基于非线性混合效应模型的蒙古栎1年生苗木生长拟合

Growth modeling of one-year-old Quercus mongolica seedlings using a nonlinear mixed-effects model

摘要

【目的】筛选适宜描述蒙古栎1年生苗木苗高与地径生长的最优非线性生长模型，并构建非线性混合效应模型解析种源与个体层次的变异来源，揭示蒙古栎苗期生长节律及种源间生长差异。【方法】以来自辽宁、吉林、黑龙江3省9个种源地的757株蒙古栎1年生实生苗为研究对象，采用定株定时测定法，在生长季内（2025年6月20日—11月20日）每30 d测定一次苗高和地径，共获得6次重复观测数据。运用Logistic、Gompertz和von Bertalanffy 3种非线性生长模型对苗高和地径的生长动态拟合，基于均方根误差（RMSE）和平均绝对误差（MAE）最小原则筛选最优基础模型。在此基础上，进一步构建非线性混合效应模型，将渐近线参数a设为随机效应，b和k设为固定效应。采用限制性极大似然法（REML）估计参数，通过似然比检验比较不同随机效应结构的拟合优度，并利用决定系数（R²）、RMSE和MAE评价模型拟合效果。【结果】（1）以RMSE最小为主要准则，Logistic模型在苗高与地径生长拟合中均表现最优：苗高RMSE范围为3.152 ~ 7.965 cm，MAE范围为2.459 ~ 6.077 cm，地径RMSE范围为0.549 ~ 0.977 mm，MAE范围为0.439 ~ 0.731 mm。（2）基础Logistic模型因未考虑数据层次结构，拟合优度偏低，苗高R²介于0.312 ~ 0.510，地径R²介于0.453 ~ 0.594。引入种源与个体嵌套随机效应后，非线性混合效应模型的拟合优度显著提升，苗高R²提升至0.611 ~ 0.852，地径R²提升至0.685 ~ 0.868，各种源R²平均提升苗高约0.356、地径约0.257；RMSE平均降低苗高约34%、地径约33%；MAE平均降低苗高约39%、地径约36%。（3）随机效应方差组分分解结果显示，苗高生长极限的变异主要来源于个体间（方差分量22.09，占总变异39.48%），种源间变异次之（方差分量14.59，占总变异26.07%）；地径生长变异同样主要来源于个体间（方差分量0.55，占总变异44.16%），种源间变异次之（方差分量0.34，占总变异27.70%）。（4）种源间生长潜力存在显著差异，抚顺、金州、龙江种源的苗高理论最大生长量（a值）较高（分别为30.779、30.038、29.346 cm），柴河、叶赫种源较低（分别为20.359、19.786 cm）；地径生长方面，龙江种源理论最大地径最高（5.037 mm），盖州种源最低（3.245 mm）。蒙古栎苗期生长呈现“先高后径”的节律特征，9个种源苗高达到最大生长速率点的时间均早于地径。【结论】Logistic模型可作为蒙古栎1年生苗木苗高与地径生长的普适性优选模型。非线性混合效应模型通过引入种源与个体嵌套随机效应，显著提升了模型拟合精度，有效解析了苗期生长性状的层次结构变异。研究结果可为蒙古栎优质苗木培育、苗期管理优化及种质资源早期评价提供科学依据和量化工具。

Abstract

[Objective] This study sought to identify the most suitable nonlinear growth model for describing height and ground diameter growth in one-year-oldQuercus mongolica seedlings, and to develop a nonlinear mixed-effects model that partitions variation at provenance and individual levels, and to characterise the seedling growth rhythm and growth differences among provenances of Q. mongolica. [Method] We used 757 one-year-old seedlings from nine provenances across Liaoning, Jilin and Heilongjiang provinces. Height and ground diameter were measured every 30 days during the growing season (20 June to 20 November 2025), giving six repeated observations per seedling. Three growth functions—Logistic, Gompertz and von Bertalanffy—were fitted to the data. The best basic model was selected by minimising root mean square error (RMSE) and mean absolute error (MAE). We then constructed a nonlinear mixed-effects model, treating the asymptotic parametera as random and parameters b and k as fixed. Parameters were estimated using restricted maximum likelihood (REML). Likelihood-ratio tests were used to compare different random-effect structures, and model performance was evaluated using the coefficient of determination (R²), RMSE and MAE. [Result] (1) The Logistic model achieved the lowest RMSE across all provenances (3.152–7.965 cm), MAE of 2.459–6.077 cm; for ground diameter, RMSE was 0.549–0.977 mm and MAE was 0.439–0.731 mm. (2) The basic Logistic model, which ignored the hierarchical data structure, exhibited poor fit:R² ranged from 0.312 to 0.510 for height and from 0.453 to 0.594 for diameter. After incorporating provenance and individual nested random effects, the mixed-effects model performed substantially better: R² increased to 0.611–0.852 for height and 0.685–0.868 for diameter. On average, R² rose by about 0.356 for height and 0.257 for diameter, RMSE decreased by about 34% for height and 33% for diameter, and MAE decreased by about 39% for height and 36% for diameter. (3) Variance component decomposition revealed that most of the variation in asymptotic height came from individuals (variance component 22.09, 39.48% of total variation), followed by provenances (14.59, 26.07%). Similarly for ground diameter, individual variation contributed 0.55 (44.16%) and provenance variation 0.34 (27.70%). (4) Growth potential differed significantly among provenances. The Fushun, Jinzhou and Longjiang provenances had higher theoretical maximum height (a values: 30.779, 30.038 and 29.346 cm, respectively), while the Chaihe and Yehe provenances had lower values (20.359 and 19.786 cm). For ground diameter, Longjiang had the highest theoretical maximum (5.037 mm) and Gaizhou the lowest (3.245 mm). Seedling growth followed a “height-first, diameter-later” rhythm: all nine provenances reached their peak height growth rate earlier than their peak diameter growth rate. [Conclusion] The Logistic model is a broadly suitable choice for modelling height and ground diameter growth in one-year-oldQ. mongolica seedlings. By incorporating provenance and individual nested random effects, the nonlinear mixed-effects model greatly improved fitting accuracy and effectively captured the hierarchical variation in seedling growth traits. These findings provide a scientific basis and quantitative tools for cultivating high-quality seedlings, optimising nursery management, and evaluating germplasm resources at an early stage.

关键词

蒙古栎 / 苗高 / 地径 / Logistic模型 / 非线性混合效应模型

Key words

Quercus mongolica / seedling height / ground diameter / Logistic model / nonlinear mixed-effects model

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基金资助

国家重点研发计划(2024YFD1501400)