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基于Logistic和Gompertz模型的小叶杨幼苗生长组合优化模型

葛会硕, 宋跃朋, 苏雪辉, 张德强, 张晓宇

葛会硕, 宋跃朋, 苏雪辉, 张德强, 张晓宇. 基于Logistic和Gompertz模型的小叶杨幼苗生长组合优化模型[J]. 北京林业大学学报, 2020, 42(5): 59-70. DOI: 10.12171/j.1000-1522.20190296
引用本文: 葛会硕, 宋跃朋, 苏雪辉, 张德强, 张晓宇. 基于Logistic和Gompertz模型的小叶杨幼苗生长组合优化模型[J]. 北京林业大学学报, 2020, 42(5): 59-70. DOI: 10.12171/j.1000-1522.20190296
Ge Huishuo, Song Yuepeng, Su Xuehui, Zhang Deqiang, Zhang Xiaoyu. Optimal growth model of Populus simonii seedling combination based on Logistic and Gompertz models[J]. Journal of Beijing Forestry University, 2020, 42(5): 59-70. DOI: 10.12171/j.1000-1522.20190296
Citation: Ge Huishuo, Song Yuepeng, Su Xuehui, Zhang Deqiang, Zhang Xiaoyu. Optimal growth model of Populus simonii seedling combination based on Logistic and Gompertz models[J]. Journal of Beijing Forestry University, 2020, 42(5): 59-70. DOI: 10.12171/j.1000-1522.20190296

基于Logistic和Gompertz模型的小叶杨幼苗生长组合优化模型

基金项目: 中央高校基础研究基金(2017ZY30、2015ZCQ-LY-01),国家自然科学基金(11501032)
详细信息
    作者简介:

    葛会硕。主要研究方向:概率论与数理统计。Email:huishuo_ge@163.com  地址:100083 北京市海淀区清华东路35号北京林业大学

    责任作者:

    张晓宇,副教授,硕士生导师。主要研究方向:概率论与数理统计。Email:xyzhang@bjfu.edu.cn  地址:同上

  • 中图分类号: S758.5

Optimal growth model of Populus simonii seedling combination based on Logistic and Gompertz models

  • 摘要:
    目的  通过对我国小叶杨幼苗的株高生长规律及生长模型的研究,为其生长预估及科学育苗提供参考。
    方法  以来自我国16个产区的小叶杨幼苗作为研究对象,通过试验地调查及测量等方法获取基础数据,以时间维度建立生长模型分析小叶杨幼苗株高的生长规律。选取具有生物学意义的生长方程,根据模型拟合优度与评价指标选取最优基础生长模型,并在最优模型的基础上构建适合不同产区幼苗生长的组合优化生长模型。
    结果  (1)不同产区的样本株高的综合最优基础模型分别为Logistic方程和Gompertz方程,其中Logistic模型的R2和预测精度分别在0.847 9和92.23%以上,Gompertz模型的R2和预测精度分别在0.891 5和92.60%以上。(2)在此基础上构建的组合优化模型对我国7个产区的小叶杨幼苗生长呈现出较大的F值(α = 0.01)和较高的预测精度,其中门源回族自治县样本的预测精度提高了0.90%,富平县和都兰县的样本预测精度分别提高了0.37%和0.34%,且模型显著相关。(3)通过组合优化模型参数发现16个产区样本平均在17.73 d达到生长速率最大点。
    结论  小叶杨幼苗的生长受地理位置和气候等条件的影响,通过建立符合不同产区小叶杨幼苗生长的模型,有利于提高生长模型的精度和适用性,不仅能为幼苗研究提供科学基础,也为进一步的全基因组关联分析奠定基础。
    Abstract:
    Objective  Through the research on the growth regularity and growth model of Populus simonii seedlings, it could provide reference for its growth prediction and scientific seedling raising.
    Method  Taking P. simonii seedlings in 16 regions of China as the research object, basic data were obtained by means of experimental field investigation and measurement, and the growth rules of plant height were analyzed from the time dimension. We used existed growth equations to select the optimal basic growth model according to the model goodness of fit and evaluation indexes, and constructed optimal growth models suitable for seedling growth in different regions with the basis of the optimal model.
    Result  (1) The comprehensive optimal basic models of sample plant height in different regions were Logistic equation and Gompertz equation, respectively. The Logistic model’s R2 and prediction accuracy were above 0.847 9 and 92.23%, and the Gompertz model ’s R2 and prediction accuracy were above 0.891 5 and 92.60%, respectively. (2) On this basis, the combinative optimization model constructed showed a larger F value (α = 0.01) and a higher prediction accuracy for the growth of P. simonii seedlings in 7 regions of China, among which the prediction accuracy of samples in Menyuan County increased by 0.90%, and the sample prediction accuracy of samples in Fuping County and Dulan County increased by 0.37% and 0.34%, respectively. (3) Through combinatorial optimization of model parameters, it was found that samples from 16 regions reached the maximum growth rate in the 17.73th day on average.
    Conclusion  The growth of P.  simonii seedlings was affected by geographical location and climate and other conditions. Establishing models suitable for the growth of seedlings of P. simonii in different regions was conducive to improve the accuracy and applicability of the models, which could not only provide a scientific basis for seedling research, but also lay a foundation for further genome-wide association analysis.
  • 植物的寿命是其生活史上一个重要的特征,准确判断植物的生长年龄对理解植物在特定环境中的发育和繁殖更新状况,评估其生长影响因子及环境适应能力,由此制定合理的栽培管理及开发利用措施意义重大[1-5]。植物生长年龄一般可以通过周年生长形态、物候及生长轮等进行分析。在木本植物中,多年生茎中的年轮解剖结构特征可以反映其实际生长年限及生长发育状况[1,6]。研究表明,多年生草本植物位于地下的宿存器官(根茎,块茎,块根和鳞茎)中存在类似树木年轮的“生长轮”可以作为其生长年限判别的依据[1]。根茎是根茎类植物营养物质的重要贮存场所[7-9],也是其自然更新和分株繁殖的主要器官,是芽与根生理整合的枢纽通道[10],对根茎的形态特征、次生结构及其生长年龄的研究是揭示其生长发育及环境适应机制的重要基础,因而受到广泛关注。目前,对根茎生长年龄的判断一方面是根据其世代繁殖更新的形态特征来确定[2-5],另一方面可以通过其生长轮来进行判断,在灰白千里光(Senecio incanus),草甸鼠尾草(Salvia pratensis )及奇异蜂斗菜(Petasites paradoxus)等植物的根茎中均发现有生长轮的存在[1]

