Pattern and process: competition causes regular spacing of individuals within plant populations

Authors


Peter Stoll, Department of Integrative Biology, Section of Conservation Biology, University of Basel, St Johanns-Vorstadt 10, CH-4056 Basel, Switzerland (tel. +41 61 267 08 56; fax +41 61 267 08 32; e-mail peter.stoll@unibas.ch).

Summary

  • 1We used simulated and experimental plant populations to analyse mortality-driven pattern formation under size-dependent competition. Larger plants had an advantage under size-asymmetric but not under symmetric competition. Initial patterns were random or clumped.
  • 2The simulations were individual-based and spatially explicit. Size-dependent competition was modelled with different rules to partition overlapping zones of influence.
  • 3The experiment used genotypes of Arabidopsis thaliana with different morphological plasticity and hence size-dependent competition. Compared with wild types, transgenic individuals over-expressed phytochrome A and had decreased plasticity because of disabled phytochrome-mediated shade avoidance. Therefore, competition among transgenics was more asymmetric compared with wild-types.
  • 4Density-dependent mortality under symmetric competition did not substantially change the initial spatial pattern. Conversely, simulations under asymmetric competition and experimental patterns of transgenic over-expressors showed patterns of survivors that deviated substantially from random mortality independent of initial patterns.
  • 5Small-scale initial patterns of wild types were regular rather than random or clumped. We hypothesize that this small-scale regularity may be explained by early shade avoidance of seedlings in their cotyledon stage.
  • 6Our experimental results support predictions from an individual-based simulation model and support the conclusion that regular spatial patterns of surviving individuals should be interpreted as evidence for strong, asymmetric competitive interactions and subsequent density-dependent mortality.

Introduction

Spatial patterns of individuals within populations are closely linked to ecological processes. Consequently, ecological processes may be deduced from spatial patterns (Watt 1947; Greig-Smith 1979; Leps 1990). A classical example comes from resource competition in plants and other sessile organisms where interactions are inherently local and thus primarily among nearest neighbours (Harper 1977). Therefore, an individual's performance (i.e. growth, survival and fecundity) may be determined by number of neighbours and distance to them. However, the numbers and distances to neighbours change substantially during stand development because of density-dependent mortality (Antonovics & Levin 1980). If competition is mainly for light and therefore one-sided or asymmetric (Ford & Newbould 1970; Cannell et al. 1984; Hara 1988; Weiner 1990; Ford & Sorrensen 1992; Cannell & Grace 1993; Schwinning & Fox 1995), strong local regular patterns of surviving individuals develop from initially random or clumped patterns. There is general agreement that such pattern formation is driven by resource competition and subsequent density-dependent mortality (Newman 1973; Ford 1975; Ford & Diggle 1981; Watkinson et al. 1983; Hughes 1988; Chapin et al. 1989; Kenkel 1988; Powell 1990; Kenkel et al. 1997).

However, phenotypic plasticity (Bradshaw 1965), particularly in terms of morphology, i.e. the ability of individuals to respond to neighbours by increased height growth and decreased branching, may reduce the asymmetry of competition for light (Schwinning & Weiner 1998; Stoll et al. 2002). Collectively known as shade avoidance syndrome (SAS), much work has focused on the specific mechanisms (Furuya 1993) and adaptive significance (Schmitt et al. 1995) of SAS (reviewed in Ballaré 1999). In brief, as plants grow and start to compete they alter light quantity and quality, in particular the red to far-red ratio (R : FR) such that crowded conditions can be perceived by individuals through their phytochrome photoreceptor system. Moreover, neighbouring plants also alter the spectral light quality of reflected light such that morphological responses have been observed even before shading actually occurred (Ballaréet al. 1987; Ballaréet al. 1990; Schmitt & Wulff 1993). Once close neighbours are perceived, the SAS allows small and suppressed individuals to avoid or at least delay shading and therefore asymmetric competition from taller neighbours. Thus, increased height growth enables shaded individuals to place at least some of their leaves in upper canopy layers and thus higher light levels. While the SAS has received considerable attention at the level of individuals, its population-level implications have rarely been tested (Ballaré & Scopel 1997). One exception is a self-thinning experiment that used the same Arabidopsis genotypes (see below) as in the current experiment and showed that competition among individuals with disabled shade avoidance is more asymmetric than among wild-type individuals (Stoll et al. 2002). The introduced gene that disables shade avoidance leads to reduced hypocotyl elongation responses to density, as well as increased size variability and density-dependent mortality.

