was terminated when the target plants were 37 days old.

Distance experiment

One target plant and one neighbour were grown in each of 150 pots. All plants were started on the same day (28 July 2000). Targets were planted in the centre of the pot and neighbours were grown at different distances from the target giving 30 replicates in each of 5 treatments (2.5, 5, 10, 20 or 40 mm). There was some variation in distance within treatments and, 5 days after sowing, the actual distance between the centre of the rosettes of the two plants in each pot was measured. At this time the mean distances for the treatments were 5.1, 7.1, 10.0, 20.1 and 40.7 mm; hereafter, distance refers to these measurements. Thirty control target plants were included, as in the previous experiment. The pots were placed in a glasshouse, with thermostat-controlled ventilation. No artificial lighting was used for this experiment. The average daily temperature in the glasshouse was 19.6 °C. The pots were arranged randomly, and were re-randomised five times during the course of the experiment. The experiment was terminated when the plants were 26 days old.


13 × 13 × 13 cm square pots were filled with Levington F2S compost. The surface was flattened and treated with imidacloprid, a residual insecticide. To reduce random variation in seedling size, for each desired plant a group of 5–10 seeds was sown at the same position. Three days after sowing, all but the largest seedling in each group was removed. The plants were kept well watered at all times, and treated with nicotine at the first sign of any insect herbivory.


Plant growth was followed by repeatedly measuring the area occupied by each plant, as a surrogate for plant weight. Seven measurements were taken from the size experiment, with a time interval of 3 days (starting on day 19, i.e. when the targets were 19 days old, i.e. at least 9 days after sowing their neighbours, and at least 6 days after thinning them out). Measurements were taken from the distance experiment 10, 12, 14, 17, 19, 24 and 26 days after sowing. The measurements were taken within the time period before obvious senescence began, the only exceptions being the neighbours sown earliest in the size experiment, and the largest plants in the distance experiment, which were beginning to senesce when the last measurements were taken.

The area occupied by plant i was defined as the area of the smallest rectangle that could contain the rosette, one side of the rectangle being parallel to the axis joining its centre to that of its neighbour, referred to hereafter as box-area, Wi (Fig. 1). For the measurement of control plants (without a neighbour), the axis joining the centre of the rosette to one (marked) corner of the pot was used. An experiment run in parallel to the size experiment showed that, for Arabidopsis, box area is linearly related to above-ground drymass (R2 = 0.91, Kirkby 2001). This relationship holds over most of the life cycle, but plants that are flowering heavily have higher dry masses than predicted by the box-area relationship (Kirkby 2001).


Figure 1. Plant size measured by box area. The rectangle (dashed line) has one axis parallel to the axis joining two plants (continuous line); the area of this rectangle is the box area, and is proportional to above-ground dry mass.

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  • Relative growth rate (RGR) of target plant i was obtained from logarithms of consecutive pairs of measurements of its box area

  • image(eqn 1)

where δt is the time interval between measurements (Hunt 1982, p. 18). We used a logarithmic transformation wi = ln(Wi/k); here k is a parameter with the same units as the box area Wi and can be thought of as a minimum plant size, such that Wi > k for all i; this makes wi dimensionless and greater than zero. We took k to be 1 mm2; seedlings were larger than this at the cotyledon stage. Table 1 summarizes the notation used in the analysis.

Table 1.  Notation used in the text
WiSize (box area) of plant i
wiScaled logarithm of size of plant i: wi = ln(Wi/k)
kMinimum plant size
RGRiRelative growth rate of plant i
F1(wi)Function for growth of an isolated plant i
α, βParameters for the Gompertz equation
F2(wi, wj, dij)Competition kernel (effect of neighbour j on target i)
dijDistance between i and j
xijResidual observed effect of j on short-term growth of i, after removing intrinsic growth of i
f(wi, wj) Size-dependent component of competitive effect, eqn 3–5
g(dij) Distant-dependent component of competitive effect, eqn 3–5
γg(dij) evaluated at a particular distance, eqn 3–5
Z(wj)Zone of influence of i, eqn 6
L(wi, wj) Fractional allocation of resources to i region of overlap, eqn 6
z(wi, wj, dij) Overlap in zones of influence of i and j, eqn 6
ρConstant for scaling box area to zone of influence, eqn 6
φParameter for asymmetry of competition
κParameter for effect of j on i when dij = 0
σParameter for attenuation of competitive effect with increasing dij