    芍药(herbaceous peony)是典型的根茎类多年生草本植物,野生遗传资源丰富[11],栽培品种繁多,形成了3大品种类群[12-15],有重要的观赏和药用价值[16-17]。目前关于芍药根茎的研究报道较少,仅对中国芍药品种群的个别品种的根茎生长发育及初生组织结构特征进行了初步的研究[18],对芍药根茎中是否存在生长轮以及不同品种间的生长轮差异尚未有报道。因此,本研究通过比较观察分析不同芍药品种群品种的根茎形态生长发育特点,对芍药根茎进行解剖学研究,观察其生长轮特点,判别其生长年限,以期为芍药合理栽培措施的制定、无性繁殖技术的优化及资源的开发利用研究提供一定的基础理论指导。研究结果对其他多年生根茎类植物的生长年龄判断及相关研究的开展也具有一定的借鉴意义。

    芍药不同品种群品种(表1),种植于国家花卉工程中心芍药种质资源圃(北京昌平区小汤山镇),供试材料为3年生分株苗。

    表  1  供试芍药品种信息
    Table  1.  Variety information of experimental materials
    编号 No.品种名称 Variety name品种群分类 Classification of cultivar groups倍性 Ploidy level
    1 ‘种生粉’ ‘Zhongshengfen’ 中国芍药品种群 Lactiflora group 2n = 2x = 10
    2 ‘粉玉奴’ ‘Fenyunu’ 中国芍药品种群 Lactiflora group 2n = 2x = 10
    3 ‘珊瑚落日’ ‘Coral Sunset’ 杂种芍药品种群 Hybrid group 2n = 3x = 15
    4 ‘乳霜之愉’ ‘Cream Delight’ 杂种芍药品种群 Hybrid group 2n = 4x = 20
    5 ‘草原风情’ ‘Prairie Charm’ 伊藤杂种品种群 Itoh hybrid group 2n = 3x = 15
    6 ‘抓狂的香蕉’ ‘Going Bananas’ 伊藤杂种品种群 Itoh hybrid group 2n = 3x = 15
    下载: 导出CSV 
    | 显示表格

    于2018年9月中旬至11月底,剪除芍药地上枯茎,将地下根茎整体起挖,去除泥土、杂物,沿根茎生长方向将其理顺并去除多余的肉质根以使根茎清晰可见,拍照观察并记录芍药根茎的生长更新特征。

    选取不同品种当年生根茎芽基部1 cm以下的成熟根茎组织,沿其横轴切取约2 mm的薄片,FAA固定液(50%乙醇∶甲醛∶冰醋酸 = 90∶5∶5,体积比)真空固定处理24 h以上,随后加入约1/5 FAA体积甘油软化处理10 d以上,随后经脱水,透明,浸蜡,包埋后切片,切片厚度16 ~ 25 μm,经固绿−番红染色后中性树胶封片,Leica EZ4HD体式显微镜观察和拍照。

    体式显微观察:按照根茎的着生规律,切取发育正常的不同生长年限的根茎,冰盒保存带至实验室,用自来水冲洗3遍,去除根茎表面的泥土及其他杂质,置于吸水纸上室温晾1 ~ 2 h左右,观察时不锈钢刀片沿其横轴方向截平,将切口晾3 ~ 5 min后Leica EZ4HD体式显微镜拍照观察。

    石蜡切片制备:按照根茎的着生规律,选取发育正常的芍药不同生长年限的根茎按1.2.2所述方法进行切片观察。

    不同芍药品种群品种植株地下的组织架构基本一致,即由根茎、着生于根茎上的根茎芽和根3部分组成,根茎与肉质根在颜色上基本一致,除顶部根茎上着生的根茎芽外,下部根茎上也宿存大量处于休眠状态的根茎芽。正常发育的芍药地下根茎发育形态具有较明显的年龄分级特征,我们把当年根茎芽萌发后形成的根茎作为1龄生根茎,则1龄生根茎所着生的上一年形成的母代根茎则为2龄生根茎,2龄生根茎所着生的上一年形成的母代根茎为3龄生根茎,以此类推,各生长年限的根茎之间以根茎上宿存的茎或者残留的茎痕为界,偶见有当年生根茎着生于2龄以上的母代根茎(图1)。

    图  1  芍药根茎结构发育示意
    1YR:1龄生根茎;2YR:2龄生根茎;3YR:3龄生根茎;4YR:4龄生根茎;5YR:5龄生根茎;RB:根茎芽;St:茎;AR:不定根。1YR, 1 year old rhizome; 2YR, 2 years old rhizome; 3YR, 3 years old rhizome; 4YR, 4 years old rhizome; 5YR, 5 years old rhizome; RB, rhizome bud; St, stem; AR, adventitious root.
    Figure  1.  Structural characteristic of rhizome of herbaceous peony

    ‘种生粉’‘粉玉奴’‘Coral Sunset’‘Prairie Charm’和‘Going Bananas’5个品种地下根茎结构形态类似:每年的纵向(长度)生长量适中且横向(直径)膨大变异较小,不同龄级的根茎组织结构易于区分,且根茎背地向上更新(图2ab)。四倍体品种‘Cream Delight’每年纵向生长量小而横向生长量较大,膨大明显,不同龄级的根茎组织结构紧凑而不易区分,且往往与地表水平方向横向更新(图2cd)。

    图  2  芍药不同品种地下根茎发育结构特征
    a、b. ‘珊瑚落日’根茎;c、d. ‘草原风情’根茎;RB:根茎芽;Rt:根;Rh:根茎;St:茎。a, b, the rhizome of‘Coral Sunset’; c, d, the rhizome of ‘Cream Delight’; RB, rhizome bud; Rt, root; Rh, rhizome; St, stem.
    Figure  2.  Structural characteristics of rhizomes of different cultivars of herbaceous peony