The main objective of our study was to analyse pattern formation in the context of size-dependent competition and morphological plasticity. We used an individual-based, spatially explicit simulation model (DeAngelis & Gross 1992) in combination with an experimental approach. In the simulation model, size-dependent competition was modelled using a dynamic ‘zone of influence’ for individual plants (Weiner et al. 2001). In the experiment, we used wild-type and transgenic phytochrome A over-expressors of Arabidopsis thaliana (L.). In the over-expressors, the normal shade avoidance response to low R : FR ratios was counteracted by over-expression of an oat phytochrome A leading to persistently high levels of phytochrome A (Boylan & Quail 1991; Whitelam et al. 1992; Whitelam et al. 1998). As a consequence, the transgenic Arabidopsis were ‘phytochrome blind’ and unable to detect neighbours. Therefore, we compared pattern formation in simulated populations under symmetric and asymmetric competition with pattern formation in experimental populations composed of symmetric (wild-type) and asymmetric (over-expressor) competing individuals. We used combined count-distance analysis (Ripley 1981) to quantify the completely mapped spatial patterns of individuals before and after density-dependent mortality. If pattern formation is indeed mainly driven by asymmetric competition for light, we expected a pronounced formation of regular patterns under asymmetric but not under symmetric competition. Because combined count-distance analysis revealed small-scale regularity of initial patterns of wild-types, we further analysed the experimental patterns using distance statistics (Diggle 1983), i.e. nearest neighbour and point to nearest neighbour distance functions.

Materials and methods

simulation model

In our zone of influence model, plants grow as circles in two dimensions and can be thought of as ellipsoids in three dimensions. The area a plant occupies, A, represents resources potentially available to the plant, and is allometrically related to the plant's biomass, B, as AB2/3. A plant's potential growth rate, that is, its growth if there are no neighbours, is sigmoidal

image

where Bmax is the maximum (asymptotic) biomass, r is the initial (maximum) relative growth rate in mass per unit area occupied (in units of mass area−1 time−1), and t is time. When plants overlap, they compete for the resources in areas of overlap. The effective area of a plant, Ae, is calculated as the area it covers minus that part of the area lost to neighbours and determines the realized growth rate of the plant during the next time interval:

image

The size dependency of competition is reflected in the rules for dividing up the overlapping areas. Under completely asymmetric competition larger individuals obtain the whole overlapping areas. In contrast, under completely symmetric competition the overlapping areas between individuals are divided equally (for implementation details see Weiner et al. 2001). We assumed that individuals die if their actual growth rate falls below 5% of their current mass. This size threshold was necessary because under symmetric competition there is no mortality without a threshold. The simulations were stochastic because there was random normal independent variation in initial size (B0 = 0.001; SD = 0.0001), initial growth rate (r0 = 2.0; SD = 0.2) and asymptotic size (Bmax = 50; SD = 5).

Plant growth was modelled from random and clumped initial patterns. Random patterns were generated as Poisson processes. Clumped patterns were generated by first distributing 110 individuals as random background and then adding 35 offspring around four parents. The distance from the parent was chosen from a normal distribution with mean 0 and standard deviation of 5 mm. The angle was chosen from a uniform distribution in the range 0–360°. This procedure imitated the experimental distributions (see below) as closely as possible.

