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Scale-invariant structure of size fluctuations in plants

A wide range of physical and biological systems exhibit complex behaviours characterised by a scale-invariant structure of the fluctuations in their output signals. In the context of plant populations, scaling relationships are typically allometric. In this study, we analysed spatial variation in the size of maize plants (Zea Mays L.) grown in agricultural plots at constant densities and found evidence of scaling in the size fluctuations of plants. The findings indicate that the scaling of the probability distribution of spatial size fluctuation exhibits non-Gaussian behaviour compatible with a Lévy stable process. The scaling relationships were observed for spatial scales spanning three orders of magnitude. These findings should provide additional information for the selection and development of empirically accurate models of pattern formation in plant populations.


Results
Data were collected from well-fertilised maize crops (Zea Mays L.) in southern Brazil (23u209S, 51u579W; approximately 450 m above sea level) with standard plant population densities: the linear density at a given stand was^4:7 plants m 21 , and the typical distance between stands was 0.9 m. Successive measures of the heights of surviving plants were recorded along linear stands [see Fig. 1(a)]. The heights of mature plants were measured from the soil surface to the pennant apex. The data sets correspond to the maize hybrids AG 9010, AS 1570, P 30F87 and P 30K75, and the number of data points are 2203, 2111, 1803 and 2109, respectively. The data for the AS 1570 hybrid are shown in Figures 1(b) and 1(c).
The probability density function (PDF) of plant size, P(S), exhibited Gaussian behaviour around the peak and a long left tail, which indicates negative skewness [ Fig. 2 (a)]. This finding is consistent with previously reported results for maize crops 21 . In contrast, the PDF of size fluctuation, P(z 1 ), was nearly symmetric, leptokurtic and non-Gaussian [ Fig. 2 (b)]. For comparison, we fitted P(z 1 ) to a symmetric Lévy distribution, Þcos qz ð Þdq, with scale factor c and index a. As shown in Fig. 2 (b), a good agreement with the Lévy distribution was observed.
We also investigated size fluctuations for larger spatial scales. We found that P(z Dk ) exhibits a common functional form for spatial scales in the range 1 , Dk , 1000. When the PDFs at different spatial scales were superimposed, all the data collapsed onto the same curve [e.g., Fig. 2 (c)]. We fitted P(z Dk ), for different values of

Discussion
According to the analysis described above, the scaling of the probability distribution of size differences in maize crops exhibits non-Gaussian behaviour that is compatible with a Lévy stable process. The heavy-tailed character of the distribution of size differences, which is consistent across different spatial scales, indicates that rare but relatively large size differences-compared to Gaussian processes-can occur along the stand. As a result, clusters of small fluctuations are connected by these rare and large fluctuations. The analysis of size differences may therefore provide an alternative method for quantifying size inequality in plant populations.   Our findings suggest that the complexity of crowded plant populations may be characterised by scaling properties of size fluctuations. The underlying mechanisms responsible for this pattern may be better understood by considering examples of heavy-tailed distributions in human dynamics. The burst nature of individual activities has been linked to a decision-making process in which individuals execute tasks based on a perceived priority. These priority-based mechanisms can be used to explain a wide range of collective human activity patterns 22,23 . Analogously, bursts and heavy tails in plant populations may reflect some fundamental and potentially generic features of individual plants. These collective patterns may be related to individual ecological strategies, such as competition for resources and plant physiology, and naturally, the effects of different strategies must be considered in tandem. For example, taller plants more readily access light and therefore possess a competitive advantage at any moment during the growth process. However, height incurs numerous costs including past investment, the maintenance of stems and support structures and disadvantages associated with the transport of water to increased heights 24 .
Size inequality in plant populations has been previously shown to be related to competition, but the underlying mechanisms are still debated. A favoured hypothesis is size-asymmetric competition: larger individuals obtain a disproportionate resource advantage (for their relative size) over smaller individuals, as in the example of competition for light 17,25,26 . Another hypothesis is related to variations in local crowding: the ability of individuals to obtain resources is affected by the size, proximity and number of neighbours [27][28][29] . Other proposed mechanisms consider functional interactions among plants that trigger morphological and physiological changes in response to the proximity of other plants. The role of informational photoreceptors in these responses suggests that light-dependent mechanisms contribute to changes in the size structure of plant populations [30][31][32][33] .
With the goal of identifying and quantifying the mechanisms responsible for size inequality, several approaches, such as reactiondiffusion models, patchy-environment models and neighbourhood models, have been applied to assess the spatiotemporal behaviour of plant populations 34,35 . However, the distribution of size fluctuation was not addressed in these studies. Our approach should provide additional information for the selection and development of empirically accurate models of pattern formation in plant populations. A natural extension of the present work would be to investigate the robustness of the results with respect to plant development from germination to maturity. Future studies should also include a parallel analysis in other natural plant populations along with a two-dimensional analysis in crop fields.