Multifractal analysis of weeds in a no-tillage system in the Pre-Amazon region 1

: Weeds have several mechanisms of dispersal and reproduction, resulting high spatial variability. The objective of this study was to assess the scale and spatial heterogeneity of weeds using multifractal analysis in a no-tillage system in the Pre-Amazon region. Sampling was conducted in a commercial soybean ( Glycine max ) production plot in the Mata Roma, Maranhão, Brazil, comprising 1,071 points marked on a 10 × 10 m grid. Data were analyzed using descriptive statistics and multifractal analysis through the box-counting method. Weeds showed varying degrees of multifractality, resulting in different scales and spatial heterogeneity in the study area. Euphorbia hirta and Turnera subulata exhibited asymmetry of branches to the left in the singularity spectrum, indicating dominance of high measurement values.

Spatial variability of weeds has been described through different methods.Chiba et al. (2010) and Siqueira et al. (2016) analyzed the spatial variability of weeds using geostatistical tools, whereas Silva et al. (2022) evaluated scale properties of weeds using multifractal analysis.Multifractal analysis characterizes the structure of a system or object, allowing for the evaluation of the distribution of measurement values at different scales, thus describing the spatial variability (Hentschel & Procaccia, 1983;Halsey et al., 1986;Evertsz & Mandelbrot, 1992;Posadas et al., 2009).Therefore, it enables the assessment of the heterogeneity of a system (Vidal-Vázquez et al., 2013;Bertol et al., 2017;Santos et al., 2019;Leiva et al., 2019;Silva & Siqueira, 2020, Leiva et al., 2021;Siqueira & Silva, 2022;Siqueira et al., 2022;Silva et al., 2022).According to Dafonte et al. (2015), it is possible to determine whether the structure of a system is monofractal or multifractal.
Therefore, the hypothesis in this work is that the spatial distribution of weeds constitutes multifractal systems.The objective of this study was to evaluate the scale and spatial heterogeneity of weeds using multifractal analysis in a notillage system in the Pre-Amazon region.

Materials and Methods
The study was conducted in a 10-hectare area (Figure 1) in Mata Roma, state of Maranhão, Brazil (3°42'26.56''S,43°11'19.56'W,and average altitude of 130 m).The soil of the area was classified as a Latossolo Vermelho-Amarelo típico by the Brazilian Soil Classification System (Santos et al., 2018), which corresponds to a Typic Hapludox (United States, 2022).The region presents an Aw climate, according to the Koppen classification, with two well-defined seasons: a dry season from June to November and a rainy season from December to May, with an annual rainfall depth of 1,835 mm, temperatures ranging from 23 to 36 °C, and a mean relative air humidity is 80%.The study area has been managed with crop rotation since 2007, with soybean [Glycine max (L.) Merrill] and maize (Zea mays L.) crops under no tillage system, with subsoiling up to a depth of 0.32 m when necessary, every five years.
Sampling was conducted when the area was cultivated with soybeans, sown on December 23, 2016.Weeds were sampled on January 12, 2017, at 1,071 points distributed in a 10 × 10 m grid (Figure 2), considering weeds within circular plots with a useful area of approximately 1 m 2 (1.126 m diameter).Weeds were counted and identified following the procedures described by Gazziero et al. (2015).A total of 21,217 individuals were identified: Euphorbia hirta L. ( 10 The ecology among species was evaluated according to Küchler et al. (1976), considering: Density = total number of individuals per species / total number of sampling points in the area (plants m -2 ); Relative Density = species density × 100 / sum of densities of all specimens (%); Frequency = number of sampling points containing the species / total number of sampling points obtained in the area; Relative Frequency = species frequency × 100 / sum of frequencies of all specimens (%); Abundance = total number of individuals per species / total number of sampling points containing the species; Relative abundance = species abundance × 100 / sum of abundances of all specimens (%); and Importance Value Index (IVI) = relative density + relative frequency + relative abundance (%).Richness was determined by the total number of taxonomic groups identified at each sampling point.