Based on the number of plants in the experiment (see below), initial densities in the simulations were set to 250 individuals, matching the average number of wild-type individuals that germinated per pot. After growth and competition, 159 wild-type individuals but only 96 over-expressors survived on average. Therefore, the final pattern of survivors was taken, after mortality levels reached approximately those of the experimental data (≤ 150 surviving individuals). The resulting pattern of survivors was then analysed by the combined-count distance analysis.

pattern experiment

Based on information from previous studies (Stoll et al. 2002), we chose an initial density of 300 A thaliana seeds per pot (quadratic 7 × 7 cm wide and 7.5 cm deep, filled with a media of TKS1 and perlite, ratio 4 : 1, Floragard Vertrieb GmbH, Oldenburg, Germany, and UFA Samen, Lyssach, Switzerland). This density was high enough to result in sufficient mortality while allowing an accurate mapping of all individuals. Before sowing, seeds of wild-type (ecotype Nossen, hereafter WT) and transgenic over-expressors (over-expressors) of A. thaliana were vernalized at 4 °C for 1 week to increase germination rate.

Before distributing the seeds, a thin layer of soil was sieved over the pots to provide a surface as smooth as possible with minimal spatial heterogeneity. For the random initial distribution, seeds were mixed with 0.7 g of sand and sown with a sieve through a funnel placed around the pot to ensure that no seeds were lost. The clumped initial pattern was realized in two steps: first, 160 seeds, mixed with 0.7 g sand, were sown as in the random pattern. Secondly, the inner plot surface (excluding a 1-cm border) was overlaid by a lattice of 25 1 × 1 cm cells. Four of these cells were randomly chosen and 35 seeds per cell were additionally sown yielding 300 seeds per pot as in the random distribution.

The experimental design was a randomized block design with two blocks. There were six replicates (three per block) per genotype × initial pattern combination. To reduce edge effects, border pots with approximately 100 plants were placed around the complete experimental unit. The pots were watered twice a day and grew in the glasshouse with 16 hours of light (sun and artificial light in the range 170–230 µΕ m−2 s−1) at 20 °C and 60% relative air humidity. Water dripping through the roof of the glasshouse partly destroyed two replicates of the clumped distribution of over-expressor seed, and these were excluded from analysis of the final pattern. Fortunately, this dripping occurred after the initial patterns were photographed (see below) and did not affect any of the other experimental pots.

One week after sowing, the seedlings were photographed (Olympus digital reflex camera 1400 L, 1280 × 1024 pixels) and the pictures digitized (Graphtec KD 4300) to map the location of every individual with a resolution of 0.5 mm. After 5 weeks, when the rosettes were flowering and fruiting, plants were cut off with scissors such that stems of 0.5 cm height and the origin of the rosettes were left in the pots. Very small, light green individuals, which were also left by this method, were treated as dead. Because the stems were too thin to be photographed they were replaced by pins, which were then photographed and digitized.

anovas using genotype and pattern as factors were used to analyse differences in germination and survival. Because WT individuals showed higher germination rates than over-expressors, we used the initial number of individuals as a covariate in the analysis of the number of surviving individuals. We did not transform the dependent variable, because distributions of residuals were reasonably normal without transformation and results with square-root or log-transformation were very similar to the results from the untransformed analysis.