Fitting dynamic competition kernels to growth of Arabidopsis was done in three steps, first finding a function to describe growth of an isolated plant, second modifying the function to allow for effects of neighbours of different sizes and distances in the short term, and third evaluating how well the modified functions accounted for size of plants in the long term at the end of the experiments. Here we describe these steps in general; details of the kernels are given in the Analysis.

Growth of an isolated plant. We used regression analysis to investigate the relationship between RGR and w, testing a variety of two-parameter models. (This statistical analysis and subsequent ones were based on multiple measures of growth from individual plants; residuals are unlikely to be fully independent and the results of statistical tests should therefore be seen as no more than rough guides.) The analysis was done independently for control target plants in the size and distance experiments, each experiment thus returning a parameterized function F1(wi) for growth of an isolated plant. An appropriate function was found to be

  • F1(wi) = α − βwi(eqn 2)

(see Analysis); in mechanistic terms, α can be thought of as intrinsic growth due to the uptake of resources, and βwi as a loss due to metabolism which becomes greater as the plant becomes larger.

Growth in presence of a neighbour

In general, a competition kernel is some function F2(wi, wj, dij) size of the target plant wi, size of the neighbour wj and distance dij between them. Our main source of functions was the extensive literature on intraspecific competition in stands of trees (e.g. Biging & Dobbertin 1992; Stoll et al. 1994; Soares & Tomé 1999). This large set of functions (called competition indices in this literature) was reduced to only four, according to the need for mathematical simplicity and ecological realism. We also investigated a function based on overlapping ZOI of the target and its neighbour, because of the importance of ZOI models in the literature on model-ling growth in plant populations (e.g. Bonan 1988, 1991; Benjamin & Sutherland 1992; Benjamin 1993, 1999; Weiner et al. 2001; Larocque 2002). Although much less tractable mathematically, ZOI models appear better suited than other types of models to capture the fundamentals of the mechanisms of plant competition (Czárán & Bartha 1992).

We introduced competition kernels into eqn 2 in four ways:

  • RGRij = α − βwi − F2(wi, wj, dij)(eqn 3)
  • image(eqn 4)
  • image(eqn 5)
  • RGRij = α[1 − F2(wi, wj, dij)] − βwi(eqn 5)

the ij indexing of RGR makes explicit that target plant i was grown with a neighbour j. Equation 3 was chosen on the grounds of simplicity, and eqn 4 includes the kernel as a term in a hyperbolic function after Weiner (1984). Equation 5 is also hyperbolic, but the competitive effect acts only on the positive term in eqn 2; this is consistent with a resource-based interpretation of eqn 2, with the neighbour reducing only the availability of resources to the target, rather than also increasing the target's metabolic losses. Equation 6 was used to investigate ZOI functions, and like eqn 5 it assumes that a neighbour reduces resources available to the target. Equation 6 is appropriate for ZOI functions because total monopolization of resources by the neighbour corresponds to a kernel value of one, resulting in a rate of resource uptake by the target of zero.

The parameters α and β were estimated by independent measurements on isolated plants. Rearranging eqn 3–6 gives an ‘observed’ value for the neighbour's effect on the target after removing intrinsic growth of the target, respectively,

  • xij = α − βwi − RGRij(eqn 7)
  • image(eqn 8)
  • image(eqn 9)
  • image(eqn 10)

Competition kernels were fitted using the statistical model xij = F2(wi, wj, dij) + ɛij, minimizing the sum of squares Σ[F2(wi, wj, dij) − xij]2 of the error ɛij by means of non-linear regression. Because this analysis operated on residuals after eliminating intrinsic growth of the plants, the variation explained by the kernels (R2 values) was relatively small.