    6个芍药品种根茎截面解剖构造均符合双子叶植物茎的次生构造,由周皮、皮层、次生韧皮部、形成层、次生木质部和中央髓组成(图3)。

    图  3  不同品种根茎解剖结构
    a、b. ‘大富贵’根茎解剖结构;c、d. ‘粉玉奴’根茎解剖结构;e、f. ‘珊瑚落日’根茎解剖结构;g、h. ‘乳霜之愉’根茎解剖结构;i、j. ‘草原风情’根茎解剖结构;k、l. ‘抓狂的香蕉’根茎解剖结构;Pe:周皮;Sp:次生韧皮部;Vc:维管形成层;Sx:次生木质部;Pi:髓。标尺 = 1 000 μm。a, b, rhizome anatomy of ‘Dafugui’; c, d, rhizome anatomy of ‘Fenyunu’; e, f, rhizome anatomy of ‘Coral Sunset’; g, h, rhizome anatomy of ‘Cream Delight’; i, j, rhizome anatomy of ‘Prairie Charm’; k, l, rhizome anatomy of ‘Going Bananas’; Pe, periderm; Sp, secondary phloem; Vc, vascular cambium; Sx, secondary xylem; Pi, pith. Scale bar = 1 000 μm.
    Figure  3.  Anatomical structure of rhizomes of different cultivars of herbaceous peony

    ‘种生粉’‘粉玉奴’‘Coral Sunset’和‘Cream Delight’4个品种根茎次生木质部显微结构类似:大小导管有规律地依次排列,口径较大的导管和周围的小导管聚集形成群团状,导管群分布较稀疏,两导管群之间的间隔明显。与‘Cream Delight’相比,‘Coral Sunset’的导管群分布较紧凑。‘Prairie Charm’和‘Going Bananas’根茎的次生木质部大小导管分布较均匀,形成较连续的环带,并不聚集形成团块状(图3)。

    芍药根茎截面在脱水后维管组织凸起,呈白色或淡黄色,间断环状分布,中央髓部组织下凹,位于不同环的维管组织从髓部向皮层呈放射状排列(图4)。

    图  4  芍药‘粉玉奴’根茎横切面结构
    a、b. 2龄生根茎;c、d. 6龄生根茎;Sx:次生木质部;Pi:髓。标尺 = 1 000 μm。a, b, 2 years old rhizome; c, d, 6 years old rhizome; Sx, secondary xylem; Pi, pith. Scale bar = 1 000 μm.
    Figure  4.  Cross section structure of rhizome of Paeonia lactiflora ‘Fenyunu’

    次生木质部显微观察结果显示,口径较大的导管及其周围的小导管聚集呈团块状,导管群切向断续排列成与形成层平行的环,形成清晰的生长轮(图5)。

    图  5  芍药‘粉玉奴’根茎横切面显微结构
    a. 2龄生根茎;b. 4龄生根茎;Vc:维管形成层;Pi:髓。标尺 = 1 000 μm。a, 2 years old rhizome; b, 4 years old rhizome; Vc, vascular cambium; Pi, pith. Scale bar = 1 000 μm.
    Figure  5.  Microstructure of the cross section of the rhizome of Paeonia lactiflora ‘Fenyunu’

    对生长发育正常的芍药不同生长年限根茎进行组织切片观察发现,一年生根茎生长轮数目为1(图6a),2年生根茎生长轮的数目为2(图6b),3年生根茎的生长轮数目为3(图6c),依此类推。生长轮的数目与其实际生长年限一致。

    图  6  芍药根茎不同生长年限生长轮观察
    a. 1龄生根茎;b. 2龄生根茎;c. 3龄生根茎;d. 4龄生根茎;e. 5龄生根茎;f、g. 6龄生根茎;h、i. 7龄生根茎;①. 第1个生长轮;②. 第2个生长轮;③. 第3个生长轮;④. 第4个生长轮;⑤. 第5个生长轮;⑥. 第6个生长轮;⑦. 第7个生长轮。标尺 = 1 000 μm。a, 1 year old rhizome; b, 2 years old rhizome; c, 3 years old rhizome; d, 4 years old rhizome; e, 5 years old rhizome; f, g, 6 years old rhizome; h, i, 7 years old rhizome; ①, the first growth ring; ②, the second growth ring; ③, the third growth ring; ④, the fourth growth ring; ⑤, the fifth growth ring; ⑥, the sixth growth ring; ⑦, the seventh growth ring. Scale bar = 1 000 μm.
    Figure  6.  Observation on the growth rings of the rhizome of herbaceous peony under a stereomicroscope

    根茎的形态及生长年限反映了植物在特定气候环境条件下的生长发育状况。准确判断根茎的年龄结构对预知植物个体乃至种群繁殖发育现状及未来更新的动态发展,由此制定合理的栽培及开发利用措施意义重大[1]。目前,对根茎类植物年龄结构的判断尚无统一标准和方法,一般是根据其实际栽培年限[8, 19-21]、营养繁殖世代特征结合颜色及直径大小等进行判断[2-3]。本研究中,芍药每年夏秋形成的当年生根茎由位于上一年形成的母代根茎芽发育而来,由此逐年进行世代更替,通过这种繁殖世代特征可以初步判断芍药根茎的年龄结构。芍药根茎芽的更新严格受控于顶端优势的调控[22],因而在发育正常的情况下,芍药的根茎一般按照实际生长年限逐级生长[18],但是,在本研究中,我们观察到在一些植株中当年生根茎由2龄或更高级年龄的母代根茎发育而来,若非经全株整体观察及长时间的持续追踪,完全按照根茎由上至下的分级次序来判别每一级根茎的生长年限往往存在一定的困难,对母代根茎的实际生长年限易造成误判。近年来兴起的草本植物生长轮研究为草本植物生长年限的研究提供了新的思路[1, 17, 21]。本研究中,芍药根茎的初生结构与茎的结构基本类似,由表皮、皮层、维管束和中央髓组成[18, 23]。与地上茎不同的是,芍药根茎的次生结构外围形成了具有保护作用的周皮组织,因而其能多年宿存生长。芍药不同龄级根茎中存在明显的生长轮,且生长轮的数目与其对应根茎的实际生长年限一致,可以作为判别芍药根茎实际生长年限的稳定依据。

    芍药根茎生长轮的组织形态和根茎的发育状况受植物本身遗传差异和栽培环境的影响。本研究中,考虑到供试样本栽培环境基本一致,不同品种根茎形态及生长轮的差异可能主要与其亲本来源不同有关。中国芍药品种群和杂种芍药品种群各品种亲本来源于芍药属(Paeonia)的多年生草本植物类群,品种群内各品种根茎生长轮的组织结构类似,木质部导管群断续排列成环,而伊藤杂种品种群内两个品种根茎次生木质部呈现连续的环带分布,主要是由于其亲本融合了芍药属亚灌木的牡丹类群的遗传信息,因而生长轮结构与牡丹茎的次生结构类似[23],据此,可以将其与其他两个品种群的品种进行区分。至于这种次生结构差异对其存活年限的影响有待进一步研究。

