Forest structure and composition result from soil and climate conditions, management, and natural disturbances. All of these drivers of forest ecosystem functioning are rapidly evolving due to global changes (Aber et al., 2001; Lindner et al., 2010; Campioli et al., 2012). While environmental and societal changes are taking place and will continue to happen in the future, their magnitude and the way they will occur locally remain largely uncertain (Lindner et al., 2014). Designing silvicultural systems and selecting tree species adapted to future conditions seem therefore to be risky bets (Ennos et al., 2019). Messier et al. (2015) proposed another vision of the forests in which they are considered as complex adaptive systems whose future dynamics are inherently uncertain. To maintain the ability of forests to provide a large range of goods and services in whatever future conditions, their resilience and adaptability must be improved by favouring an uneven-aged structure and tree species mixture (Thompson et al., 2009; Oliver et al., 2015). As the combinations of site conditions, climate projections, stand structures and tree species compositions are nearly infinite, all of the management options that could potentially enhance the resilience and adaptive capacity of forests cannot be tested in situ (Cantarello et al., 2017). Furthermore, such silvicultural trials provide results only for the long run, given the life span of trees, and cannot anticipate future conditions. Scenario analyses based on model simulations are therefore useful for selecting the most promising management strategies and evaluating their long-term sustainability. To explore the forest response to new silvicultural practices and yet unexperienced climate conditions in a realistic way, one needs new process-based models that are able to deal with mixed and structurally complex stands and to incorporate uncertainties in future conditions (Berger et al., 2008; Bravo et al., 2019).

In connection with the traditional forestry, viewing forests as stable systems that can be controlled, many empirical models were developed to predict tree growth in monocultures, considering that past conditions will remain unchanged in the future. On the other hand, scientists developed process-based ecophysiological models to better understand the short- and long-term forest ecosystem responses to multiple and interacting environmental changes (Dufrêne et al., 2005). This can indeed not be done through direct experimentation because the multisite and multifactorial experiments required for doing so would be too complex and too expensive (Aber et al., 2001; Boisvenue and Running, 2006). Most experiments of environment manipulation focus on single or few factors during a limited period of time, which precludes to properly take into account interactions, feedbacks and acclimation. To simplify the mathematical formalization of ecophysiological processes (e.g. radiation interception) and limit the calculation time, these process-based models were first designed for pure even-aged stands without considering the spatial heterogeneity of stand structure.

With the increasing interest in uneven-aged stands and tree species mixtures, cohort and tree-level models were also developed. Pretzsch et al. (2015) reviewed 54 forest growth models to show how they represent species mixing. Among those models, 36 were process-based with nine at the stand, 11 at the cohort and 16 at the tree level. While cohort models allow a description of the vertical structure of the stand, tree-level models are generally necessary to consider the spatial heterogeneity in the horizontal dimensions. To represent stand structure in three dimensions, the model must not only operate at the individual level but also consider the tree position. In the review of Pretzsch et al. (2015), 11 process-based models were individual-based and spatially explicit, but only three of them accounted simultaneously for radiation transfer, water cycling and phenology (i.e. BALANCE, EMILION and MAESPA). Since it describes canopy and water balance processes using a state-of-the-art approach (based on a fine crown discretization), MAESPA is a very useful tool for analysing outcomes of ecophysiological experiments (Duursma and Medlyn, 2012). MAESPA is however not suitable for multiyear simulations since it contains no routine for carbon allocation, respiration and tree dimensional growth. EMILION is also restricted to 1-year simulation (no organ emergence) and is specific to pine species with a quite detailed structural approach (Bosc et al., 2000). In contrast, tree dimensional growth is well described in BALANCE, which possesses a fine representation of tree structure (Grote and Pretzsch, 2002). In BALANCE, radiation interception by trees and water cycling are based on simpler ecophysiological concepts compared to MAESPA, and photosynthesis is calculated with a 10 d time step using the routine of Haxeltine and Prentice (1996). As the Forest v5.1 model (Schwalm and Ek, 2004), BALANCE has the advantage of merging two traditions, conventional growth and yield models together with process-based approaches, providing outputs familiar to foresters (classical tree and stand measurements obtained from forest inventory) as well as carbon fluxes and stocks. Among the three models, BALANCE is the only one that considers mineral nutrition through the impact of nitrogen (N) availability on tree growth. Some soil chemistry processes (e.g. ion exchange or mineral weathering) are however not described although they are essential to estimate bioavailability of the major nutrients other than N (P, K, Mg and Ca). Not considered in the review of Pretzsch et al. (2015), iLand is another individual-based model that describes the ecophysiological processes with an intermediate level of detail using simplified ecophysiological concepts (such as the radiation use efficiency approach) in order to also simulate forest dynamics at the landscape scale. Later, Simioni et al. (2016) developed the NOTG 3-D model to study water and carbon fluxes in Mediterranean forests using an individual-based approach to account for the spatial structure of the stand. This model is more suited for short-term simulations (a few years) than long-term (a rotation) simulations since tree dimensions are updated based on fixed empirical relationships between diameter at breast height (dbh) and tree height or crown radius.