pattern analysis

The function λK(t) of the combined count-distance analysis (Ripley's K(t)) is defined as the expected number of plants within a distance t of a randomly chosen plant in the population. If individuals of intensity λ are randomly distributed (Poisson expectation), then the expected number of neighbours within radius t of a randomly chosen plant is λπt2 and, hence, K(t) = πt2. K̂(t) is an unbiased estimator of K(t) obtained with an appropriate edge correction (Ripley 1977). Before plotting against t, K̂(t) is usually transformed to L̂(t) =t − {K̂(t) /π}1/2. The advantage of using L̂(t) for measuring spatial patterns is that the square root transformation stabilizes the variance (see Besag in the discussion in Ripley 1977). L̂(t) has zero expectation if the point pattern is random at scale t, whereas positive values of L̂(t) indicate regularity and negative values clumping. For the simulated and experimental data, L̂(t) was calculated up to a scale of t = 35 mm in intervals of Δt = 0.5 mm. Initial patterns were tested for departures from complete spatial randomness by simulating 99 realizations of random patterns and plotting the minimum and maximum L̂(t) values as confidence envelopes (Diggle 1983). Observed L̂(t)values falling outside the confidence envelopes reveal regular (above) or clumped (below) spatial initial pattern and are deemed significant. A different randomization was used to test the final patterns for departures from random mortality (Kenkel et al. 1997). The same number of individuals that actually died in the experimental replicates was randomly removed from the initial pattern and 99 realizations of this random mortality were generated for each replicate with the appropriate number of individuals. The maximum and minimum L̂(t) values of the randomizations were again used as confidence envelopes. Observed L̂(t) values outside the envelopes are deemed significant and indicate more regular or clumped final patterns than observed if mortality would have been random.

To further quantify different aspects of the patterns, we used empirical distribution functions (EDF) of nearest neighbour distances (w), Ĝ1(w), and ‘point to nearest event’ distances (y), 1(y), as described in Diggle (1983). However, we denote point to nearest event distances with E (instead of F as used by Diggle) to avoid confusion with F-values for variance ratios used in the analysis of variance tables. The regular grid used to calculate Ê1(y) consisted of 26 × 26 = 676 points regularly spaced at intervals of 2.5 mm. Because confidence envelopes only provide approximate results, formal significance tests of cumulative Gs and Es were performed as follows: (i) calculate s = 99 sets of n random coordinates with n as in the corresponding experimental replicate; (ii) calculate the mean (i(w), Ēi(y), i = 2 ... s) of the appropriate functions of the simulations with random coordinates; (iii) calculate a test statistic, u1, as the sum of the squared differences between observed cumulative Ĝ1(w) and Ê1(y), and the mean of the 99 simulations (i(w), Ēi(w)) under complete spatial randomness; (iv) calculate the uis of each of the 99 simulations with their mean; (v) rank u1 amongst the uis and divide the rank of u1 by s+ 1 = 100 to obtain the error probability.

Finally, we performed analyses of variance on the mean of nearest neighbour distances () and the mean of point to nearest event distances (Ē) of the experimental patterns to further test for differences in pattern formation among genotypes and patterns. Because nearest neighbour distances and point to nearest event distances depend on the density of the pattern, the analyses of variance with and Ē as dependent variables were performed with the number of individuals as covariate.

All statistical and pattern analyses were done using GENSTAT 7th Edition, Release 7.1.

Results

There was a clear difference in density-dependent mortality amongst the genotypes independent of the initial pattern (Table 1). Over-expressors showed a much higher mortality rate than WTs. This difference was significant whether or not the initial number of individuals was used as covariate (Table 2). On average less than 100 WT individuals but more than 130 over-expressors died during the 5 weeks of the experiment.

Table 1.  Analysis of variance of the effects of genotype (wild type vs. transgenic, phytochrome A over-expressor) and pattern (random vs. clumped) on final number of surviving Arabidopsis thaliana individuals after 5 weeks. Because wild types had a slightly higher germination rate than over-expressors, initial density was used as covariate. The interaction (not shown) was not significant. Abbreviations: d.f. = degree of freedom; F = variance ratio
Source of variationd.f.FP
  1. NS = not significant; ***P < 0.001.