Evaluation of competition kernels

Although R2 values may be small, the cumulative effect of different kernels on plant size in the long term can still be large. As a further test of the kernels, we examined how well they could predict size of target plants at the end of each experiment. From its definition, RGR ≈ dw/dt, so growth of a target i and its neighbour j can be described by a coupled pair of differential equations. For instance, if growth has the dependencies shown in eqn 3, the equations are:

  • image(eqn 11)
  • image(eqn 12)

Similar coupled equations can be constructed for eqn 4–6. For a given kernel and given initial values of wi and wj, we used numerical integration to predict the size of targets and neighbours at the end of each experiment. Initial box areas were set at 2 mm2 for seedlings at age 1.78 days in the size experiments, and at age 1.94 days in the distance experiments, because eqn 2 then gave the correct final size of control (i.e. isolated) plants. The fit between observed O{Wk} and predicted E{Wk} final sizes was calculated as inline image Following Biging & Dobbertin (1992) the predictive ability of each function was quantified by expressing the fit as a percentage of the MSE of a standard model, such as the isolated growth model (eqn 2) in which the neighbour is ignored; low values of percentage MSE thus indicate a large improvement over the standard model.


  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

Target plants with neighbours that were sown later, i.e. with neighbours that were smaller throughout the experiment, grew to a larger size than targets with neighbours that were sown earlier (Fig. 2a, one-way anova, F = 642.9, degrees freedom (error) = 164, P < 0.001). In the treatment with the neighbours that were sown latest, targets still grew to a much smaller size than control plants (Fig. 2a), even though the neighbours were considerably smaller than the targets throughout the experiment.


Figure 2. Mean size of target plants (±1 standard error). (a) Size experiment: target size on day 37 vs. neighbour size on day 19. (b) Distance experiment: target size on day 26 vs. distance to neighbour. Filled circles (labelled C) are control plants with no neighbour.

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Target plants with neighbours at a larger distance tended to grow to a larger final size than those with closer neighbours (Fig. 2b, one-way anova, F = 5.3, degrees freedom (error) = 130, P = 0.001).


Growth of an isolated plant

Growth of control target plants (i.e. isolated plants) was close to, but slightly less than, exponential (Fig. 3a). Figures 3b and c show that RGRi was negatively correlated with wi in the control target plants from both the size and the distance experiment (Spearman rank correlation, rs = –0.790, –0.715, respectively, P < 0.001 for each). Of the two-parameter models tested (including logarithmic, exponential, inverse and power functions), a linear relationship between RGRi and wi (eqn 2) gave the best fit for both experiments. For the distance experiment, plants larger than 2900 mm2 were not used in this regression, since there was some indication that these plants were starting to senesce (Fig. 3c). The estimated parameter values were greater in the distance experiment (α = 0.686, β = 0.0698, R2 = 0.704) than in the size experiment (α = 0.508, β = 0.0420, R2 = 0.603), presumably because of differences in light levels in the glasshouse at different times of the year.


Figure 3. Growth of isolated plants. (a) Mean target size (±1 standard error) vs. time since sowing, using control plants from the size experiment (a straight line on this plot indicates exponential growth); (b) and (c) RGR vs. plant size, using control plants from the size experiment and distance experiment, respectively; lines obtained by linear regression using eqn 2.

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Because RGR ≈ dwi/dt, the relationship between RGR and wi (eqn 2) is in effect a differential equation for plant growth. Integrated over time, it gives the Gompertz equation, describing sigmoidal growth to an equilibrium plant size ŵi = α/β[or Ŵi= exp(α/β)] (Hunt 1982; page 128). The ratio α/β was lower for control target plants from the distance experiment (1.9 × 104 mm2) than for control target plants from the size experiment (1.8 × 105 mm2), suggesting that the former plants were approaching a lower equilibrium size. However, this should be treated with caution, because Wi is sensitive to small changes in α and β, and the plants may undergo ontogenetic changes (specifically, diverting resources to seed production) before equilibrium is reached. Nevertheless, the plants from the distance experiment were smaller at time of flowering than those from the size experiment.