    植物多倍体往往具有营养器官大、抗逆性强、生长迅速等特点[24-27]。与二倍体品种相比,芍药多倍体品种往往也表现出茎秆粗壮、直立性强等生长优势[15,28-29]。本研究中,从根茎生长表现来看,四倍体品种‘Cream Delight’相同龄级的根茎体量明显大于二倍体及三倍体品种,由于根茎每年生长量大,加之向地伸展空间有限,因而多呈水平状横向更新。而三倍体品种根茎形态并未表现出与二倍体品种明显的生长差异,可能原因及调控机制有待进一步研究。从根茎生长轮的组织结构特征来看,同一品种群内相同倍性的品种间根茎生长轮特征基本类似,而不同倍性的品种间差异较大;而品种群间不同品种染色体倍性与其根茎形态无明显关联,‘Coral Sunset’‘Prairie Charm’和‘Going Bananas’3个品种均为三倍体,但是根茎次生结构差异明显。因而,仅根据遗传倍性不能区分各品种的根茎生长轮特征。当然,由于生长轮的形成和发育受环境条件影响较大,加之芍药品种遗传背景复杂,关于生长轮的发育特性在芍药中的更普遍规律需要结合更多样本开展更进一步的研究。

    根茎由于随着生长年限的增大,受限于材料的大小以及软硬程度的差异,采用组织切片的方法不能一一鉴别且花费时间较长,因而徒手切片结合体式显微观察可以作为多年生根茎年龄判别的快速方法。在生产实践中,我们可以通过上述徒手切片的一般操作方式快速地区分根茎与根,鉴定根茎的年龄结构。

    芍药不同品种地下根茎组织架构特征基本一致,且存在明显的龄级特征。二倍体及三倍体品种根茎形态发育特征相似,而与四倍体品种不同。不同芍药品种根茎次生结构均由周皮、皮层、次生韧皮部、形成层、次生木质部和中央髓组成,中国芍药及杂种芍药品种群品种根茎生长轮结构相似而与伊藤杂种差异明显,杂种芍药品种群内三倍体及四倍体品种根茎生长轮结构差异较大,根茎生长轮结构特征与其品种倍性无关。芍药根茎中存在生长轮,且其数目能够反映芍药的实际生长年限。

  • 图  1   我国16个产区的小叶杨幼苗样本生长曲线对比图

    A. 河北赤城产区;B. 陕西富民产区;C. 陕西富县产区;D. 青海贵德产区;E. 陕西高陵产区;F. 宁夏中宁产区;G. 陕西岚皋产区;H. 陕西麟游产区;I. 青海门源产区;J. 内蒙古包头产区;K. 河南嵩县产区;L. 北京陶然亭产区;M. 青海都兰产区;N. 青海兴海产区;O. 河南伊川产区;P. 河北张家口产区。表示4月29日、5月20日和6月4日小叶杨幼苗实际株高;表示6月17日小叶杨幼苗实际株高。A, Chicheng production area of Hebei; B, Fumin production area of Shaanxi; C, Fu production area of Shaanxi; D, Guide production area of Qinghai; E, Gaoling production area of Shaanxi; F, Zhongning production area of Ningxia; G, Langao production area of Shaanxi; H, Linyou production area of Shaanxi; I, Menyuan production area of Qinghai; J, Baotou production area of Inner Mongolia; K, Songxian production area of Henan; L, Taoranting production area of Beijing; M, Dulan production area of Qinghai; N, Xinghai production area of Qinghai; O, Yichuan production area of Henan; P, Zhangjiakou production area of Hebei. indicates the actual plant height of P. simonii seedlings on April 29, May 20 and June 4; indicates the actual plant height of P. simonii seedlings on June 17.

    Figure  1.   Comparison in growth curves of P. simonii seedling samples in 16 production areas of China

    表  1   小叶杨株高的描述性统计分析

    Table  1   Descriptive statistical analysis on seedling height of P. simonii

    产地
    Production area
    4月29日
    April 29th
    6月17日
    June 17th
    产地
    Production area
    4月29日
    April 29th
    6月17日
    June 17th
    河北赤城
    Chicheng County, Hebei Province
    15.52 ± 0.61 29.30 ± 0.55 河南嵩县
    Song County, Henan Province
    14.69 ± 1.25 28.37 ± 1.33
    陕西富平
    Fuping County, Shaanxi Province
    17.16 ± 1.02 43.51 ± 0.89 陕西麟游
    Linyou County, Shaanxi Province
    15.07 ± 1.00 29.84 ± 0.78
    陕西富县
    Fu County, Shaanxi Province
    16.67 ± 0.88 35.11 ± 1.20 青海都兰
    Dulan County, Qinghai Province
    13.41 ± 0.79 25.11 ± 0.56
    青海贵德
    Guide County, Qinghai Province
    11.15 ± 1.23 18.39 ± 0.82 青海兴海
    Xinghai County, Qinghai Province
    14.12 ± 0.73 28.35 ± 0.24
    陕西高陵
    Gaoling District, Shaanxi Province
    16.86 ± 0.45 27.02 ± 0.76 陕西岚皋
    Langao County, Shaanxi Province
    15.07 ± 1.57 27.35 ± 1.44
    宁夏中宁
    Zhongning County, Ningxia Hui Autonomous Region
    15.19 ± 0.90 28.73 ± 0.67 河北张家口
    Zhangjiakou City, Hebei Province
    12.35 ± 1.22 27.65 ± 1.03
    青海门源
    Menyuan Hui Autonomous County, Qinghai Province
    12.31 ± 0.43 20.24 ± 0.56 内蒙古包头
    Baotou City, Inner Mongolia Autonomous Region
    12.87 ± 0.57 22.56 ± 0.43
    河南伊川
    Yichuan County, Henan Province
    16.73 ± 0.68 29.01 ± 0.78 北京陶然亭
    Taoranting, Beijing
    18.10 ± 0.60 26.73 ± 0.48
    下载: 导出CSV