As the models accounting for both the functional and spatial complexity are rare, we developed a new model (HETEROFOR) using a spatially explicit approach to describe individual tree growth based on resource use (light, water and nutrients) in heterogeneous forests. While the BALANCE and iLand models existed and responded roughly to our expectations, we decided to build a new model for several reasons. First, we thought that another model of this particular type would not be redundant if it was based on other concepts. Instead of calculating an index of light availability, we chose to estimate radiation interception for all trees using a ray tracing approach. For calculating photosynthesis and tree transpiration, we selected the Farquhar model with a shorter time step than in BALANCE in order to account for hourly variations in climate and soil water conditions. While we used a slightly more complex approach for the water balance module (Darcy approach instead of bucket model for soil water dynamics and rainfall partitioning when passing through the canopy), our model rests on a simpler representation of tree structure than BALANCE. Second, we aimed at incorporating a detailed tree nutrition and nutrient cycling module since we realized the necessity to integrate nutritional constraints in forest growth modelling, especially for predicting the response to climate change (Fernandez-Martinez et al., 2014; Jonard et al., 2015). Finally, we wanted to develop the model within the frame of a collaborative modelling platform dedicated to tree growth and stand dynamics. Among the various platforms, Capsis was the only one allowing multimodel integration and providing a user-friendly interface (Dufour-Kowalski et al., 2012). HETEROFOR was therefore progressively elaborated through the integration of various modules (light interception, phenology, water cycling, photosynthesis and respiration, carbon allocation, mineral nutrition, and nutrient cycling) within Capsis. The advantage of such a platform is to use common development environment, model execution system, user interface and visualization tools and to share data structures, objects, methods and libraries.

To simulate the response of forests to management and changing environmental conditions, structuring and integrating the existing knowledge into the process-based models is essential but not sufficient. These models must also be documented and evaluated in order to know exactly their strengths and limits when analysing their outputs. The objectives of this paper are to (i) describe the carbon-related processes of HETEROFOR (photosynthesis, respiration, carbon allocation and tree dimensional growth), (ii) evaluate the model ability in reconstructing tree growth in three broadleaved stands of different species composition and compare various options for describing photosynthesis, respiration and crown extension, and (iii) illustrate its potentialities by simulating tree growth dynamics in an oak and beech stand under various IPPC (Intergovernmental Panel on Climate Change) climate scenarios.

Few tree-level, process-based and spatially explicit models have been developed, and these often contain only some of the modules necessary to estimate resource availability (solar radiation, water and nutrients). While descriptions of these models are generally available in the literature, doing an evaluation by making a comparison with tree growth measurements is not always accessible, or such an evaluation has been carried out based on stand-level variables. We have therefore very little information to compare the performances of HETEROFOR at the tree level with those of similar models. Simioni et al. (2016) faced the same problem with the NOTG 3-D model.