With the parameters α and β as estimated above on isolated (control) plants from the size experiment, the mean value of the ‘observed’ competitive effect xij was significantly greater than zero in eqn 7–10 (respectively, mean: 0.028, 0.490, 0.08, 0.056; standard error: 0.002, 0.05, 0.005, 0.004; P < 0.001 for each assuming normality). Furthermore, xij was positively correlated with neighbour size, indicating that larger neighbours had a greater competitive effect (for eqn 7–10, respectively: rs = 0.150, 0.167, 0.150, 0.150, n = 1014, P≤ 0.001).

This non-independence of residuals left after eliminating effects of intrinsic growth justifies a more detailed search for functions describing how neighbours affect targets. In the size experiment, because the distance between targets and neighbours was fixed, the non-ZOI kernels could be simplified to

  • F2(wi, wj, dij) = γf(wi, wj),(eqn 13)

where γ is a constant, confining attention to functions describing size effects alone. For the ZOI model the effects of size and distance cannot be separated, because they interact non-linearly to determine the region of overlap between the two ZOIs, z(wi, wj, dij) (see below).

Four explicit functions were used for f(wi, wj) as listed (A–D) in Table 2. Each function was tested in eqn 3–5, giving 12 functions in all. In addition, a ZOI kernel was used in eqn 6; this function was

Table 2.  Tests of the effect of plant size on growth of target and neighbour plants. The tests use non-linear regression on calculated values of competitive effect xij (eqn 7–10) in the size experiment. A–D are functions defined at the end of the Table; D is an APA (area potentially available) function of Soares & Tomé (1999); ρ is a parameter used in the ZOI function, not estimated in the regression. The term γ is a parameter estimated by regression; its standard error is also given. R2 measures the proportion of variation in xij explained by the kernel. The percentage MSE measures how close the predicted and observed final plant sizes are, relative to the sizes predicted from growth of an isolated plant (eqn 2); lower percentage MSE values indicate a better predictive ability
Function γStandard errorR2Percentage MSE (× 10−6)
From eqn 3: α − βwi  γf(wi, wj)A0.004380.000300.0229.07
From eqn 4:
  • image
From eqn 5:
  • image
ZOI from eqn 6: α(1 − F2 βwjρ    
Definitions of f(wi, wj):A wi C wi, wj  
B wi, wj D wj/(wi, wj)  
  • image(eqn 14)

where the area of the ZOI of individual i, Z(wi), is proportional to the area of the largest circle that could be placed inside a square of area equal to i's box area:

  • image(eqn 15)

The constant ρ scales the relation between box area and ZOI and four values (0.5, 1, 1.5, 2; Table 2) were used in fitting the kernel. The function z(wi, wj, dij) is the area of overlap of the ZOIs of i and j, determined by the rules of trigonometry (details not shown), and L(wi, wj) describes the fractional allocation of resources to j in this region of overlap

  • image(eqn 16)

Initially, we set φ = 1 corresponding to perfect size symmetry, where contested resources are divided between competitors in proportion to their sizes (Schwinning & Weiner 1998); effects of asymmetric competition are considered below.

Regression analysis thus involved fitting a single parameter γ for each function. Because the analysis was based on residuals after removing effects of intrinsic growth of targets, R2 values were relatively small (Table 2). The analysis shows the highest R2 values came from models of the form of eqn 5, suggesting that eqn 5 is the most appropriate way to modify the Gompertz equation to include the competitive effect of a neighbour.