Multifractal analysis of weeds in a no-tillage system in the Pre-Amazon region
In heterogeneous systems, the contents of the grids can be quantified following the scale properties (Eq. 1 -Table 1), using a probability distribution (P), which enables the estimation of the scale properties (ε) in the i th region or spatial location (Posadas et al., 2009); thus, the Hölder exponent (α i ) can vary in the interval (α -∞ , α +∞ ).The partition function µ(q, e) (Eq.2) of order q is confirmed based on the scale properties, where N(ε) is the number of segments with size ε and the statistical moments q are defined by -∞< q <∞.Furthermore, the partition function is dimensioned as ε τ(q) (Eq.3), where the exponent τ(q) is the moment correlation exponent of order q, also known as the mass function (Halsey et al., 1986).In this sense, multifractal sets are characterized through the generalized dimension (D -Eq.4) for moments of order q in a Dq distribution (Hentschel & Procaccia, 1983); when q is replaced by 0, 1, and 2, it is possible to determine the dimensions of capacity (Eq.5), information (Eq.6) and correlation (Eq.7), respectively.
The dimension spectra or singularity spectra (q) are defined by Equations 8 and 9 (Chhabra & Jensen, 1989), which specify that the scale properties of the partition function reflect the contribution of individual segments.The degree of multifractality (Δ -Equation 10) and asymmetry (AI -Equation 11) of the data were determined considering the values of D q and α where: D is the generalized dimension at times q = -5 and q = 5; AI is the asymmetry of the system; α0 is the value of f(α) in interval 0; α 5 is the value of f(α) in the interval q = 5; and α -5 is the value of f(α) in the interval q = -5 (Halsey et al., 1986).

Result and Discussion
The diversity parameters presented in Table 2 are indicators used to characterize the ecology among species.Density, frequency, and abundance express the participation of different species, the spatial distribution of each specimen, and concentration of species in the study area, respectively (Küchler et al., 1976;Siqueira et al., 2016;Caetano et al., 2018;Castro et al., 2021).The importance value index (IVI) is intended to characterize which species have a greatest influence within the weed community (Caetano et al., 2018), thus, the higher the IVI (Table 2), the higher the positive species rate at the sampling points.
The weed species with the highest density (D) and relative density (RD -Table 2)  The species with the highest frequency were E. hirta (F = 1) and S. verticillata (F = 1); frequency close to one denotes uniformity in the distribution of weeds in the study area; E. hirta and S. verticillata were found in all sampling points.The species with the highest abundance was E. hirta (A = 9.36 -Table 2), with a high concentration of weed plants, resulting in a higher importance value index (IVI = 102.69).According to Freitas et al. (2021) Euphorbiacea is a family of species with short cycles, tiny inflorescences, and a high potential for seed production.This explains the high incidence of E. hirta in the study area, as described by the ecological parameters (Table 2).
The three species with the lowest IVI [S.rhombifolia (8.16%), D. tortuosum (8.06%), and S. dulcis (5.40%)] were grouped into the category Other Weeds (OW).According to Carvalho & Carvalho (2009) and Gazziero et al. (2015), species in the OW category have later germination relative to the soybean cycle, explaining the occurrence of lower IVI for these species.
The statistical parameters of weeds in the study area are shown in Table 3.The species E. hirta had the highest mean number of individuals (x ̅ = 9.36 plants per m 2 ), followed by S. anthelmia (4.52 plants per m 2 ) and OW (x ̅ = 2.85 plants per m 2 ).These are similar results to those found in other studies.Samuel & Rastogi (2022)  The coefficients of variation (CV%; Table 3) of the evaluated weeds were classified as moderate (12 % < CV < 60 %), according to Warrick & Nielsen (1980); except for the OW category (CV = 81%) which presented a high CV (> 60%).The OW category encompassed three weed species [D.tortuosum, S. dulcis, and S. rhombifolia], with high heterogeneity in the study area and cluster distribution (Chiba et al., 2010;Siqueira et al., 2016;Castro et al., 2021;Silva et al., 2022), as well as distinct ecological processes (Gazziero et al., 2015).