    表  2   生长模型表达式

    Table  2   Expression of growth model

    模型 Model逻辑斯谛 Logistic理查兹 Richards韦布尔 Weibull冈珀茨 Gompertz
    表达式 Expression y=k1+exp(abt) y=a(1expct)b y=a(1expbtc) y=(aexpbexpct)
    注:k代表最大生长量;a代表初始株高;b代表生长率;c代表最大生长天数;t代表时间。Notes: k represents the maximum growth amount; a represents initial plant height; b represents growth rate; c represents the maximum growth days; t represents time.
    下载: 导出CSV

    表  3   16个产区样本株高的Logistic模型参数估计

    Table  3   Logistic model parameter estimation of plant height in 16 production areas

    产地
    Production area
    k/cma/cmb/%最大生长速率点
    Maximum growth rate point
    下渐近线
    Lower asymptote
    河北赤城 Chicheng County, Hebei Province 28.403 6 2.118 2 0.118 0 (17.95, 14.20) 3.048 8
    陕西富平 Fuping County, Shaanxi Province 40.705 2 2.422 2 0.117 5 (20.61, 20.35) 3.317 2
    陕西富县 Fu County, Shaanxi Province 27.633 4 1.399 7 0.093 7 (14.93, 13.81) 5.467 6
    青海贵德 Guide County, Qinghai Province 18.479 6 2.465 5 0.143 8 (17.14, 9.23) 1.447 1
    陕西高陵 Gaoling District, Shaanxi Province 25.859 0 1.344 1 0.081 8 (16.43, 12.92) 5.348 5
    宁夏中宁
    Zhongning County, Ningxia Hui Autonomous Region
    30.556 7 2.776 6 0.150 1 (18.49, 15.27) 1.790 6
    陕西岚皋 Langao County, Shaanxi Province 26.462 2 2.037 8 0.120 8 (16.86, 13.23) 3.050 8
    陕西麟游 Linyou County, Shaanxi Province 30.480 9 1.875 9 0.104 2 (18.00, 15.24) 4.049 7
    青海门源
    Menyuan Hui Autonomous County, Qinghai Province
    19.417 6 1.834 5 0.126 1 (14.54, 9.70) 2.673 8
    内蒙古包头
    Baotou City, Inner Mongolia Autonomous Region
    22.159 1 1.885 5 0.106 5 (17.70, 11.07) 2.919 6
    河南嵩县 Song County, Henan Province 28.768 0 1.648 3 0.077 0 (21.40, 14.38) 4.641 3
    北京陶然亭 Taoranting, Beijing 35.709 8 0.478 8 0.026 7 (17.93, 17.85) 13.660 0
    青海都兰 Dulan County, Qinghai Province 23.386 5 2.933 5 0.163 9 (17.89, 11.69) 1.181 5
    青海兴海 Xinghai County, Qinghai Province 24.462 6 2.997 6 0.172 3 (17.39, 12.23) 1.162 8
    河南伊川 Yichuan County, Henan Province 26.129 0 1.545 3 0.093 0 (16.61, 13.06) 4.592 5
    河北张家口 Zhangjiakou City, Hebei Province 28.173 6 2.070 8 0.104 2 (19.87, 14.08) 3.154 5
    注:下渐近线为k/(1 + exp(a)),最大生长速率点是(a/b, k/2),a/b是生长天数,k/2是相应的株高。Notes: k/(1 + exp(a)) is the lower asymptote, the maximum growth rate point is (a/b, k/2), a/b is the growth days, k/2 is the corresponding plant height.
    下载: 导出CSV

    表  4   我国16个小叶杨幼苗样本产区的地理特征

    Table  4   Geographical characteristics of P. simonii seeding samples of 16 production areas in China

    产地
    Production area
    经度
    Longitude
    纬度
    Latitude
    海拔
    Altitude/m
    年均温度
    Annual average temperature/℃
    年均降雨量
    Average annual rainfall/mm
    河北赤城 Chicheng County, Hebei Province 115º15′ ~ 116º16′E 40º18′ ~ 41º13′N 945 6.8 404.9
    陕西富平 Fuping County, Shaanxi Province 102°12′ ~ 102°28′E 25°04′ ~ 25°21′N 400 15.8 846.5
    陕西富县 Fu County, Shaanxi Province 108°17′ ~ 109°25′E 35°26′ ~ 36°13′N 944 16.9 500.0
    青海贵德 Guide County, Qinghai Province 109°27′ ~ 117°12′E 20°05′ ~ 25°18′N 3 000 2.3 1 777.0
    陕西高陵 Gaoling District, Shaanxi Province 109°04′ ~ 110°13′E 34°31′ ~ 36°18′N 400 15.6 732.9
    宁夏中宁
    Zhongning County,Ningxia Hui Autonomous Region
    105°15′ ~ 106°04′E 37°05′ ~ 37°30′N 2 955 9.5 202.1
    陕西岚皋 Langao County, Shaanxi Province 108°53′ ~ 109°01′E 32°01′ ~ 32°33′N 1 500 16.0 1 050.0
    陕西麟游 Linyou County, Shaanxi Province 108°43′ ~ 108°52′E 32°03′ ~ 32°30′N 1 486 17.0 1 000.0
    青海门源
    Menyuan Hui Autonomous County, Qinghai Province
    107°11′ ~ 108°01′E 34°19′ ~ 34°34′N 1 271 9.1 680.0
    内蒙古包头
    Baotou City, Inner Mongolia Autonomous Region
    102°00′ ~ 102°03′E 37°03′ ~ 38°00′N 2 866 0.8 520.0
    河南嵩县 Song County, Henan Province 111º24′ ~ 112º22′E 33º35′ ~ 34º21′N 1 966.6 14.0 800.0
    北京陶然亭 Taoranting, Beijing 115°07′ ~ 117°04′E 39°04′ ~ 41°06′N 44 12.6 483.9
    青海都兰 Dulan County, Qinghai Province 98°06′ ~ 98°34′E 36°19′ ~ 37°08′N 3 180 2.7 179.1
    青海兴海 Xinghai County, Qinghai Province 99°01′ ~ 100°43′E 34°37′ ~ 36°10′N 3 924 1.5 359.0
    河南伊川 Yichuan County, Henan Province 112°36′ ~ 114°01′E 34°36′ ~ 35°52′N 277 15.0 620.0
    河北张家口 Zhangjiakou City, Hebei Province 113°42′ ~ 116°19′E 39°21′ ~ 42°07′N 1 400 10.0 400.0
    下载: 导出CSV

    表  5   16个产区样本株高的Logistic模型的评价指标

    Table  5   Evaluating indices of Logistic model for sample plant height in 16 production areas