To evaluate the predictive power of the functions in the long term, we compared the plant sizes observed at the end of the experiment with plant sizes predicted by numerical integration using the functions (see for example eqn 11 and 12). Regardless of the choice of function, incorporating the effect of the neighbour improved the prediction of final size by several orders of magnitude as measured by the small percentage MSE values (Table 2, Fig. 4). Fits to the observed data were good, bearing in mind that the equations contain only three parameters (only one more than the equation for isolated growth), and that over the time period in question plant size increased by a factor of 2000–4000. Equation 5 gave percentage MSE values somewhat lower than eqn 3, 4 and 6, as can be seen from visual inspection of the graphs (Fig. 4a). Equation 5 also gave better predictions for the neighbour sizes than did eqn 3 and 4 (Table 2, Fig. 4b). Equation 6 was insensitive to the scaling from box area to ZOI (parameter ρ), and was relatively unsuccessful in predicting target size unless the target and neighbour plants were similar in size (Fig. 4a). None of the equations gave good predictions for neighbours sown earliest, probably because these plants were senescing. The results suggest a ranking Equation 5, 3, 6, 4 in terms of their success incorporating competition into the Gompertz equation.


Figure 4. (a) Final target plant size, and (b) final neighbour size in the size experiment: observations (±1 standard error); predicted for an isolated plant, from numerical integration of the isolated growth model (IGM, eqn 2); and predicted from numerical integration of coupled equations with different competition kernels. The functions shown are those carried over to the analysis of the distance experiment and, except for the ZOI model, do not contain an asymmetry parameter. Functions A and B are as defined in Table 2. The ZOI model shown is from Table 3, with ρ = 1.5. Neighbour sowing dates are relative to the target plant, negative numbers indicating neighbours sown before the target.

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To test for asymmetric competition between plants of different sizes, we introduced a second parameter φ, replacing w with wφ in the functions (the ZOI model already included the asymmetry parameter φ, see eqn 16). Effectively a value of φ > 1 increases the differential in size between competing plants, making the interaction more asymmetric, whereas φ < 1 reduces the size differential and makes the interaction less asymmetric. The regression analysis was repeated with two parameters: γ and φ (Table 3).

Table 3.  Tests of asymmetric competition between target and neighbour plants. Regression analysis is the same as in Table 2 except that an asymmetry parameter φ is estimated as well as γ. The percentage MSE in this case measures how close the predicted and observed final plant sizes are in the two-parameter models, relative to the one-parameter models in Table 2; values < 100% (shown in boldface) indicate that the asymmetry parameter improves the prediction
 γStandard error γφStandard error φR2Percentage MSE γ only
From eqn 3:
From eqn 4:
From eqn 5:
From eqn 6:ρ      

The extra parameter φ did little to improve the fit. In eqn 3–5, the estimated value of φ was most often greater than one, implying that competition was more asymmetric than was implied by the functions without φ. In the ZOI model eqn 6, it is striking how close to zero estimates of φ were, implying competition close to complete symmetry, where contested resources are divided equally between competitors regardless of size (Schwinning & Weiner 1998). The percentage MSE values show that prediction of final size of the plants was not improved by allowing for asymmetry in eqn 3–5: often asymmetry actually made the prediction worse, and in view of this we assume φ = 1 when using these equations below. However, the asymmetry parameter did provide some improvement in the fit of the ZOI model (Table 3), so we retain the extra parameter φ in this case. Nevertheless, even with the inclusion of the asymmetry parameter φ, the predictions of final size from the ZOI are still inferior to the non-ZOI functions with symmetric competition (Fig. 4).

On the basis of these results, we carry forward the following five equations for analysis of the distance experiment:

  • image(eqn 17)
  • image(eqn 18)
  • RGRij = α−βwi−γ(wj/wi)(eqn 19)
  • RGRij = α−βwi−γwj(eqn 20)
  • image(eqn 21)

Equations 17 and 18 are from eqn 5 with competition functions B and A, respectively (Table 2); eqn 19 and 20 are from eqn 3 with functions B and A, respectively (Table 2); eqn 21 is the ZOI model as described by eqn 14–16.

The R2 and percentage MSE values indicate that the best equation overall was eqn 17. Equation 18 is also kept, as it contains a particularly simple form of the function f(wi, wj) and, at the same time, gave quite accurate predictions. We retain eqn 19 and 20 because these gave predictions not greatly inferior to eqn 17 and 18 and include the effect of neighbours in an especially simple way. Lastly we keep the ZOI model eqn 21, in view of its interest in plant ecology.


