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Wang L, He K, Hui C, Ratkowsky DA, Yao W, Lian M, Wang J, Shi P. Comparison of four performance models in quantifying the inequality of leaf and fruit size distribution. Ecol Evol 2024; 14:e11072. [PMID: 38435001 PMCID: PMC10905244 DOI: 10.1002/ece3.11072] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/14/2023] [Revised: 01/28/2024] [Accepted: 02/09/2024] [Indexed: 03/05/2024] Open
Abstract
The inequality in leaf and fruit size distribution per plant can be quantified using the Gini index, which is linked to the Lorenz curve depicting the cumulative proportion of leaf (or fruit) size against the cumulative proportion of the number of leaves (or fruits). Prior researches have predominantly employed empirical models-specifically the original performance equation (PE-1) and its generalized counterpart (GPE-1)-to fit rotated and right-shifted Lorenz curves. Notably, another potential performance equation (PE-2), capable of generating similar curves to PE-1, has been overlooked and not systematically compared with PE-1 and GPE-1. Furthermore, PE-2 has been extended into a generalized version (GPE-2). In the present study, we conducted a comparative analysis of these four performance equations, evaluating their applicability in describing Lorenz curves related to plant organ (leaf and fruit) size. Leaf area was measured on 240 culms of dwarf bamboo (Shibataea chinensis Nakai), and fruit volume was measured on 31 field muskmelon plants (Cucumis melo L. var. agrestis Naud.). Across both datasets, the root-mean-square errors of all four performance models were consistently smaller than 0.05. Paired t-tests indicated that GPE-1 exhibited the lowest root-mean-square error and Akaike information criterion value among the four performance equations. However, PE-2 gave the best close-to-linear behavior based on relative curvature measures. This study presents a valuable tool for assessing the inequality of plant organ size distribution.
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Affiliation(s)
- Lin Wang
- Department of Applied Mathematics, College of ScienceNanjing Forestry UniversityNanjingChina
| | - Ke He
- Architectural Design and Research InstituteShenzhen UniversityShenzhenChina
| | - Cang Hui
- Department of Mathematical Sciences, Centre for Invasion BiologyStellenbosch UniversityStellenboschSouth Africa
- Mathematical and Physical Biosciences, African Institute for Mathematical SciencesCape TownSouth Africa
| | - David A. Ratkowsky
- Tasmanian Institute of AgricultureUniversity of TasmaniaHobartTasmaniaAustralia
| | - Weihao Yao
- Bamboo Research Institute, College of Ecology and EnvironmentNanjing Forestry UniversityNanjingChina
| | - Meng Lian
- Department of Applied Mathematics, College of ScienceNanjing Forestry UniversityNanjingChina
| | - Jinfeng Wang
- Bamboo Research Institute, College of Ecology and EnvironmentNanjing Forestry UniversityNanjingChina
| | - Peijian Shi
- Department of Applied Mathematics, College of ScienceNanjing Forestry UniversityNanjingChina
- Bamboo Research Institute, College of Ecology and EnvironmentNanjing Forestry UniversityNanjingChina
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Yao W, Niinemets Ü, Yao W, Gielis J, Schrader J, Yu K, Shi P. Comparison of Two Simplified Versions of the Gielis Equation for Describing the Shape of Bamboo Leaves. PLANTS (BASEL, SWITZERLAND) 2022; 11:plants11223058. [PMID: 36432787 PMCID: PMC9699164 DOI: 10.3390/plants11223058] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/02/2022] [Revised: 11/08/2022] [Accepted: 11/10/2022] [Indexed: 05/05/2023]
Abstract
Bamboo is an important component in subtropical and tropical forest communities. The plant has characteristic long lanceolate leaves with parallel venation. Prior studies have shown that the leaf shapes of this plant group can be well described by a simplified version (referred to as SGE-1) of the Gielis equation, a polar coordinate equation extended from the superellipse equation. SGE-1 with only two model parameters is less complex than the original Gielis equation with six parameters. Previous studies have seldom tested whether other simplified versions of the Gielis equation are superior to SGE-1 in fitting empirical leaf shape data. In the present study, we compared a three-parameter Gielis equation (referred to as SGE-2) with the two-parameter SGE-1 using the leaf boundary coordinate data of six bamboo species within the same genus