HETEROFOR first estimates the key phenological dates, the radiation interception by trees and the hourly water balance (de Wergifosse et al., 2019). Then, based on the absorbed PAR radiation, individual gpp is calculated with a PUE approach or with the photosynthesis routine of CASTANEA (Dufrêne et al., 2005). Whatever the option retained for calculating tree respiration and crown extension, the photosynthesis routine of CASTANEA and the PUE efficiency approach performed similarly (Table 3). It is quite encouraging that the process-based approach for estimating photosynthesis provided predictions of the same quality as the empirical approach fitted with tree growth data taken on the study site. If no extrapolation to future climate is required, the PUE approach remains however still valuable, especially when hourly meteorological data are lacking. For the three stands in Baileux, we related the npp reconstructed from successive tree inventories with the gpp predicted based on the CASTANEA approach (Fig. 2). The good linear relationships (Pearson's correlation > 0.89) obtained for both oak and beech make us confident in the adaptation of the photosynthesis routine of CASTANEA to the tree level. Furthermore, since the parameters of the photosynthesis routine were taken directly from CASTANEA and not calibrated specifically for HETEROFOR, one can expect that the agreement between the predicted gpp and the reconstructed npp could still be improved.

When comparing the two options available in HETEROFOR for converting gpp into npp, model performances were systematically better with the npp to gpp ratio approach than with the temperature-dependent routine for the maintenance respiration calculation (Table 3). This can be partly explained because the error in the maintenance respiration calculation results from various sources. At the tree compartment level, uncertainties in the estimation of biomass, sapwood proportion, nitrogen concentration and temperature are multiplied (Eq. 11). Then, the errors made on all of the tree compartments are summed up. Among these uncertainty sources, the inaccuracy in the estimation of the sapwood proportion could explain why the maintenance respiration routine provided better results for the sessile oak than for European beech (Table 3). Since the sapwood of sessile oak can easily be distinguish from the heartwood based on the colour change, we had a lot of sapwood measurements available to fit a relationship. For the European beech, this was not the case; instead, we used a sapwood relationship obtained based on sap flow measurements (Jonard et al., 2011). This relationship could certainly be improved by direct measurements of sapwood made after staining the wood to highlight the living parenchyma. Another way to improve these relationships is to consider the social status of the trees since dominant trees have a higher sapwood depth than the suppressed ones (Rodríguez-Calcerrada et al., 2015). We tried to account for this by estimating the sapwood area based on the tree growth rate, but it did not significantly increase the quality of the predictions. The poor performances obtained with the maintenance respiration option also indicate that the processes at play are still poorly understood and that further research is needed on this topic.

The process-based approach for estimating maintenance respiration explicitly accounts for the temperature effect through a Q10 function. With the npp to gpp ratio approach, temperature is considered more indirectly by assuming that it affects respiration and photosynthesis in the same proportion, which is valid only in a given range of temperature (< 20 ∘C) and for non-stressing conditions. Indeed, the optimum temperature for photosynthesis is between 20 and 30 ∘C, while the optimum temperature for respiration is just below the temperature of enzyme inactivation (> 45 ∘C). Therefore, between 30 and 45 ∘C, photosynthetic rates decrease, but the respiration rate could continue to increase (Yamori et al., 2013). This reasoning however does not consider that the base rate of respiration acclimate to new mean temperature conditions and that this acclimation process tends to maintain the npp to gpp ratio more stably (Collalti and Prentice, 2019). In addition, while water stress reduces both photosynthesis and respiration, its effect on the two processes is not necessarily equivalent (Rodríguez-Calcerrada et al., 2014). The main argument in favour of the npp to gpp ratio approach is the tight coupling between respiration and photosynthesis since the substrate for respiration originates from photosynthesis. The npp to gpp ratio is unfortunately neither universal nor constant. It may vary with tree development stage, climate, soil fertility and competition conditions (Collalti and Prentice, 2019). The alternative option based on maintenance respiration calculation is theoretically more appropriate to simulate the impact of climate change, but this is at the expense of less accurate predictions at the tree level. The ideal is to compare the two options to evaluate the prediction uncertainty associated with the modelling of respiration. In the future, the two approaches could be improved. Applying the reconstruction procedure of HETEROFOR on a large diversity of sites would allow us to estimate the npp to gpp ratio in many different situations, to create a function predicting the npp to gpp ratio based on its main drivers and to subsequently use it in the model. In parallel, the respiration calculation could be refined by accounting for thermal acclimation, such as is done in 3D-CMCC (Collalti et al., 2018).