We now use growth of pairs of plants in the distance experiment to find functions that describe the effect of spatial separation of the target and neighbour on growth of the target. Once again, we computed the residual xij left after removing the effect of intrinsic growth of the plants: Equation 9 (corresponding to eqn 17 and 18), eqn 7 (corresponding to eqn 19 and 20), and eqn 10 (corresponding to eqn 21). Values of xij were negatively correlated with dij, irrespective of which equation was used (rs = −0.092 and P = 0.006 for each). Because the distance between the target and its neighbour was the main remaining variable, there is a justification for a more detailed search for functions that can best describe the effect of this distance.

To deal with distance in eqn 17–20, the parameter γ was replaced by functions of distance g(d)ij, giving kernels of the form

  • F2(wi, wj, dij) = g(d)ijf(wi, wj);(eqn 22)

four simple decay functions for g(d)ij were used to describe this dependence (Table 4). The g(d)ij functions had two parameters: the maximum deleterious effect κ of a neighbour on the flux in size of the target when the plants are at the same location (dij = 0), and the distance σ at which the competitive effect is half of the maximum value. These functions were applied to each of eqn 17–20 giving 16 functions as listed in Table 4. This separation of effects of distance and size was not possible in the ZOI function, eqn 21; here the two parameters γ and φ were retained from the size experiment and, as before, we examined four values of the constant ρ in the ZOI function.

Table 4.  Tests of some functions to describe the effect of neighbour distance on growth of target plants. The tests use non-linear regression on calculated values of the competitive effect of a neighbour, xij. Functions describing size dependence from the size experiment are given by the right-hand sides of eqn 17–20. Functions describing distance dependence are given by A–D at the end of the Table; these replace γ in eqn 17–20. Two parameters κ and s are estimated; SEκ and SEσ are standard errors of the estimates. Equation 21 is the ZOI model, with parameters γ and φ estimated as for the size experiment. R2 measures the proportion of the variation in xij explained by the kernel. The percentage MSE measures the deviation between the observed final plant size and the size predicted from the kernel, as described in the text; lower percentage MSE values indicate a better predictive ability
g(dij): κSEκσ (mm)SEσR2Percentage MSE
From eqn 17:
 A0.07640.01028.912.1 0.01311.23
 B0.06440.006438.4 9.5 0.01111.41
 C0.09210.02117.011.0 0.01411.05
 D0.06600.007236.412.0 0.01111.05
From eqn 18:
 A0.01300.001831.313.1 0.01211.82
 B0.01100.003940.210.0 0.01111.95
 C0.01570.003418.311.5 0.01311.62
 D0.01120.001239.513.0 0.01111.94
From eqn 19:
 A0.03070.006133.822.9 0.004 8.95
 B0.02690.003939.114.5 0.004 9.00
 C0.03290.009029.429.7 0.004 8.94
 D0.02720.004339.620.0 0.004 9.01
From eqn 20:
 A0.005800.001133.319.8 0.015 9.18
 B0.004980.0006839.913.4 0.015 9.26
 C0.006470.001724.621.6 0.015 9.12
 D0.005040.0007440.918.6 0.015 9.30
From eqn 21:
 0.50.2060.034 0.509 0.77−0.00210.33
 1.00.1490.021 0.576 0.68 0.010 9.17
 1.50.1310.018 0.452 0.59 0.017 8.86
 2.00.1200.016 0.361 0.56 0.017 8.86
Definitions of g(dij):
A: Exponentialinline image C: Hyperbolic Iκ[1 + (dij/σ)]−1   
B: Gaussianinline image D: Hyperbolic IIκ[1 + (dij2)]−1   

We carried out a non-linear regression analysis to fit the two parameters of the functions (Table 4). As before, the R2 values were rather small, because we were dealing with the residuals after removing the effect of intrinsic growth of targets. For the predictions of growth in the long term, each function provided a substantial improvement over the isolated growth model (eqn 2), although the percentage MSEs remained much larger than those from the size experiment (Table 2). These larger percentage MSE values are not surprising, because there were more sources of variation in the distance experiment (wi, wj, dij varied with treatment and time), and the observed mean target sizes did not increase monotonically with dij (Fig. 5). The values for R2 and percentage MSE were largely independent of the function chosen for g(d)ij (Table 4).