that have representative long lanceolate leaves, with >300 leaves for each species. We sampled 2000 data points at approximately equidistant locations on the boundary of each leaf, and estimated the parameters for the two models. The root−mean−square error (RMSE) between the observed and predicted radii from the polar point to data points on the boundary of each leaf was used as a measure of the model goodness of fit, and the mean percent error between the RMSEs from fitting SGE-1 and SGE-2 was used to examine whether the introduction of an additional parameter in SGE-1 remarkably improves the model’s fitting. We found that the RMSE value of SGE-2 was always smaller than that of SGE-1. The mean percent errors among the two models ranged from 7.5% to 20% across the six species. These results indicate that SGE-2 is superior to SGE-1 and should be used in fitting leaf shapes. We argue that the results of the current study can be potentially extended to other lanceolate leaf shapes.
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Affiliation(s)
- Weihao Yao
- Bamboo Research Institute, College of Biology and the Environment, Nanjing Forestry University, Nanjing 210037, China
| | - Ülo Niinemets
- Institute of Agricultural and Environmental Sciences, Estonian University of Life Sciences, 51006 Tartu, Estonia
- Estonian Academy of Sciences, 10130 Tallinn, Estonia
| | - Wenjing Yao
- Bamboo Research Institute, College of Biology and the Environment, Nanjing Forestry University, Nanjing 210037, China
- Correspondence: (W.Y.); (P.S.)
| | - Johan Gielis
- Department of Biosciences Engineering, University of Antwerp, B-2020 Antwerp, Belgium
| | - Julian Schrader
- School of Natural Sciences, Macquarie University, Sydney, NSW 2109, Australia
| | - Kexin Yu
- Bamboo Research Institute, College of Biology and the Environment, Nanjing Forestry University, Nanjing 210037, China
| | - Peijian Shi
- Bamboo Research Institute, College of Biology and the Environment, Nanjing Forestry University, Nanjing 210037, China
- Correspondence: (W.Y.); (P.S.)
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Wang L, Miao Q, Niinemets Ü, Gielis J, Shi P. Quantifying the Variation in the Geometries of the Outer Rims of Corolla Tubes of Vinca major L. PLANTS 2022; 11:plants11151987. [PMID: 35956465 PMCID: PMC9370681 DOI: 10.3390/plants11151987] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/04/2022] [Revised: 07/21/2022] [Accepted: 07/26/2022] [Indexed: 11/16/2022]
Abstract
Many geometries of plant organs can be described by the Gielis equation, a polar coordinate equation extended from the superellipse equation, r=a|cosm4φ|n2+|1ksinm4φ|n3−1/n1. Here, r is the polar radius corresponding to the polar angle φ; m is a positive integer that determines the number of angles of the Gielis curve when φ ∈ [0 to 2π); and the rest of the symbols are parameters to be estimated. The pentagonal radial symmetry of calyxes and corolla tubes in top view is a common feature in the flowers of many eudicots. However, prior studies have not tested whether the Gielis equation can depict the shapes of corolla tubes. We sampled randomly 366 flowers of Vinca major L., among which 360 had five petals and pentagonal corolla tubes, and six had four petals and quadrangular corolla tubes. We extracted the planar coordinates of the outer rims of corolla tubes (in top view) (ORCTs), and then fitted the data with two simplified versions of the Gielis equation with k = 1 and m = 5: r=acos54φn2+sin54φn3−1/n1 (Model 1), and r=acos54φn2+sin54φn2−1/n1 (Model 2). The adjusted root mean square error (RMSEadj) was used to evaluate the goodness of fit of each model. In addition, to test whether ORCTs are radially symmetrical, we correlated the estimates of n2 and n3 in Model 1 on a log-log scale. The results validated the two simplified Gielis equations. The RMSEadj values for all corolla tubes were smaller than 0.05 for both models. The numerical values of n2 and n3 were demonstrated to be statistically equal based on the regression analysis, which suggested that the ORCTs of V. major are radially symmetrical. It suggests that Model 1 can be replaced by the simpler Model 2 for fitting the ORCT in this species. This work indicates that the pentagonal or quadrangular corolla tubes (in top view) can both be modeled by the Gielis equation and demonstrates that the pentagonal or quadrangular corolla tubes of plants tend to form radial symmetrical geometries during their development and growth.