The differences in prediction quality between the two methods of crown extension modelling (distance-dependent approach vs. distance-independent approach) were quite small, probably because the length of the simulation was not sufficient to drastically affect the crown dimensions, which had been initialized based on measurements. Describing mechanisms that govern crown development in interaction with neighbours (mechanical abrasion and crown interpenetration) is however crucial to capturing the nonadditive effect of species mixtures (Pretzsch, 2014). By accounting for crown plasticity, our distance-dependent approach could help us to better understand how uneven-aged and mixed stands optimize light interception by canopy packing and how they increase productivity (Forrester and Albrecht, 2014; Jucker et al., 2015). To better evaluate the relevance of this approach, the predicted crown development should be compared with precise crown measurements that are repeated over several decades and taken in a large diversity of stand structures. When the model will be calibrated for a larger number of tree species, long-term simulations could also be performed to evaluate to what extent the model is able to reconstruct the empirical relationships that describe tree allometry variations in response to intra- and inter-specific competition. Such relationships were established by del Río et al. (2019) using data from the Spanish National Forest Inventory.

Based on the current evaluation, the process-based variant performs similarly to the more empirical one for photosynthesis and crown extension but not for respiration; this is probably because the processes are better known for photosynthesis. For the best combination of options using the CASTANEA photosynthesis (npp to gpp ratio with distance-dependent crown extension), the Pearson's correlation between measurements and predictions of individual basal increment amounted to 0.83 and 0.63 for European beech and sessile oak, respectively. By comparison, Grote and Pretzsch (2002) obtained a correlation of 0.60 for the individual volume of beech trees with the BALANCE model. This lower correlation can partly be explained by the integration of the uncertainty in tree height in the volume estimations.

Individual npp and retranslocated carbon are allocated first to foliage and fine roots and then partitioned between above- and below-ground structural compartments. Based on the derivative and rearrangement of a biomass allometric equation, the increment in aboveground structural biomass is used to estimate the combined increment in dbh and height. This results in a system of one equation with two unknowns (increment in dbh and height). We decided to resolve it by fixing the height growth based on a relationship that takes into account tree size (dbh or height), the height growth potential (height increment if all of the remaining carbon was allocated to height growth) and a light competition index. An intermediate level of sophistication was adopted to describe height growth, between the simple height–dbh allometry and the fine description of tree architecture of functional–structural models. Height–dbh relationships provide a static picture in which age and neighbour effects are confounded and are not suitable to describe individual growth trajectories (Henry and Aarsen, 1999). More sophisticated relationships considering age and dominant height can be used for even-aged stands (Le Moguédec and Dhôte, 2012) but are hardly applicable in uneven-aged stands for which tree age is unknown. On the other hand, the functional–structural models that are based on resource availability at the organ level and use a short time step can only be applied to a limited number of trees given the high computational demand (Letort et al., 2008).

Our individual height growth model was fitted with height data measured 10 years apart (Appendix E). A large uncertainty was however associated to these data. First, height measurements were obtained to the nearest metre given the difficulty to clearly identify the top of the trees in closed canopy forests. Second, as the height increment was obtained based on repeated height measurements, the error for this variable is the sum of those made on the height measurements. Consequently, the uncertainty was more or less of the same order of magnitude as the expected height growth in 10 years. Despite these uncertainties, a substantial part of the variability was explained by the model (72 % for European beech; 43 % for oak). Among the variables tested, the height growth potential had the main effect, which is not surprising since this height growth potential contains the information on height increment. We were also able to depict the effect of light competition. For a same-height growth potential, trees undergoing stronger light competition seem to invest more carbon for height growth than for dbh increment (Fig. E1 in Appendix E), which is corroborated by results of other studies (e.g. Lines et al., 2012). This strategy aims at minimizing overtopping by neighbours and maximizing light interception (Jucker et al., 2015). Trouvé et al. (2015) found similar results and showed the positive effect of stand density on height growth in the allocation between height and diameter increment in even-aged stands of sessile oak trees. The decrease in the red : far red ratio of incident light promotes apical dominance and internode elongation through the phytochrome system (shade avoidance reaction; Henry and Aarsen, 1999). By considering the light availability effect on height growth at the tree level, HETEROFOR adapts tree allometry to intra- and inter-specific competition, which is crucial to accounting for mixing effects in structurally complex stands (del Río et al., 2019).

