Vallet, P. | Search bioRxiv

bioRxiv2026

An efficient hyperbolic equation for modelling environmental constraints in ecology

The influence of environmental factors on the dynamics of tree species can imply non-linear relationships. Some of them exhibit threshold effects. Hyperbolic functions effectively represent ecological processes that display threshold behaviours. However, the mathematical formulation of the hyperbola is complex, which makes its use challenging and its parameters difficult to interpret. In this article, we propose an efficient mathematical formulation for the hyperbola, one in which all the parameters are independent and easily interpretable. We also provide an R script and a Python script to facilitate the implementation of this hyperbolic formulation in modelling studies. We then modelled the influence of edaphic and climatic factors on the growth of 18 forest tree species widely distributed across Europe based on a dataset of 8,330 plots from the French National Forest Inventory. Our hyperbolic function allowed us to identify the threshold effects of summer climatic constraints on forest growth for several species. In particular, we found negative effects for soil water deficit and maximum summer temperature, although for several species these effects only appear beyond a certain level of constraint. Accounting for such threshold effects is crucial to improve our ability to understand and predict forest ecosystem responses in the context of climate change.

ecology↗

bioRxiv2025

Robust height-diameter allometries for 42 European tree species: stand characteristics and structure matter

Tree height is a key variable for assessing forest functioning and resources (e.g volume, carbon stocks) at both tree and stand levels. However, direct field measurements are costly and time-consuming. Developing accurate and unbiased height-diameter allometries applicable over large spatial scales is therefore crucial for forest research and management. Based on the French National Forest Inventory, with 269,460 tree-height observations on 49,120 plots measured, we developed generalized, species-specific height-diameter allometries that integrate stand dendrometrical characteristics and stand structure. The models cover 41 European tree species and the four main stand structures: even-aged, uneven-aged, coppice-with-standards and coppice. To enhance local accuracy, we developed an optional recalibration method at the plot level, assessing how many additional trees should be measured and which ones to select to maximize the improvement of the model. By integrating basal area as stand density and quadratic mean diameter as an indicator of stand development stage, our models enable us to assess the influence stand dendrometrical characteristics on these allometries and evaluate how different species responded to competition. Results indicate that with increasing competition, tree height tends to be higher for a given diameter, and that stand structure significantly influences 28 out of the 41 species. Local recalibration showed that measuring just one to six trees (among the largest and thinnest diameter) per plot reduced prediction error by 10 to 70%, depending on species. This study provides a useful, robust and scalable tool for forest research and management, for the most widespread species in Europe, while offering precision for local applications.

ecology↗