TreeMig is an intermediate-complexity DVM simulating local stand dynamics of multiple competing tree species. In addition to local dynamics, TreeMig allows for the explicit simulation of tree species' migration, with the rare advantage of including seed production, seed dispersal and subsequent regeneration processes . TreeMig simulations require time series of three different annual bioclimate drivers: the minimum winter temperature, the sum of daily mean temperatures above 5.5∘C, and an index denoting the severity of droughts . These drivers are derived from monthly averaged temperatures and monthly precipitation sums (see e.g. Supplement Sect. S2). TreeMig's state variables are population densities of tree species in a constant number of height classes per grid cell . Additionally, TreeMig stores the density of currently available seeds per tree species, representing the seed bank of a cell.

The TreeMig two-layer implementation described in the following – TreeMig-2L – is implemented in Fortran and based on TreeMig-Netcdf 2.0 . TreeMig-Netcdf 2.0 includes all processes described in the original TreeMig version with the growth response curve amendments described in , the minimum population density thresholds described in and the climate extrapolation method preserving spatial autocorrelation described in .

Most of the processes simulated with TreeMig do not require information on the position of the cell relative to other cells on the grid: competition for light, growth, mortality and production of seeds in a cell only use information that is independent of other cells. Therefore, these processes can be simulated on the new non-spatial layer (see Fig. ). In TreeMig-Netcdf 2.0, the only process which requires information on the position of a cell is seed dispersal. Other processes that would require information on the neighbourhood are, for example spatially connected disturbances, such as snow avalanches e.g.. In TreeMig-Netcdf 2.0, however, spatially connected disturbances are not represented. Thus, seed dispersal is the only process in TreeMig-2L that has to be simulated on the two-dimensional layer. Yet, to enable efficient simulations on two layers, associations between cells on the two-dimensional layer and elements on the non-spatial layer need to be as long lived as possible. This means that re-merging of elements that resulted from a very recent split needs to be prevented, and splits should only be conducted if they entail actual changes in species' compositions. The availability of seeds of a species does not necessarily have to lead to changes in the species' composition, because a species might not be able to regenerate in a cell. Therefore, the entire regeneration process was assigned to the two-dimensional layer (Fig. ) in order to only induce splits when newly dispersed seeds actually establish.

When designing the architecture of TreeMig-2L the two main requirements were an efficient organisation of the elements on the non-spatial layer and a fast exchange of status information between the two layers.

One of the challenges for an efficient organisation of the elements on the non-spatial layer was that the number of elements is not known in advance, because it is an emergent property of the simulation. To allow for an arbitrary number of elements on the non-spatial layer, the elements are stored in linked lists, instead of using an array structure with a predetermined fixed size, as it is usually done in space-discrete DVMs. However, using one large linked list, and comparing all elements with each other, would be very inefficient and would lead to a large organisational overhead. To reduce the organisational overhead required for the comparison of elements during runtime, TreeMig-2L uses several linked lists, and only elements in the same list are compared. To define the number of linked lists, the fact that the bioclimate drivers are an input to TreeMig is used, and that these drivers can thus be utilised to pre-structure a simulation area (see Fig.3 for an example, and Supplement Sect.S1.2 for additional information). To pre-structure a simulation area, grid cells are classified into “bioclimate types”. Each bioclimate driver is thereby discretised according to a set of pre-defined “bioclimate bins” (see e.g. Table ). To limit the number of possible bioclimate types and to save computation time, the discretisation is only applied for temporal averages of the bioclimate drivers on a pre-defined set of “supporting periods”. For each cell, one average per driver and supporting period is calculated. Two cells are classified into the same bioclimate type, if their averages fall into the same bioclimate bin for each driver and supporting period, whereby the bioclimate bins can differ in between supporting periods. For each existing bioclimate type, i.e. each type with which at least one cell is associated, an own linked list is used. The bioclimate drivers for a bioclimate type are calculated as the averages of all associated cells.

An element can be associated with a changing number of cells, therefore, an element only tracks how many but not which cells are currently associated with it (see Fig. ). The information exchange between the layers is induced by the cells. Each cell on the two-dimensional layer accesses required information from its currently associated element. For seed dispersal calculations the density of produced seeds per species is required. For regeneration, additionally, the current bioclimatic conditions and the light distribution of the lowest height class need to be accessed, because they influence the density of newly germinated seeds see. Newly germinated seeds are used to determine if splits are necessary (see Sect. ). After possibly required dynamic changes in associations between the layers, cells push their germinated seed densities to the currently associated element and each element on the non-spatial layer calculates the average of the germinated seed densities received from its associated cells. Averaged densities of germinated seeds falling below a pre-specified presence threshold are thereby set to zero.

In order to account for spatial processes resulting from seed dispersal, associations between the two-dimensional and the non-spatial layer need to be dynamic. In a TreeMig-2L simulation, elements on the non-spatial layer can be split up or merged. For details on the execution sequence of a TreeMig-2L simulation see Supplement Sect. S1.3.