Genotype 133.1***
Pattern 1 1.3NS
Initial density 139.2***
Residual18  
Total21  
Table 2.  Average initial (1 week after sowing) and final (5 weeks after sowing) number of wild type and transgenic, phytochrome A over-expressor individuals of Arabidopis thaliana per pot (7 × 7 cm). The last column (covar. adj.) gives the final number of individuals adjusted for the initial number. There were n = 12 replicates per genotype, except for the over-expressor, which at the end of the experiment had only 10 replicates (because of an accident in the glasshouse)
 Number of individuals
GenotypeInitialFinalFinal (covariate adjusted)
  1. LSD (5%): least significant difference for P < 0.05.

Wild type246159155
Over-expressor228 96101
LSD (5%) 12 18 20

Visual inspection of the spatial pattern for the simulated plant populations showed that under symmetric competition, pairs of individuals survived starting from a random pattern and clumps were still visible if the initial pattern was clumped (Fig. 1). In contrast, a regular final pattern developed quickly from a random pattern under asymmetric competition. Similarly, initial clumps were quickly thinned leading to a relatively regular pattern of survivors. The spatial patterns of a selected replicate from the experimental plant populations (Fig. 2) differed among genotypes. Compared with WTs, final patterns of over-expressors were more regular if the initial pattern was random. Finally, clumps were still visible in the final pattern of WTs but disappeared almost completely in the over-expressors.

Figure 1.

Initial and final spatial patterns of the simulated populations under symmetric and asymmetric competition. In the final pattern, simulations were stopped as soon as less or equal to n = 150 survivors were reached. Time indicates the number of iterations with time steps = 0.1 needed to reach the stopping point.

Figure 2.

Initial (1 week after sowing) and final (5 weeks after sowing) spatial pattern of wild type and transgenic, phytochrome A over-expressors of Arabidopsis thaliana for a representative replicate in the experiment (n = number of individuals).

Random initial patterns of simulated plant populations of both types were within the confidence envelopes derived from complete spatial randomness (Fig. 3). Experimental wild types, however, were initially more regular than completely random patterns for distances less than about 2–3 mm. For over-expressors, initial patterns were within the envelopes for distances less than about 2 mm but deviated towards clumping for distances of 6 mm and greater (not shown). Starting from a random initial pattern, the final pattern under symmetric competition was not different to a pattern derived from random mortality, except for the simulated populations that deviated towards clumping at short distances. In contrast, density-dependent mortality under asymmetric competition resulted in a pattern of survivors that deviated substantially from that expected under random mortality (Fig. 3). This deviation towards regularity was found for simulated as well as experimental plant populations.

Figure 3.

Combined count-distance analysis (&#x004c;̂(t) vs. distance) for initially random spatial patterns. Simulations were performed with an individual-based zone of influence model under symmetric and asymmetric competition. Experimental plant populations (two representative replicates out of six are shown) consisted of either wild type Arabidopsis thaliana with intact shade avoidance or transgenic, phytochrome A over-expressors with disabled shade avoidance. Over-expressors have been shown to compete more asymmetrically than wild types. The continuous, black lines give &#x004c;̂(t) values for the initial patterns, and the dashed lines confidence envelopes from 99 simulations of complete spatial randomness for the initial pattern. The grey lines give &#x004c;̂(t) values for the final patterns, and the thin lines confidence envelopes from 99 simulations of random mortality.

Clumped initial patterns of simulated plant populations were well outside the confidence envelopes derived from complete spatial randomness (Fig. 4). Wild types, were slightly more regular at very short distances (< 1 mm), but were clumped for distances greater than 2 mm. For over-expressors they were within the envelopes for very short distances but were outside the envelopes for greater distances. Starting from a clumped initial pattern, the final pattern under symmetric competition was not substantially different to a pattern derived from random mortality. Asymmetric competition, again, resulted in a pattern of survivors that deviated substantially from that expected under random mortality as evidenced by &#x004c;̂(t) values of final patterns lying well outside the confidence envelopes derived from random mortality (Fig. 4). This deviation towards regularity was found for simulated as well as experimental plant populations. The final patterns of simulated symmetrically competing and experimental WTs never reached the confidence envelopes for complete spatial randomness beyond 3 mm. In contrast, the final pattern of simulated asymmetrically competing and experimental over-expressors was above or within the envelopes of complete spatial randomness up to 6 mm.