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Affiliation(s)
- Lin Wang
- College of Science & College of Biology and the Environment, Nanjing Forestry University, Nanjing 210037, China; (L.W.); (Q.M.)
| | - Qinyue Miao
- College of Science & College of Biology and the Environment, Nanjing Forestry University, Nanjing 210037, China; (L.W.); (Q.M.)
| | - Ülo Niinemets
- Institute of Agricultural and Environmental Sciences, Estonian University of Life Sciences, 51006 Tartu, Estonia
- Estonian Academy of Sciences, 10130 Tallinn, Estonia
- Correspondence: (Ü.N.); (P.S.); Tel.: +86-25-8542-7231 (P.S.)
| | - Johan Gielis
- Department of Biosciences Engineering, University of Antwerp, B-2020 Antwerp, Belgium;
| | - Peijian Shi
- College of Science & College of Biology and the Environment, Nanjing Forestry University, Nanjing 210037, China; (L.W.); (Q.M.)
- Correspondence: (Ü.N.); (P.S.); Tel.: +86-25-8542-7231 (P.S.)
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Shi P, Gielis J, Quinn BK, Niklas KJ, Ratkowsky DA, Schrader J, Ruan H, Wang L, Niinemets Ü. 'biogeom': An R package for simulating and fitting natural shapes. Ann N Y Acad Sci 2022; 1516:123-134. [PMID: 35879250 DOI: 10.1111/nyas.14862] [Citation(s) in RCA: 10] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/20/2023]
Abstract
Many natural objects exhibit radial or axial symmetry in a single plane. However, a universal tool for simulating and fitting the shapes of such objects is lacking. Herein, we present an R package called 'biogeom' that simulates and fits many shapes found in nature. The package incorporates novel universal parametric equations that generate the profiles of bird eggs, flowers, linear and lanceolate leaves, seeds, starfish, and tree-rings, and three growth-rate equations that generate the profiles of ovate leaves and the ontogenetic growth curves of animals and plants. 'biogeom' includes several empirical datasets comprising the boundary coordinates of bird eggs, fruits, lanceolate and ovate leaves, tree rings, seeds, and sea stars. The package can also be applied to other kinds of natural shapes similar to those in the datasets. In addition, the package includes sigmoid curves derived from the three growth-rate equations, which can be used to model animal and plant growth trajectories and predict the times associated with maximum growth rate. 'biogeom' can quantify the intra- or interspecific similarity of natural outlines, and it provides quantitative information of shape and ontogenetic modification of shape with important ecological and evolutionary implications for the growth and form of the living world.