Splitting, and thus introduction of new elements on the non-spatial layer is required as soon as the density of germinated seeds among any two cells associated with the same element is considered not similar enough. In TreeMig-2L, similarity in the density of germinated seeds can be defined by a set of thresholds. The simplest possible set consists of a single threshold that defines presence, i.e. species with a germinated seed density below this threshold are assumed to not have germinated. However, other thresholds are also possible, for example dividing sparse occurrences from more frequent ones. If the density of germinated seeds of any two cells associated with the same element fall on different sides of any of these thresholds for a single species, a split is required. One determinant for the efficiency of a TreeMig-2L simulation is assumed to be the number of considered splits. When k-1 thresholds are specified and n species are simulated, kn different combinations could possibly entail splits. In TreeMig-2L this trade-off between accuracy and possible splits is approached by not considering splitting for all species, but only for a set of species previously specified as species to be tracked; i.e. for tracked species differences in the number of germinated seeds among grid cells associated with the same element can lead to splits. For untracked species, in contrast, the number of germinated seeds is not compared among grid cells and deviations, thus, cannot lead to splits. Apart from the splitting step, all species are treated the same, in particular the number of germinated seeds on an element is calculated as the average over all associated cells for each species, no matter if tracked or untracked. As a consequence, untracked species might be represented less accurately, which can feedback on other species via competition. Therefore, species that are expected to change their distribution in the course of the simulation should be defined as tracked species.

Merging of elements is possible as soon as two elements belonging to the same bioclimate type are similar enough. The state variables stored in the elements on the non-spatial layer are population densities per species for a fixed set of height classes. In TreeMig-2L, two elements belonging to the same bioclimate type are regarded similar enough, when deviations between each pair of their population densities do not exceed a similarity threshold pre-specified for each of the height classes. To avoid immediately re-merging of elements that were recently split up, newly germinated seed densities are also compared. As opposed to splitting, merging of elements on the non-spatial layer potentially decreases accuracy. Moreover, there is a trade-off between computational expenses involved with merging and the reduction of repetitive calculations caused by similar elements. Therefore, merging is only performed after a pre-defined number of iterations in TreeMig-2L.

Previous TreeMig simulations were conducted with spatial resolutions ranging from cell side lengths of 25 m to 1 km and spatial extents ranging from single cells up to 77 000 km2. The number of simulated interacting tree species ranged from one up to 31 species and simulation areas had differing spatio-temporal complexity see e.g.. Such differences in the simulation settings are expected to influence the benefit of the D2C implementation, because they control two of the three aspects assumed to be most important (see Sect. ): (1) the non-reducible base load due to processes simulated on the two-dimensional layer, and (2) the ratio of grid cells and elements on the non-spatial layer, whereby the latter is strongly influenced by the number of bioclimate types.

The number of bioclimate types is, for example expected to be large when the simulation area has a high spatio-temporal complexity, i.e. a high heterogeneity in its bioclimate drivers. The number of bioclimate types will also be influenced by the spatial extent of a simulation area: a larger extent, up to a certain point, potentially leads to a larger number of types; however, there might be a threshold beyond which the number of already contained bioclimate types exceeds the number of newly added types. A similar effect is expected for the spatial resolution; bioclimate drivers in a cell are always averages and with finer resolution fewer bioclimate extremes might be smoothed out, leading to a larger number of required bioclimate types. However, especially in areas with a homogeneous bioclimate, a finer resolution might entail a larger number of similar grid cells, increasing the expected benefit of a D2C implementation. A finer resolution, on the other hand, will increase the computational expenses for seed dispersal. In TreeMig, seed dispersal is simulated from the perspective of the source cell, providing seeds to sink cells according to a pre-calculated truncated probabilistic density function see Supplement of. The number of sink cells thereby depends on the spatial resolution. A fine resolution implies a large number of sink cells, causing large computational expenses on the two-dimensional layer, and therefore a large base load. A fine resolution thus can diminish the benefit of a D2C implementation. Further aspects that can influence the benefit of the D2C implementation are the number of tracked species, the splitting and merging thresholds, and the number of iterations after which merging is considered.

TreeMig-2L simulations were conducted for two different application scenarios (A1 and A2), pertaining to different regions of Switzerland. Figure  shows the location of their simulation areas. Scenario A1, in the flatter areas of northern Switzerland, stems from a study with a preliminary TreeMig-2L version without dynamic associations between the layers . Scenario A2, a north–south transect across the Swiss Alps, stems from a study investigating the influence of interannual bioclimate variability in simulations of the northwards migration of Ostrya carpinifolia Scop. (European Hop Hornbeam) with TreeMig-Netcdf 1.0 . An in-depth description of the scenarios can be found in Supplement Sect. S2.

The two scenarios were selected because they strongly differ in their simulation settings (see Table  for their main characteristics). Application scenario A1 has a more homogeneous simulation area and a finer spatial resolution than A2. Therefore, on one hand, a more beneficial ratio between the number of cells on the two-dimensional layer and the number of bioclimate types is expected for A1. On the other hand, due to the finer spatial resolution a higher base load is expected for A1 than for A2.