Figure 4.

Combined count-distance analysis (&#x004c;̂(t) vs. distance) for initially clumped spatial patterns. Simulations were performed with an individual-based zone of influence model under symmetric and asymmetric competition. Experimental plant populations (two representative replicates out of six are shown) consisted of either wild type Arabidopsis thaliana with intact shade avoidance or transgenic, phytochrome A over-expressors with disabled shade avoidance. Over-expressors have been shown to compete more asymmetrically than wild types. The continuous, black lines give &#x004c;̂(t) values for the initial patterns, and the dashed lines confidence envelopes from 99 simulations of complete spatial randomness for the initial pattern. The grey lines give &#x004c;̂(t) values for the final patterns, and the thin lines confidence envelopes from 99 simulations of random mortality.

Summarizing the results across all replicates revealed, for example, that final patterns of all six wild type replicates with a random starting pattern were within the envelopes of random mortality at 3.5 mm (Table 3). By contrast, none of the over-expressor replicates were within the envelopes of random mortality, i.e. they were all more regular than a pattern derived from random mortality. Similarly, starting from a clumped pattern, five out of six wild type replicates were within the envelopes of random mortality at 2.0 mm. However, all over-expressor replicates were more regular than a pattern derived from random mortality.

Table 3.  Pattern analysis of wild type and transgenic, phytochrome A over-expressing Arabidopsis thaliana. The entries give the number of replicates out of six (except for over-expressors in the clumped starting pattern where only four remained) that were within the envelopes of 99 simulations with random mortality
Distance (mm)Innitial pattern
RandomClumped
Wild type (n = 6)Over- expressor (n = 6)Wild type (n = 6)Over- expressor (n = 4)
0.53464
1.06541
1.55120
2.05150
2.54040
3.04020
3.56020
4.06120
4.56110
5.06110
5.56210
6.06310

After adjusting for the number of individuals, nearest neighbour distances (G) did not differ between initial or final patterns and genotypes (Table 4). In contrast, point to nearest event distances (E) decreased (F1,37 = 36.4, P < 0.001) from the initial (2.9 mm) to the final pattern (2.5 mm) and differed between genotypes (F1,37 = 5.0, P = 0.032). The effect of the starting pattern was highly significant for nearest neighbour distances and point to nearest event distances. The time–genotype interaction was significant for nearest neighbour distances. In WTs, covariate adjusted nearest neighbour distances decreased slightly (from 2.8 to 2.7 mm) but increased from 2.4 to 3.4 mm in over-expressors. For the point to nearest event distances the time–genotype interaction was also significant (F1,37 = 5.1, P = 0.030) and indicated highest regularity for the final pattern of over-expressors. The three-way interaction time × genotype pattern was significant for nearest neighbour distances (F1,37 = 5.9, P = 0.020) and point to nearest event distances (F1,37 = 6.8, P = 0.013).

Table 4.  Analysis of variance of the effects of time (initial vs. final), genotype (wild type vs. transgenic, phytochrome A over-expressor) and starting pattern (random vs. clumped) on nearest neighbour distances (G) and point to nearest event distances (E). The number of individuals was used as covariate. Abbreviations: d.f. = degree of freedom; F = variance ratio
Source of variationd.f.Nearest neighbour distance (G)Point to nearest event (E)
FPFP
  1. NS = not significant; *P < 0.05; ***P < 0.001.