A first simulation experiment was achieved to assess how tree biomass production and its allocation to tree compartments respond to climate conditions and thinning. The results of these simulations are in line with the basic principles of silviculture; thinning favoured individual tree growth (especially that of the smaller trees) by redistributing stand biomass production on a smaller number of trees. At the stand level, thinning slightly reduced biomass production since its intensity was substantial and the simulation lasted only 1 year, which was not sufficient to allow the remaining trees to fill the gaps by extending their crown. Drought conditions reduced the biomass increment of structural components, and this effect was more pronounced on big trees than on small trees. Indeed, when soil water availability decreases, smaller trees maintain a higher stomatal conductivity because of their lower position in the stand canopy. Functional compartments were less influenced by climate because carbon is allocated to them in priority in the model. We could improve the allocation routine by making the fine root to foliage ratio and the root-to-shoot ratio dependent on the mean soil water availability (Thurm et al., 2017).

We were also quite satisfied to observe that the model was able to reproduce the size–growth relationship. This approach describes the growth partitioning among trees in a stand, which is useful to estimate the mode of competition. For the three studied stands and the two tree species, the competition was partially size asymmetric, with a resource partitioning in favour of the larger trees (Carl et al., 2018). Within the studied stands, the European beech trees can be classified in two groups: a group of small suppressed trees whose radial growth was close to 0 and which were just surviving and the rest of the trees (beyond a girth threshold) whose radial growth linearly increased with girth. Regarding sessile oaks, nearly all of the trees were in the second group, which can be related to the fact that sessile oak is a less shade-tolerant species than European beech. In the this mixed stand, the nearly perfect match between the predicted and observed relationships indicates that the model was able to reproduce growth partitioning among trees of different tree species and size. These very good results can be ascribed to the fact that the extinction coefficient and the respiration parameters were calibrated with data from this stand. In the beech-dominated stand, the model slightly underestimated the radial growth of the small oak trees and overestimated that of the big beech trees. In this case, the model seems to allocate too many resources to the big beech trees which shade the small oak trees. This could be improved by a model calibration partly specific to this stand (for the npp to gpp ratio) or by a calibration with data covering a much larger range of stand structures.

To illustrate one possible application of HETEROFOR, a second simulation experiment was completed and allowed us to compare the radial growth predicted for 2076–2099 according to three IPCC scenarios with that of the one simulated for an historical period (1976–1999). When the atmospheric CO2 concentration was kept constant (380 ppm), differences among scenarios remained nonsignificant, except for sessile oak displaying a slightly higher basal area increment for the RCP2.6 scenario than for the historical scenario (Fig. 6). Analysing the model outputs in depth, we found that this lack of effects resulted from a balance between negative and positive impacts of climate change. While the increase in air temperature (+0.9 and 3.7 ∘C for RCP2.6 and RCP8.5) and in the vegetation period length (+8 and 37 days for RCP2.6 and RCP8.5) favoured photosynthesis, the more frequent and intense water stress negatively affected it (de Wergifosse et al., 2020). The positive and negative effects of climate change were of the same magnitude for both tree species and offset each other. For the simulations with a variable atmospheric CO2 concentration, the differences among scenarios were much larger, highlighting a strong CO2 fertilization effect for both sessile oak and European beech (Fig. 6). These results are in agreement with Reyer et al. (2014), which used the 4C model to predict productivity change in Europe according to a large range of climate change projections. They found npp increases in most European regions (except a few cases in Mediterranean mountains) when considering persistent CO2 effects by using a variable atmospheric CO2 concentration. Assuming an acclimation of photosynthesis to CO2 (by maintaining constant atmospheric CO2), they predicted increases in northern Europe, decreases in southern Europe and ambivalent responses elsewhere in Europe. Similar response patterns were also obtained by Morales et al. (2007). Rötzer et al. (2013) used the BALANCE model to compare the impact of future and current climate conditions on the productivity of beeches in Germany and showed a 30 % decrease in npp without considering the rise in atmospheric CO2 concentration. After evaluating CASTANEA against eddy covariance and tree growth data in a few highly instrumented sites, Davi et al. (2006) simulated the trend in gpp and net ecosystem productivity (NEP) in these sites from 1960 to 2100. For sessile oak and European beech, they obtained a 53 % and 67 % increase in gpp and NEP, respectively.