Figure 5. Final target plant size in the distance experiment: observations (±1 standard error); predicted for an isolated plant from the numerical integration of the isolated growth model (IGM, eqn 2); and predicted from numerical integration of coupled equations with different competition kernels (for the non-ZOI kernels, the results shown are from using ‘hyperbolic I’ for g(dij), Table 4). Distance refers to the distance from the target to the neighbour.

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The results show that the largest R2 values, and the greatest improvement over the isolated growth model (lowest percentage MSEs) was given by the ZOI model (eqn 21) with scaling parameter ρ = 1.5 (Table 4). The fitted value for the asymmetry parameter φ was close to 0.5, in contrast to the φ-values close to zero for the ZOI model fitted to the size experiment (Table 3): apparently, differences in the conditions in the glasshouse at different times of year induced differences in the asymmetry of competition. The best non-ZOI function was eqn 21, which gave R2 and percentage MSE values close to the ZOI model (Table 3). The predictions from the different functions for the final size of a target grown with a neighbour at distance zero ranged from 1724 to 2465 mm2 (results not shown). These values are reasonable, given that isolated plants grew to 3380 mm2 in the same period (Fig. 2). [A prediction for final target size with a neighbour at distance zero of much less than half the size of an isolated plant of the same age would correspond to overcompensating density dependence, which has rarely been observed in plant populations (Silvertown & Lovett-Doust 1993).]


  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

The analysis presented here shows how dynamic competition kernels, consistent with observed patterns of plant growth, can be derived from simple experiments. With the exception of the ZOI model, the kernels are novel: evidently the experimental approach can reveal new rules governing plant growth. We do not claim that the form of the functions, or the parameter values derived from the experiments, will hold in general: kernels are bound to be conditional on the morphology and physiology of the particular species involved. However, we do suggest that functions derived empirically would help understanding of plant competition and its effects on population dynamics.


The results of fitting different functions to the size experiment and the distance experiment did not indicate a single best function: for the size experiment, the best function was eqn 17, whereas for the distance experiment, the ZOI model eqn 21 gave the best results. However, the size experiment showed that the ZOI model was only appropriate when the competing plants were similar in size (as was the case in the distance experiment). In contrast, the simplest of all of the functions examined here, eqn 21 with γ replaced by a distance function g(d)ij, gave reasonable predictions for both the size experiment and the distance experiment, even though it was not the best function in either case. Based on the performance of this function in the two experiments, and its mathematical simplicity, a model of growth of the following form would be a good compromise for Arabidopsis:

  • image(eqn 23)

in which the dynamic competition kernel F2(wi, wj, dij) is given by g(dij)wj. Here g(dij) could take a number of different forms such that the effect of the neighbour attenuates with distance, because Table 4 indicates that the exact choice of function is not critical.

Equation 23 extends readily to an arbitrary number of neighbours

  • image(eqn 24)

There are two important assumptions to bear in mind about this extension. First, it assumes that neighbours act additively on target growth; this is a non-trivial matter that would need to be tested with experiments with three or more competing plants. Second, it ignores the angular dispersion of neighbours, which has been shown to affect the final target size (Bergelson 1993; Kirkby 2001). However, this long-term effect could result simply from the compounding of short-term competition between the neighbours themselves (if the neighbours are near to each other, they become smaller through time, and therefore affect the target less): it is not known whether the angular dispersion of neighbours affects the way their competitive effects on the target combine in the short term. We remind the reader also that the model deals with a period of growth before effects of senescence become evident (in practise one might expect growth to be much reduced when resource allocation switches to reproduction, and to stop altogether at a size smaller than that given by the equilibrium point of the kernel), and that the variable wi is the logarithm of box area, box area being proportional to above-ground dry mass.