    产地
    Production area
    R2赤池信息准则
    An information criterion (AIC)
    贝叶斯信息准则
    Bayesian information criterion (BIC)
    MSEF
    河北赤城 Chicheng County, Hebei Province 0.929 6 − 18.745 7 − 17.973 1 0.291 6 198.77
    陕西富平 Fuping County, Shaanxi Province 0.903 8 1.934 7 2.707 2 1.062 3 145.49
    陕西富县 Fu County, Shaanxi Province 0.936 5 − 26.658 2 − 25.885 6 0.177 8 220.42
    青海贵德 Guide County, Qinghai Province 0.945 7 − 38.140 8 − 37.368 3 0.086 7 257.71
    陕西高陵 Gaoling District, Shaanxi Province 0.910 0 − 21.812 2 − 21.039 6 0.240 8 155.54
    宁夏中宁
    Zhongning County, Ningxia Hui Autonomous Region
    0.986 0 − 32.686 7 − 31.914 1 0.122 0 1 000.40
    陕西岚皋 Langao County, Shaanxi Province 0.939 5 − 25.353 9 − 24.581 3 0.192 9 231.45
    陕西麟游 Linyou County, Shaanxi Province 0.921 0 − 15.111 9 − 14.339 3 0.366 0 177.20
    青海门源
    Menyuan Hui Autonomous County, Qinghai Province
    0.958 5 − 46.320 3 − 45.547 7 0.052 0 337.58
    内蒙古包头
    Baotou City, Inner Mongolia Autonomous Region
    0.925 1 − 26.512 1 − 25.739 5 0.179 5 186.96
    河南嵩县Song County, Henan Province 0.847 9 − 5.651 4 − 4.878 8 0.661 2 92.07
    北京陶然亭 Taoranting, Beijing 0.340 2 12.550 9 13.323 5 2.062 6 21.21
    青海都兰 Dulan County, Qinghai Province 0.167 0 30.447 2 31.219 8 6.312 2 5.24
    青海兴海 Xinghai County, Qinghai Province 0.949 9 − 30.416 3 − 29.643 8 0.140 6 279.36
    河南伊川 Yichuan County, Henan Province 0.922 9 − 22.971 8 − 22.199 2 0.223 9 181.49
    河北张家口 Zhangjiakou City, Hebei Province 0.903 9 − 11.891 6 − 11.119 0 0.447 6 145.63
    注:R2 = 1 − (SSE/SST),其中SST为总平方和,SSE为残差平方和。MSE = SSE/nk,其中n是样本的数目,k是参数个数. Notes: R2 = 1 − (SSE/SST), where SST is the sum of total squares, SSE is the sum of residual squares. MSE = SSE/nk, where n is the number of samples and k is the number of parameters. AIC = ln(SSE/n) + 2k; BIC = ln(SSE/n) + k·ln(n); Fα = 8.86 (P < 0.01); Fα = 4.16 (P < 0.05).
    下载: 导出CSV

    表  6   16个产区样本株高的Gompertz模型参数估计

    Table  6   Gompertz model parameter estimation of sample plant height in 16 production areas

    产地
    Production area
    k/cma/cmb/%最大生长速率点
    Maximum growth rate point
    下渐近线
    Lower asymptote
    河北赤城Chicheng County, Hebei Province 28.498 8 1.518 5 0.102 9 (14.75, 14.24) 0.296 5
    陕西富平 Fuping County, Shaanxi Province 40.956 1 1.659 6 0.098 3 (16.88, 20.47) 0.213 3
    陕西富县 Fu County, Shaanxi Province 27.778 5 0.911 3 0.081 3 (11.20, 13.88) 2.308 7
    青海贵德 Guide County, Qinghai Province 18.497 3 1.938 5 0.130 6 (14.84, 9.24) 0.017 7
    陕西高陵 Gaoling District, Shaanxi Province 26.153 8 0.791 9 0.067 8 (11.67, 13.07) 2.876 0
    宁夏中宁
    Zhongning County, Ningxia Hui Autonomous Region
    31.960 0 2.057 9 0.105 0 (19.59, 15.98) 0.012 7
    陕西岚皋 Langao County, Shaanxi Province 26.524 4 1.498 7 0.107 2 (13.98, 13.26) 0.301 8
    陕西麟游 Linyou County, Shaanxi Province 30.660 6 1.265 9 0.088 8 (14.25, 15.33) 0.883 9
    青海门源
    Menyuan Hui Autonomous County, Qinghai Province
    19.438 5 1.416 3 0.115 6 (12.25, 9.71) 0.315 1
    内蒙古包头
    Baotou City, Inner Mongolia Autonomous Region
    22.271 4 1.291 3 0.091 5 (14.11, 11.13) 0.586 1
    河南嵩县 Song County, Henan Province 29.587 9 0.915 1 0.058 4 (15.66, 14.79) 2.435 9
    北京陶然亭 Taoranting, Beijing 40.219 3 0.095 0 0.016 9 (5.61, 20.10) 13.39 2
    青海都兰 Dulan County, Qinghai Province 23.398 8 2.377 5 1.150 0 (2.06, 11.69) 0.000 4
    青海兴海 Xinghai County, Qinghai Province 24.470 4 2.487 1 0.159 6 (15.58, 12.23) 0.000 1
    河南伊川 Yichuan County, Henan Province 26.318 5 0.991 4 0.079 0 (12.54, 13.15) 1.777 6
    河北张家口 Zhangjiakou City, Hebei Province 28.412 9 1.362 9 0.086 3 (15.79, 14.20) 0.570 8
    下载: 导出CSV

    表  7   16个产区样本株高的Gompertz模型的评价指标

    Table  7   Evaluating indices of Gompertz model for sample plant height in 16 production areas