The weeds had a lognormal frequency distribution (Table 3), according to the Kolmogorov-Smirnov normality test (D-KS, p < 0.01), which is consistent with the median to high CV and the asymmetry and kurtosis values.
Table 4 presents the results of the multifractal analysis for moments of order q in the interval from q = -5 to q = +5.In multifractal systems, the capacity dimension (D 0 ), information dimension (D 1 ), and correlation dimension (D 2 ) follow the pattern: D 0 > D 1 > D 2 (Posadas et al., 2009;Vidal-Vázquez et al., 2013;Dafonte et al., 2015;Bertol et al., 2017;Leiva et al., 2019;Silva & Siqueira, 2020;Leiva et al., 2021;Siqueira et al., 2022;Silva et al., 2022).Therefore, the variables represent multifractal systems, as they present the pattern D 0 > D 1 > D 2 , except for OW, which represents a monofractal system.According to Dafonte et al. (2015), a monofractal system present the following dimension pattern: D 0 ≈ D 1 ≈ D 2 .OW represents a monofractal system due to the characteristics of the species in this class, as there are three species that occur in the study area with independent spatial patterns, resulting in high variability (CV = 81.00%)and comprising a chaotic system.
The capacity dimension (D 0 = 1.995) remained constant for all evaluated weed species, with values close to 2, indicating that almost all boxes/scales are filled with measurement values, as described by Posadas et al. (2009).The information dimension (D 1 ) measures the degree of heterogeneity in the system (Siqueira et al., 2022), and values close to 2 represent a relatively uniform distribution of measurement values across scales, whereas values close to 1 represent sets that have concentrated irregularities (Leiva et al., 2019;Silva et al., 2022).The highest and lowest D 1 values (Table 4) were found for Richness (D 1 = 1.992) and for C. benghalensis (D 1 = 1.913); despite the numerical difference in D 1 , in both cases, the values are close to 2, indicating uniformly distributed systems.Mathematically, the values of correlation dimension (D 2 -Table 4) are associated with the correlation function (Hentschel & Procaccia, 1983) and describe how measurements are distributed in boxes/ scales.The results showed that D 2 values ranged from 1.818 (C.benghalensis) to 1.988 (Richness), indicating low irregularity in the data series.
The degree of multifractality (Δ -Table 4) describes systems with higher or lower heterogeneity (Vidal-Vázquez et al., 2013;Siqueira & Silva, 2022) 4), indicating that this species occurs in the study area at a low density (D = 0.86 -Table 2).In the present study, the multifractality values (Δ) reflected heterogeneous systems with greater or lesser complexity for the biological systems under study (Silva et al., 2022).
The species with the highest occurrence in the study area showed a lower degree of multifractality (Δ) due to the homogenous distribution of these species in the experimental plot.Silva et al. (2022) evaluated the multifractality of weeds and reported that the degree of multifractality represents the complexity of the ecological dynamics of weeds, reinforcing the findings of the present study.
The diversity indices Richness and Abundance showed the lowest degree of multifractality (Δ = 0.071 and Δ = 0.086, respectively), as expected, since these indices represent measures with certain uniformity across boxes.
C. benghalensis had the highest Hölder exponent (α 0 = 2.055) and asymmetry (AI = 2.324) values.Thus, this species system had the highest multifractality/heterogeneity.Overall, the Hölder exponent (α 0 ) values found for the other weed species varied slightly from one species to another, indicating that the colonization process by weeds in the study area is structured, however, with different spatial variability scales among species.Contrastingly, asymmetry (AI) values showed a high variation in the study area, with the highest value found for C. benghalensis (AI = 2.324) and the lowest for C. echinatus (AI = 0.102).The presence of positive asymmetry (AI) indicates greater variability at scales corresponding to low measurement values (Siqueira et al., 2022;Silva et al., 2022), denoting a more frequent occurrence of low measurement values throughout the study area.