Given the magnitude of the CO2 fertilization effect (leading to a 72 % increase in basal area increment in 100 years for RCP8.5), we conducted retrospective simulations to check that HETEROFOR reproduces well the increase in productivity observed by Bontemps et al. (2011) for beech forests in the north-east of France (data not shown). Based on historical atmospheric CO2 concentrations, we simulated radial growth during two periods (1879–1910 vs. 1979–2010) using the same climate data (obtained by reanalysis for 1979–2010). These simulations showed a productivity increase of 12 % over 100 years. By comparison, Bontemps et al. (2011) reported productivity increases ranging from 10 to 70 % over 100 years, depending on the nitrogen status of the forest. The increase in radial growth simulated with HETEROFOR for the mixed stand in Baileux (Fig. 6) seems therefore plausible but assumes unchanged nutritional status. Increased productivity generates however higher nutrient demand by trees, which is not systematically satisfied by larger soil nutrient supply, especially in the poorest sites. Consequently, the augmentation of forest productivity will most likely be constrained by nutrient availability and give rise to a deterioration of the nutritional status as already observed across Europe (Jonard et al., 2015). To improve our predictions, nutritional constraints must be taken into account. From this perspective, a mineral nutrition and nutrient cycle module was incorporated in HETEROFOR. As it was developed in parallel to the water balance, some adaptations are needed for the coupling of the two modules (e.g. change from an annual to a monthly time step for soil chemistry update). A complete description and evaluation of the nutrient module will be provided in a future study.

Our ambition was to develop a model that is responsive to both management actions and environmental changes and would be particularly well adapted to mixed and uneven-aged stands. We thought that this model had to be tree-level and spatially explicit and had to consider radiation transfer, water balance and nutrient cycling with a process-based approach. Such models were very scarce in the literature. The only exceptions were BALANCE, iLand and more recently NOTG 3-D. To fill this gap, we elaborated the HETEROFOR model based on concepts quite different from those used for BALANCE, iLand and NOTG 3-D. In this study, a first evaluation of the model performances showed that HETEROFOR was able to reproduce size–growth relationships in three oak and beech stands in the Belgian Ardennes. We also noticed that the npp to gpp ratio option for describing maintenance respiration provides the best results, while the process-based and empirical approaches perform similarly for photosynthesis and crown extension. As this model evaluation was limited to two tree species and one climate, it only provides a first impression of the model potential.

Here, only the core of HETEROFOR was described. The water balance and phenology modules are presented and evaluated in a companion paper (de Wergifosse et al., 2019), while the nutrient module will be described later. For the next steps, we plan to couple HETEROFOR with existing libraries such as the ones for regeneration, genetics and economics. As HETEROFOR was developed within the Capsis platform, it is continually improving thanks to the collaborative dynamics among modellers.

A broader assessment of the model performances will be carried out based on forest monitoring plots distributed all over Europe. Indeed, HETEROFOR was designed to be particularly suitable for the level II plots of ICP Forests. The processes were described at a scale that facilitates the comparison between model predictions and observations. Many data collected in these plots can be used to initialize and run the model or to calibrate and evaluate it. HETEROFOR can also be seen as a tool for integrating forest monitoring data and quantifying non-measured processes. While it is now calibrated for oak and beech forests, HETEROFOR will be parameterized for a large range of tree species in order to use it for testing and reproducing identity and diversity effects.

Given all of the uncertainties related to climate change impacts, it is unrealistic to believe that a model will accurately predict the future dynamics of forest growth. However, models such as HETEROFOR can be very useful to compare scenarios. Among others purposes, HETEROFOR can be used to select the management options that maximize ecosystem resilience or to quantify uncertainty in the response of forest ecosystem to climate change.