Compared to the other competition kernels examined here, the ZOI model is easier to relate to the mechanisms underlying plant competition, because it is based on an explicit consideration of the uptake of resources in space (Czárán & Bartha 1992). A further advantage is that the ZOI model extends naturally to competition among more than two individuals. This is because its most basic assumption is that contested resources (i.e. resources that fall within the ZOI of more than one individual) are shared according to some rule, which does not depend on the number of individuals competing for the resources. The overlaps become quite intricate to calculate with more than two individuals, and need to be obtained algorithmically in discrete space (Weiner et al. 2001).

However, we found that the ZOI model, in its currently accepted form, only worked well as a description of competition between two plants when they were similar in size. A possible problem with the ZOI model is its assumption that resources are taken up evenly within a ZOI centred on each plant. In reality, the distribution of tissue across a plant's ZOI is unlikely to be uniform and, in a rosette-forming species like Arabidopsis, the density of tissue most likely decreases near the boundary. Functions that allow the effect of neighbours to attenuate with distance might be expected to work better in these circumstances.


The log transformed plant size emerges from the data as a natural state variable, and gives the growth functions some simple properties. For example, a constant positive dwi/dt corresponds to exponential growth of Wi, and a constant negative dwi/dt corresponds to exponential decay. Because dwi/dt is equivalent to RGRi, any kernel built around RGR will share these properties (e.g. Weiner 1984). For consistency with the log transformed size of target plants, we made the competitive effect of a neighbour dependent on wj rather than Wj (Table 2, functions A–D). To our knowledge, functions based on the logarithm of neighbour size have not been considered in the literature before, and the predictive ability of the kernels tested here suggests that this approach is worth investigating. An experimental study of the relationship between neighbour size and target plant growth in Kochia scoparia also found that the logarithm of plant size was the best predictor of competitive effect (J. Weiner, pers. comm.).


The degree of asymmetry of competition is a major theme running through the ecological literature on monocultures (Weiner 1990; Schwinning & Fox 1995; Schwinning & Weiner 1998), and the kernels used in modelling monocultures have usually included an asymmetry parameter (Weiner & Thomas 1986; Bonan 1988, 1991; Miller & Weiner 1989; Yastrebov 1996; Yokozawa et al. 1998; Yokozawa & Hara 1999). In the size experiment, pairs of plants that differed greatly in size grew in competition with each other, but in general the inclusion of asymmetry parameters did not improve the fit (expect for the ZOI, but this gave poor predictions for the size in any case; Table 3). This is not to say that competition was symmetric: all other things being equal, a model that uses logarithm of plant size reduces the effect of large plants relative to small ones. Rather, the message from the size experiment is that, more important than asymmetry parameters, first, how different kernels included the competitive effect of neighbours (i.e. the choice from eqn 3–6), and second, how the competitive effect depended on the neighbour and target sizes (i.e. A–D in Table 2). However, the degree of asymmetry in competition will no doubt depend on environmental conditions (Schwinning & Weiner 1998), and in some cases the asymmetry of competition may be critical in determining growth patterns in populations (Weiner et al. 2001).


The bottom-up approach used here has suggested some novel kernels consistent with competition between pairs of plants. It has yet to be seen whether such kernels lead to behaviour of populations consistent with reality, such as the law of constant final yield (Kira et al. 1953). An appropriate model should recover such population behaviour which would also help in discriminating between potential functions. The ideal model of growth of populations of plants would be built upon a competition function that both describes growth correctly at the level of individual plants, and also recreates the population-level behaviours seen in real populations. To find such a model requires a combination of theoretical and empirical work, at the level of both the individual and the population, and has the potential to increase greatly our understanding of plant population dynamics.


  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

We thank L.D. Llambi, D.J. Murrell and J. Pitchford for discussion and comments on earlier manuscripts; and NERC and the Andrew W. Mellon Foundation for funding (DWP).


  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
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