The generalized dimension (Figure 3A) at moment q (D q ) is a decreasing function with a sigma curve shape.The graph for C. benghalensis demonstrates that at times q = 0 to q = 5 there is a difference from the other weeds, showing higher heterogeneity, which is consistent with the result found for the degree of multifractality (Table 4 -Δ = 0.388).Richness had the lowest variation for both positive (q = 0 to q = 5) and negative (q = 0 to q = -5) moments, denoting greater system homogeneity and lower multifractality, as shown by the degree of multifractality (Table 4 -Δ = 0.071).
The mass exponent or Rényi graph (Figure 3B) shows multifractal behavior for all evaluated variables.According to Santos et al. (2019), linear graphs do not represent multifractal patterns, whereas nonlinear functions correspond Table 4. Multifractal parameters of the attributes of the study area images Richness -Species richness per point; Abundance -Abundance of weeds per point; OW -Other weeds; Δ -degree of multifractality; D 0 -Capacity dimension; D 1 -Information dimension; D 2 -Correlation dimension; α -5 , α 5 , α 0 are the singularity spectra for the moments q = -5, q = 5 and q = 0 respectively; AI -Asymmetry Figure 3. Generalized dimension graph (D q vs q -A) and mass exponent graph (τ(q) vs q -B).q = statistical moment; D q = generalized dimension for moments of order q; τ(q) = correlation exponent at the moment of order q to multifractal systems, i.e., mass exponent graphs do not exhibit linear functions, but they present a certain curvature.
Weed plants (C.enchinatus, S. verticillate, and S. anthelmia) and ecological variables (Richness and Abundance) (Figure 4 2022), who evaluated the uniqueness spectrum and described the predominance of high and low values related to the left and right branches of the spectrum, respectively.The singularity spectrum (Figure 4) displays descending and concave parabolas (Bertol et al., 2017) confirming the multifractal behavior of the data (Dafonte et al., 2015).
The results showed that the promising use of multifractal analysis for studying weed plants, as it was possible to identify multifractal patterns related to ecological processes of the different species under study, including seed dispersal ability, dormancy period, and reproduction with a high disseminule production capacity (Gazziero et al., 2015;Freitas et al., 2021).The generalized dimension graph provides information on the spatial variability of measurement values, describing greater and lesser heterogeneity in the system (Posadas et al., 2009;Leiva et al., 2019).The singularity spectrum graph proved to be effective in evaluating the domain of measurement values (low or high), highlighting spatial distribution patterns that would not be characterized by other spatial analysis methods.The identification of spatial distribution patterns of weeds on a multifractal scale enables the development of increasingly precise management strategies.
. The highest degree of multifractality OW -Other weeds [Sida rhombifolia L., Desmodium tortuosum (Sw.)DC, and Scoparia dulcis L.]; x̅ -mean (plants per m 2 ); SD -Standard deviation; CV -Coefficient of variation (%); D-KS -Maximum deviation from the normal distribution using the Kolmogorov-Smirnov test at p ≥ 0.01 Table 3. Descriptive statistics for the number of weed plants was found for C. benghalensis (Δ = 0.388 -Table ) have a multifractal spectrum with asymmetry of the branches to the right, indicating dominance of low measurement values.The uniqueness spectrum for C. benghalensis, E. hirta, and T. subulata exhibits asymmetry to the left, indicating dominance of high measurement values.Information on the domain of weed values allows for effective rate of production inputs, avoiding waste, thus preserving the environment.The results found in this study are consistent with findings from other studies, such as those by Vidal-Vázquez et al. (2013), Silva & Siqueira (2020), and Silva et al. (