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Affiliation(s)
- Peijian Shi
- Co-Innovation Centre for Sustainable Forestry in Southern China, Bamboo Research Institute, College of Biology and the Environment, Nanjing Forestry University, Nanjing, China
| | - Johan Gielis
- Department of Biosciences Engineering, University of Antwerp, Antwerp, Belgium
| | - Brady K Quinn
- St. Andrews Biological Station, Fisheries and Oceans Canada, St. Andrews, New Brunswick, Canada
| | - Karl J Niklas
- School of Integrative Plant Science, Cornell University, Ithaca, New York, USA
| | - David A Ratkowsky
- Tasmanian Institute of Agriculture, University of Tasmania, Hobart, Tasmania, Australia
| | - Julian Schrader
- School of Natural Sciences, Macquarie University, Sydney, New South Wales, Australia.,Biodiversity, Macroecology & Biogeography, University of Göttingen, Göttingen, Germany
| | - Honghua Ruan
- Co-Innovation Centre for Sustainable Forestry in Southern China, Bamboo Research Institute, College of Biology and the Environment, Nanjing Forestry University, Nanjing, China
| | - Lin Wang
- Co-Innovation Centre for Sustainable Forestry in Southern China, Bamboo Research Institute, College of Biology and the Environment, Nanjing Forestry University, Nanjing, China
| | - Ülo Niinemets
- Institute of Agricultural and Environmental Sciences, Estonian University of Life Sciences, Tartu, Estonia.,Estonian Academy of Sciences, Tallinn, Estonia
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Yu K, Reddy GVP, Schrader J, Guo X, Li Y, Jiao Y, Shi P. A nondestructive method of calculating the wing area of insects. Ecol Evol 2022; 12:e8792. [PMID: 35386866 PMCID: PMC8975793 DOI: 10.1002/ece3.8792] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/22/2021] [Revised: 03/10/2022] [Accepted: 03/17/2022] [Indexed: 12/30/2022] Open
Abstract
Most insects engage in winged flight. Wing loading, that is, the ratio of body mass to total wing area, has been demonstrated to reflect flight maneuverability. High maneuverability is an important survival trait, allowing insects to escape natural enemies and to compete for mates. In some ecological field experiments, there is a need to calculate the wing area of insects without killing them. However, fast, nondestructive estimation of wing area for insects is not available based on past work. The Montgomery equation (ME), which assumes a proportional relationship between leaf area and the product of leaf length and width, is frequently used to calculate leaf area of plants, in crops with entire linear, lanceolate leaves. Recently, the ME was proved to apply to leaves with more complex shapes from plants that do not have any needle leaves. Given that the wings of insects are similar in shape to broad leaves, we tested the validity of the ME approach in calculating the wing area of insects using three species of cicadas common in eastern China. We compared the actual area of the cicadas’ wings with the estimates provided by six potential models used for wing area calculation, and we found that the ME performed best, based on the trade‐off between model structure and goodness of fit. At the species level, the estimates for the proportionality coefficients of ME for three cicada species were 0.686, 0.693, and 0.715, respectively. There was a significant difference in the proportionality coefficients between any two species. Our method provides a simple and powerful approach for the nondestructive estimation of insect wing area, which is also valuable in quantifying wing morphological features of insects. The present study provides a nondestructive approach to estimating the wing area of insects, allowing them to be used in mark and recapture experiments.
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Affiliation(s)
- Kexin Yu
- College of Biology and the Environment Bamboo Research Institute Nanjing Forestry University Nanjing China
| | - Gadi V. P. Reddy
- USDA‐ARS‐Southern Insect Management Research Unit Stoneville Mississippi USA
| | - Julian Schrader
- School of Natural Sciences Macquarie University Sydney New South Wales Australia
- Biodiversity, Macroecology and Biogeography University of Göttingen Göttingen Germany
| | - Xuchen Guo
- College of Biology and the Environment Bamboo Research Institute Nanjing Forestry University Nanjing China
| | - Yirong Li
- College of Biology and the Environment Bamboo Research Institute Nanjing Forestry University Nanjing China
| | - Yabing Jiao
- College of Biology and the Environment Bamboo Research Institute Nanjing Forestry University Nanjing China
| | - Peijian Shi
- College of Biology and the Environment Bamboo Research Institute Nanjing Forestry University Nanjing China
- Tropical Silviculture and Forest Ecology University of Göttingen Göttingen Germany
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