Key message A piecewise polynomial function already used to represent the size-density trajectories of pure even-aged stands of beech, oak, and Douglas-fir proved its ability to represent the size-density trajectories of a new species, ash. The widespread ash dieback caused departures from the expected size-density trajectories. These abnormalities can be used to detect an extra level of mortality due to infection by Hymenoscyphus fraxineus in pure even-aged ash stands . Context The size-density trajectories allow quantifying more precisely the density of stands and can help the forest manager to decide of the opportunity of thinnings. This study helped to quantify extra mortality in pure even-aged stands by using the size-density trajectories established for stands evolving at maximum density. Aims This study was conducted to establish size-density trajectories of pure even-aged ash stands and compare them with those recently established for beech and oak in France, in particular concerning the onset of density-dependent (regular) mortality. The additional effect of ash dieback on mortality was also an issue. Material and methods We used permanent and semi-permanent unthinned ash plots installed in the north of France and where inventories of trees were performed at more or less regular intervals: measurements included tree status (dead or alive) and diameter or girth at breast height for all trees and total height for a sample of living trees. The size-density trajectories of plots describing the course of the number of living trees in relation with the mean stand girth, in logarithmic scales, were modeled with a piecewise polynomial function fitted with a mixed-effects model. A permanent sample of trees was also selected for ash dieback and extra mortality monitoring. Results The piecewise polynomial function already used proved its ability to represent the size-density trajectories of even-aged ash stands of various initial densities and fertility levels. As for beech and oak, the trajectories were modeled so that mortality onset occurred at a constant relative density. This level appeared to be much higher for ash (RDI = 0.58), revealing that ash survived with less growing space than beech and oak and appeared to be more efficient. Ash dieback caused an additional mortality in the experimental ash stands studied, and this excess of mortality appeared predictable on the basis of observed departures from the expected size-density trajectories. Conclusion A single parameter function family could be used to predict the size-density trajectories of even-aged ash stands, on the basis of the results obtained previously on oak and beech. Mortality onset and space requirements of ash could be compared with those of beech and oak and show that ash can survive at higher densities and is a more efficient species. Predicted size-density trajectories proved also useful to detect and quantify the excess of mortality due to H. fraxineus on ash. This approach could be extended to other diseases and species with predictable size-density trajectories.

中文翻译：

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法国的等龄灰烬 (Fraxinus excelsior L.) 的尺寸密度轨迹建模。评估 Chalara 灰枯萎影响的基线

关键信息 已用于表示纯偶龄山毛榉、橡树和花旗松林的大小密度轨迹的分段多项式函数证明了其表示新物种白蜡树的大小密度轨迹的能力。广泛的灰烬枯死导致偏离预期的尺寸密度轨迹。这些异常可用于检测由于纯偶龄灰林中 Hymenoscyphus fraxineus 感染而导致的额外死亡率。背景 尺寸-密度轨迹允许更精确地量化林分的密度，并可以帮助森林管理者决定间伐的机会。这项研究通过使用为最大密度进化的林分建立的大小密度轨迹帮助量化纯偶龄林分的额外死亡率。目的 本研究旨在建立纯偶龄白蜡林的大小密度轨迹，并将其与最近为法国山毛榉和橡木建立的轨迹进行比较，特别是关于密度相关（常规）死亡率的发生。灰烬枯死对死亡率的额外影响也是一个问题。材料和方法 我们使用了安装在法国北部的永久性和半永久性未稀释灰烬地块，在那里或多或少地定期进行树木清查：测量包括树木状态（死的或活的）和胸高处的直径或周长活树样本的所有树木和总高度。地块的大小密度轨迹，以对数尺度描述与平均林木周长相关的活树数量的变化过程，用分段多项式函数拟合混合效应模型。还选择了永久性树木样本用于灰烬枯死和额外的死亡率监测。结果 已经使用的分段多项式函数证明了它能够表示不同初始密度和肥力水平的偶龄灰分林的大小密度轨迹。至于山毛榉和橡木，轨迹被建模，以便死亡率发生在一个恒定的相对密度。灰烬的这个水平似乎要高得多 (RDI = 0.58)，表明灰烬存活下来的生长空间比山毛榉和橡木少，而且似乎更有效。在所研究的实验灰烬林中，灰烬枯死导致额外的死亡率，根据观察到的与预期尺寸密度轨迹的偏离，这种过度的死亡率似乎是可以预测的。结论 基于先前在橡木和山毛榉上获得的结果，可以使用单个参数函数族来预测偶龄灰分林的尺寸-密度轨迹。白蜡木的死亡率开始和空间需求可以与山毛榉和橡木进行比较，表明白蜡木可以在更高的密度下存活并且是一种更有效的物种。事实证明，预测的尺寸密度轨迹也可用于检测和量化灰烬上白蜡木造成的过量死亡率。这种方法可以扩展到其他具有可预测尺寸密度轨迹的疾病和物种。结论 基于先前在橡木和山毛榉上获得的结果，可以使用单个参数函数族来预测偶龄灰分林的尺寸-密度轨迹。白蜡木的死亡率开始和空间需求可以与山毛榉和橡木进行比较，表明白蜡木可以在更高的密度下存活并且是一种更有效的物种。事实证明，预测的尺寸密度轨迹也可用于检测和量化灰烬上白蜡木造成的过量死亡率。这种方法可以扩展到其他具有可预测尺寸密度轨迹的疾病和物种。结论 基于先前在橡木和山毛榉上获得的结果，可以使用单个参数函数族来预测偶龄灰分林的尺寸-密度轨迹。白蜡木的死亡率开始和空间需求可以与山毛榉和橡木进行比较，表明白蜡木可以在更高的密度下存活并且是一种更有效的物种。事实证明，预测的尺寸密度轨迹也可用于检测和量化灰烬上白蜡木造成的过量死亡率。这种方法可以扩展到其他具有可预测尺寸密度轨迹的疾病和物种。事实证明，预测的尺寸密度轨迹也可用于检测和量化灰烬上白蜡木造成的过量死亡率。这种方法可以扩展到其他具有可预测尺寸密度轨迹的疾病和物种。事实证明，预测的尺寸密度轨迹也可用于检测和量化灰烬上白蜡木造成的过量死亡率。这种方法可以扩展到其他具有可预测尺寸密度轨迹的疾病和物种。