Time 1   1.8NS 36.4***
Genotype 1   1.0NS  5.0*
Pattern 1  63.0***136.2***
Time × Genotype 1 124.4***  5.1*
Time × Pattern 1   1.3NS  4.6*
Genotype × Pattern 1   6.5*  3.6NS
Time × Genotype × Pattern 1   5.9*  6.8*
Covariate 11243.0***524.9***
Residual37    
Total45    

In WTs of both starting patterns, covariate adjusted nearest neighbour distances decreased from initial to final pattern (Fig. 5) but the differences were not significant. On the other hand, in over-expressors nearest neighbour distances significantly increased from initial to final pattern independent of starting pattern. The biggest change was found for the over-expressor starting from a clumped initial pattern.

Figure 5.

Box plots of nearest neighbour distances (mm) of wild type and transgenic, phytochrome A over-expressing Arabidopsis thaliana. Initial patterns (open boxes) were taken 1 week after sowing, final patterns (grey boxes) were taken 5 weeks after sowing. To correct for the different densities among initial and final patterns of different genotypes, nearest neighbour distances are predicted values from an analysis of variance with number of individuals as covariate. Boxes that do not share the same letter are significantly different based on (average) least significant differences (P < 0.05) of the three-way interaction (time × genotype × pattern) in the analysis of variance.

Formal significance tests of the patterns of single replicates showed that patterns did not change much in WTs (Table 5). For example, in WTs starting from a clumped initial pattern, the same number of replicates was classified as random or clumped. In contrast, most replicates of over-expressors changed from initially clumped to regular final patterns. This result was obtained for both distance measures.

Table 5.  Pattern analysis based on nearest neighbour distances (G) and point to nearest event distances (E) of wild type and transgenic, phytochrome A over-expressing Arabidopsis thaliana. The entries give the number of replicates out of six (except for the final patterns of over-expressors in the clumped starting pattern where only four remained) classified as regular, random or clumped based on randomization tests. The test statistics were calculated as the sum of the squared differences between observed cumulative nearest neighbour distance functions or cumulative point to nearest event distance functions (with 26 × 26 = 676 points regularly spaced at intervals of 2.5 mm) and the mean of 99 simulations under complete spatial randomness with the same number of individuals as the corresponding replicate. This value was subsequently ranked within that of the sum of the squared differences between each of the 99 simulations and their mean (for further explanations see pattern analysis in Materials and methods). Entries with ‘0's were left blank
GenotypeStarting patternInitial pattern (1 week after sowing)Final pattern (5 weeks after sowing)
RegularRandomClumpedRegularRandomClumped
GEGEGEGEGEGE
Wild typeRandom66    65 1  
Clumped  1 56  1 56
Over-expressorRandom 11451462   
Clumped    66341   

Discussion

Independent of the starting pattern, simulated final patterns under symmetric competition and experimental final patterns of WTs did not deviate much from a pattern expected under random mortality. On the other hand, simulated patterns under asymmetric competition and experimental patterns of over-expressors showed substantial deviations from a pattern expected under random mortality. Density-dependent mortality caused by asymmetric competition resulted in regular patterns of survivors in both simulated and experimental populations. Moreover, the formation of regular patterns under asymmetric competition was independent of the initial spatial pattern. Thus, different initial patterns converged to regular final patterns under asymmetric competition. In contrast, initial spatial patterns were not much changed under symmetric competition.

However, experimental populations of WTs showed regular patterns at very local distances even in initial patterns before the onset of density-dependent mortality. Such regular patterns were not observed in simulated populations presumably because the simulation model did not implement early shade avoidance and early shade avoidance was disabled in over-expressors. Because this interpretation is speculative, it will be formulated as a hypothesis in the second part of the discussion. In the first part, we concentrate on the development of regular patterns under asymmetric competition and density-dependent mortality. The important difference between spatial regularity of WTs and over-expressors was that WTs developed spatial regularity at local scales before the onset of mortality, whereas the reason for regularity observed in over-expressors was density-dependent mortality.