    产地
    Production area
    R2AICBICMSEF
    河北赤城 Chicheng County, Hebei Province 0.999 8 − 110.733 0 − 109.960 0 0.000 9 89 725.28
    陕西富平 Fuping County, Shaanxi Province 0.999 8 − 87.620 2 − 86.847 6 0.003 9 58 988.08
    陕西富县 Fu County, Shaanxi Province 0.999 5 − 99.736 9 − 98.964 3 0.001 8 29 681.81
    青海贵德 Guide County, Qinghai Province 1.000 0 − 153.014 0 − 152.241 0 6.613 3 47 580.73
    陕西高陵 Gaoling District, Shaanxi Province 0.999 0 − 87.153 6 − 86.381 1 0.004 0 13 339.15
    宁夏中宁 Zhongning County, Ningxia Hui Autonomous Region 0.891 5 8.561 7 9.334 3 1.607 4 129.05
    陕西岚皋 Langao County, Shaanxi Province 0.999 9 − 119.867 3 − 119.094 0 0.000 5 19 685.19
    陕西麟游 Linyou County, Shaanxi Province 0.999 7 − 96.046 1 − 95.273 5 0.002 3 40 480.97
    青海门源Menyuan Hui Autonomous County, Qinghai Province 0.999 9 − 144.849 3 − 144.076 0 0.000 1 2 272.73
    内蒙古包头 Baotou City, Inner Mongolia Autonomous Region 0.999 7 − 109.285 0 − 108.513 0 0.001 0 47 610.94
    河南嵩县 Song County, Henan Province 0.998 0 − 68.400 9 − 67.628 3 0.013 0 7 053.93
    北京陶然亭 Taoranting, Beijing 0.990 6 − 60.268 9 − 59.496 3 0.021 7 1 483.19
    青海都兰 Dulan County, Qinghai Province 0.167 0 30.447 2 31.219 8 6.312 2 5.24
    青海兴海 Xinghai County, Qinghai Province 1.000 0 − 171.298 0 − 170.525 0 0.000 02 95 643.44
    河南伊川 Yichuan County, Henan Province 0.999 4 − 95.756 1 − 94.983 5 0.002 3 24 605.57
    河北张家口 Zhangjiakou City, Hebei Province 0.999 6 − 92.047 2 − 91.274 6 0.002 9 32 600.28
    下载: 导出CSV

    表  8   16个产区样本株高的组合优化模型参数估计和拟合优度值

    Table  8   Parameter estimation of combinatorial optimization model for sample plant height in 16 production areas

    产地
    Production area
    标准差1
    Standard deviation 1 (σ1)
    标准差2
    Standard deviation 2 (σ2)
    权重1
    Weight 1 (w1)
    权重2
    Weight 2 (w2)
    R2
    河北赤城 Chicheng County, Hebei Province 0.038 3 0.054 6 0.456 0 0.544 0 0.988 8
    陕西富平 Fuping County, Shaanxi Province 0.151 2 0.236 2 0.444 9 0.555 1 0.986 2
    陕西富县 Fu County, Shaanxi Province 0.098 1 0.127 7 0.467 1 0.532 9 0.988 6
    青海贵德 Guide County, Qinghai Province 0.002 3 0.003 2 0.458 7 0.541 3 0.991 0
    陕西高陵 Gaoling District, Shaanxi Province 0.233 3 0.307 3 0.465 7 0.534 3 0.983 8
    宁夏中宁
    Zhongning County, Ningxia Hui Autonomous Region
    0.005 9 0.008 7 0.451 9 0.548 1 0.940 1
    陕西岚皋 Langao County, Shaanxi Province 0.021 5 0.029 6 0.459 9 0.540 1 0.989 9
    陕西麟游 Linyou County, Shaanxi Province 0.108 4 0.152 8 0.457 4 0.542 6 0.987 3
    青海门源
    Menyuan Hui Autonomous County, Qinghai Province
    0.004 5 0.005 8 0.468 2 0.531 8 0.992 4
    内蒙古包头
    Baotou City, Inner Mongolia Autonomous Region
    0.046 6 0.065 3 0.458 1 0.541 9 0.987 8
    河南嵩县 Song County, Henan Province 0.770 7 1.085 1 0.457 5 0.542 5 0.974 9
    北京陶然亭 Taoranting, Beijing 2.526 6 2.711 7 0.491 2 0.508 8 0.772 1
    青海都兰 Dulan County, Qinghai Province 0.001 1 0.001 6 0.455 3 0.544 7 0.167 0
    青海兴海 Xinghai County, Qinghai Province 0.000 5 0.000 7 0.458 5 0.541 5 0.991 6
    河南伊川 Yichuan County, Henan Province 0.123 7 0.166 4 0.463 1 0.536 9 0.986 7
    河北张家口 Zhangjiakou City, Hebei Province 0.135 6 0.201 3 0.451 1 0.548 9 0.985 5
    下载: 导出CSV

    表  9   16个产区样本株高的组合优化模型评价指标

    Table  9   Evaluating indices of combinatorial optimization model for sample plant height in 16 production areas

    产地 Production areaAICBICMSEF
    河北赤城 Chicheng County, Hebei Province − 44.890 2 − 44.117 6 0.056 9 1250.38
    陕西富平 Fuping County, Shaanxi Province − 25.299 7 − 24.527 1 0.193 6 1 011.90
    陕西富县 Fu County, Shaanxi Province − 51.182 5 − 50.409 9 0.038 4 1 226.29
    青海贵德 Guide County, Qinghai Province − 63.943 7 − 63.171 1 0.017 3 1 555.40
    陕西高陵 Gaoling District, Shaanxi Province − 46.053 7 − 45.281 1 0.052 9 865.28
    宁夏中宁 Zhongning County, Ningxia Hui Autonomous Region − 42.869 4 − 3.514 3 0.720 0 233.53
    陕西岚皋 Langao County, Shaanxi Province − 50.971 3 − 50.198 7 0.038 9 1 388.53
    陕西麟游 Linyou County, Shaanxi Province − 40.957 4 − 40.184 8 0.072 7 1 099.15
    青海门源 Menyuan Hui Autonomous County, Qinghai Province − 70.957 4 − 70.184 8 0.011 1 1 844.85
    内蒙古包头 Baotou City, Inner Mongolia Autonomous Region − 52.303 8 − 51.531 2 0.035 8 1 151.26
    河南嵩县 Song County, Henan Province − 30.787 2 − 30.014 7 0.137 4 557.67
    北京陶然亭 Taoranting, Beijing − 9.7482 3 − 8.975 6 0.511 8 61.42
    青海都兰 Dulan County, Qinghai Province 30.447 2 31.219 8 6.312 2 5.24
    青海兴海 Xinghai County, Qinghai Province − 56.062 9 − 55.290 3 0.028 3 1 659.63
    河南伊川 Yichuan County, Henan Province − 47.888 1 − 47.115 5 0.047 1 1 051.28
    河北张家口 Zhangjiakou City, Hebei Province − 38.504 1 − 37.731 5 0.084 8 966.69
    下载: 导出CSV