The development of regular patterns within single species even-aged populations of plants has been reported (e.g. Cooper 1961; Laessle 1965; Ford 1975) and attributed to the effects of competition. Kenkel et al. (1997) convincingly demonstrated that shifts in spatial patterns caused by density-dependent mortality were not due to random mortality but could be interpreted as a consequence of mortality driven by competition and in particular asymmetric competition for light (Kenkel 1988; Kenkel et al. 1997). We observed quick formation of regular patterns under asymmetric competition in both simulated and experimental populations. Hence, our experimental results provide further evidence for this interpretation.

Another simulation study compared pattern formation among plants with or without compensatory growth (Brisson & Reynolds 1997). They modelled morphological plasticity as asymmetric areas of influence of individual plants. If compensation, i.e. the preferential growth in areas without neighbours, was disabled, plants competed more asymmetrically compared with plants with the potential of compensatory growth. In their simulation model, non-compensatory plants developed more regular patterns. In populations of compensatory plants the shift from clumped to regular patterns as a result of density-dependent mortality was delayed. Their simulation results were very similar to ours and support the conclusion that regular spatial patterns of surviving individuals should be interpreted as evidence for strong, asymmetric competitive interactions.

Surprisingly, the analysis of the initial pattern of wild type individuals revealed regularity at very local scales although the seeds were randomly distributed using a sand seed mixture. However, over-expressors were sown identically and did not show regularity at very local scales. We hypothesize that the reason for this difference was early shade avoidance of wild type individuals (Ballaréet al. 1987). That is, the position was digitized as the growing meristem between the two cotyledons, which could, because of early shade avoidance, well have deviated slightly (3 mm) from the position where individuals rooted. From another experiment with the same genotypes of Arabidopsis we know that hypocotyls on average reach about 6 mm 1 week after germination (Stoll et al. 2002). Thus, a dislocating bending movement of 30° already would lead to a difference of 3 mm between rooting point and digitizing point (Fig. 6). To provide empirical evidence for such a hypothesis one would have to digitize the rooting points as well as the growing meristems. A similar phenomenon, however, was used to explain clumped patterns of tree stems and more regular patterns of crown-cover centres in natural mixed forests (Ishizuka 1984). Similarly, Laessle (1965) noted that the tendency of clumped trees to diverge from the aggregation centre gives a more even-spaced pattern when distances are measured at breast height rather than at ground level.

Figure 6.

Sketch of Arabidopsis thaliana seedling in the cotyledon stage 1 week after germination to illustrate the potential effect of early shade avoidance evoked by reflected far-red light. A bending of the hypocotyl of 30° may lead to a difference of 3 mm between rooting point and digitizing point between the cotyledons.

Conclusions

The final pattern in the model and the experiment could only result from competition because environmental heterogeneity was minimized as much as possible. In nature, however, spatial pattern of individuals reflects recruitment and the modification of initial patterns by various mortality factors in heterogeneous environments. Competition, especially competition for light, is just one mortality factor that may act at very local scales. A great mixture of ordering mechanisms and randomizing biotic as well as abiotic processes influences spatial structure at various scales. For example, environmental heterogeneity at local scales, uneven age and size distributions, differences in germination time, herbivory and mutualistic organisms such as mycorrhizas are just some natural processes which may mask the expected pattern formation due to competition. Obviously, the effects of competition on pattern formation we found cannot be directly translated into natural systems. Nevertheless, our study clearly indicated that asymmetric and symmetric competition both play different roles in structuring spatial patterns. Further studies are needed to elucidate the relative importance of competition and other mechanisms (as well as their interactions) in pattern formation of more complex and natural systems.

Acknowledgements

We thank Professor G. Whitleham for providing the A. thaliana seeds, Professor D. McC Newbery, Dr G. Armbruster and two anonymous referees for critical and helpful comments on the draft. Financial support from the Japanese Society for the Promotion of Science (JSPS) and the Swiss National Science Foundation (SNF) to PS and the British Ecological Society to EB are also acknowledged.

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