    表  10   16个产区样本株高的Logistic模型、Gompertz模型和组合优化模型评价指标对比

    Table  10   Comparison in evaluation indices of Logistic model, Gompertz model and combinatorial optimization model in 16 production areas

    产地
    Production area
    逻辑斯蒂
    Logistic
    冈珀茨
    Gompertz
    组合优化模型
    Combinatorial optimization model
    均方误差
    MSE
    均方绝对误差
    MAE
    均方误差
    MSE
    均方绝对误差
    MAE
    均方误差
    MSE
    均方绝对误差
    MAE
    河北赤城 Chicheng County, Hebei Province 3.312 4 0.143 4 0.008 0 0.089 6 0.013 0 0.114 1
    陕西富平 Fuping County, Shaanxi Province 40.960 0 3.150 1 9.075 9 3.012 6 9.448 4 3.073 8
    陕西富县 Fu County, Shaanxi Province 9.363 6 1.037 1 0.949 7 0.974 5 1.007 6 1.003 8
    青海贵德 Guide County, Qinghai Province 1.000 0 0.432 7 0.176 9 0.420 6 0.181 6 0.426 2
    陕西高陵 Gaoling District, Shaanxi Province 6.812 1 0.073 1 0.030 3 0.174 0 0.016 1 0.127 0
    宁夏中宁
    Zhongning County, Ningxia Hui Autonomous Region
    0.028 9 0.818 6 0.708 8 0.841 9 0.691 2 0.831 4
    陕西岚皋 Langao County, Shaanxi Province 5.760 0 1.092 8 1.116 2 1.056 2 1.151 8 1.073 2
    陕西麟游 Linyou County, Shaanxi Province 1.345 6 1.215 6 1.697 2 1.302 8 1.594 9 1.262 9
    青海门源
    Menyuan Hui Autonomous County, Qinghai Province
    4.202 5 1.405 3 1.938 8 1.392 4 1.955 7 1.398 5
    内蒙古包头
    Baotou City, Inner Mongolia Autonomous Region
    5.062 5 0.644 7 0.346 4 0.588 6 0.377 4 0.614 3
    河南嵩县 Song County, Henan Province 34.222 0 1.408 5 1.384 9 1.176 8 1.646 2 1.283 0
    北京陶然亭 Taoranting, Beijing 16.000 0 1.097 9 1.389 6 1.178 8 1.297 5 1.139 1
    青海都兰 Dulan County, Qinghai Province 3.865 2 3.036 5 2.084 0 1.443 6 2.096 2 1.447 8
    青海兴海 Xinghai County, Qinghai Province 0.950 6 0.566 4 0.314 0 0.560 3 0.317 1 0.563 1
    河南伊川 Yichuan County, Henan Province 6.105 8 0.238 5 0.025 3 0.159 0 0.038 4 0.195 9
    河北张家口 Zhangjiakou City, Hebei Province 3.482 0 0.754 8 0.754 2 0.868 5 0.667 7 0.817 1
    下载: 导出CSV

    表  11   16个产区样本株高的Logistic模型、Gompertz模型和组合优化模型预测精度和时间复杂度对比

    Table  11   Comparison in prediction accuracy and time complexity of Logistic model, Gompertz model and combinatorial optimization model in16 production areas

    产地
    Production area
    逻辑斯蒂
    Logistic
    冈珀茨
    Gompertz
    组合优化模型
    Combinatorial optimization model
    预测精度
    Prediction accuracy/%
    时间
    Time/s
    预测精度
    Prediction accuracy/%
    时间
    Time/s
    预测精度
    Prediction accuracy/%
    时间
    Time/s
    提高百分比
    Percentage increase/%
    河北赤城
    Chicheng County, Hebei Province
    0.994 9 0.402 0 0.996 8 0.736 8 0.996 0 1.060 9
    陕西富平
    Fuping County, Shaanxi Province
    0.922 3 0.403 9 0.926 0 0.622 8 0.929 7 0.985 9 0.37
    陕西富县
    Fu County, Shaanxi Province
    0.962 2 0.399 5 0.964 6 0.702 4 0.964 8 1.123 5 0.02
    青海贵德
    Guide County, Qinghai Province
    0.976 6 0.406 5 0.977 2 0.647 0 0.977 5 1.068 4 0.03
    陕西高陵
    Gaoling District, Shaanxi Province
    0.997 1 0.396 1 0.993 2 0.611 3 0.995 0 1.011 4
    宁夏中宁
    Zhongning County, Ningxia Hui Autonomous Region
    0.973 2 0.398 7 0.972 5 0.652 4 0.972 0 1.040 3
    陕西岚皋
    Langao County, Shaanxi Province
    0.958 6 0.406 8 0.960 0 0.600 3 0.961 0 1.052 5 0.10
    陕西麟游
    Linyou County, Shaanxi Province
    0.959 9 0.411 5 0.957 1 0.558 1 0.956 6 1.077 2
    青海门源
    Menyuan Hui Autonomous County, Qinghai Province
    0.927 5 0.418 3 0.928 3 0.672 3 0.932 8 1.023 6 0.90
    内蒙古包头
    Baotou City, Inner Mongolia Autonomous Region
    0.970 8 0.407 2 0.973 4 0.569 9 0.972 9 1.066 2
    河南嵩县
    Song County, Henan Province
    0.949 7 0.409 2 0.958 3 0.539 7 0.956 4 1.074 0
    北京陶然亭
    Taoranting, Beijing
    0.961 2 0.406 0 0.958 5 0.639 2 0.958 1 0.978 3
    青海都兰
    Dulan County, Qinghai Province
    0.937 9 0.409 4 0.938 3 0.610 3 0.941 7 1.008 1 0.34
    青海兴海
    Xinghai County, Qinghai Province
    0.976 8 0.414 7 0.977 1 0.553 4 0.977 5 0.993 0 0.04
    河南伊川
    Yichuan County, Henan Province
    0.990 8 0.406 6 0.993 9 0.598 4 0.992 5 0.958 3
    河北张家口
    Zhangjiakou City, Hebei Province
    0.973 0 0.404 0 0.969 1 0.527 1 0.970 0 1.091 5
    下载: 导出CSV
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  • 收稿日期:  2019-07-14
  • 修回日期:  2019-12-29
  • 网络出版日期:  2020-05-10
  • 发布日期:  2020-06-30

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