The tracked species in scenario A2 is naturally the investigated migrating species (O. carpinifolia). The other 21 competing species simulated in this application scenario were not tracked. For A1 4 out of the 31 simulated competing species were selected as tracked species: Quercus pubescens, O. carpinifolia, Larix decidua and Pinus sylvestris. These species were selected because they have the highest drought tolerance indices in TreeMig. With increasing drought severity (and increasing temperatures), these species are therefore expected to extend their spatial distributions, whilst the other species might be less effected or have declining distributions. For both scenarios, merging was considered every 100 simulation years and the same splitting and merging thresholds were used (for further details and sensitivity tests see Supplement Sect. S3)

Both application scenarios were driven by bioclimate time series derived from SRESA1B scenario projections, though from different models and downscaled with different observational data (see Supplement Sect. S2). To cover the entire simulation time span of the examples, a stochastic extrapolation method accounting for the spatial correlation of bioclimate fluctuations was used .

Bioclimate types for both application scenarios were derived with the same sets of bioclimate bins: E1, E2 and E3 (Table ). For both scenarios three supporting periods were averaged: the first 30 (A1: 1961–1991; A2: 1901–1931) and the last 30 years (both: 2071–2100), as well as the whole time span (A1: 1961–2100; A2: 1901–2100).

Simulations were conducted with all three sets of bioclimate types (Table ). To account for the stochasticity involved with the extrapolation of the bioclimate drivers multiple repetitions of the simulations were performed. Simulations with application scenario A1 were repeated 5 times. Simulations with scenario A2, having much less grid cells and requiring only one-tenth of the CPU time of scenario A1

The average CPU time with the original one-layer approach (1L-ORG) was 50 353 s for A1 (average of five simulations with σ=784 s) and 5195 s for A2 (average of 100 simulations with σ=38 s).

The performance of the D2C implementation was assessed by two means: an accuracy measure and the required CPU time

CPU time was measured with the intrinsic Fortran procedure CPU_TIME. All simulations were conducted on 2.8 GHz AMD Opteron CPUs.

The accuracy of TreeMig-2L was evaluated by comparing biomass distributions from simulations using different model versions with a similarity coefficient (SC) ranging from 0.0 (no similarity) to 1.0 (identical biomass distributions in space). Comparisons generally led to SCs in the upper range for all output variables (ranging from about 0.8 to about 1.0 – Table ) and for both application scenarios. The level of the SC was thereby largely determined by the resolution of the applied set of bioclimate bins; the coarser the resolution is, the smaller the SCs for all variables (differences in the SCs of up to 0.14 – Table ).

To be able to assess the relevance of deviations from 1.0 in the SCs, results from 1L-ORG simulations using different pseudo-random number streams to extrapolate the bioclimate driver were compared with each other, i.e. the deviation from 1.0 in the SC due to interannual bioclimatic variability was calculated (see Supplement Sect. S3.2). The only SCs that were markedly larger than the SCs resulting from these 1L-ORG intra-comparisons were SCs from comparisons with E3 bioclimate types (compare Table  and Table S3.1).

In all cases, SCs from comparisons between 1L-ORG and 2L simulations were very close to SCs from comparisons of 1L-ORG and 1L-PB simulations (differences in the SCs ≤0.01). This indicates that the deviation in 2L simulations are mainly due to the averaging of the bioclimate drivers. SCs comparing 2L and 1L-ORG were, in particular, larger than SCs comparing 2L-NDA and 1L-ORG (differences in SCspec up to 0.04 and in SCOC up to 0.21), indicating the importance to track migrating species. The four tracked drought tolerant species in application scenario A1 and Ostrya carpinifolia in application scenario A2 seem to be good indicators to test for required splits. Sensitivity tests showed that fewer tracked species increased the error (up to 0.03 smaller SCspec values) and more species only slightly decreased it (only ∼0.01 larger SCspec values – see Supplement Fig. S3.13). The sensitivity tests further showed that the choice of merging and splitting thresholds had only limited impact on the SCs. Strong differences in the SC were only observed for simulations that tested elements for merging after a decade instead of after a century (see Supplement Sect. S3).

The temporal development of SCspec was comparable among simulations with the three sets of bioclimate types (E1–E3) for all model versions (2L, 1L-PB and 2L-NDA) and for both application scenarios. SCspec decreased in the transient phase of climate change (from around 2000 on) and stabilised after a few centuries (Fig.  and Supplement Sect. S3). The stabilisation on a lower level is mainly due to a stronger impact of differences in the drought index between bioclimate types and single cells due to overall larger drought indices in the second half of the 20th century (see Supplement Figs. S3.4 and S3.5). A comparable effect resulted for inter-comparisons of 1L-ORG simulations (see Supplement Sect. S3.2). While the temporal development in SCspec was comparable among all model versions, trajectories resulting from 2L-NDA simulations differed from those resulting from 2L and 1L-PB simulations for SCOC (Fig. ). Having no dynamic associations, SCOC for 2L-NDA simulations mainly reflects changes in the bioclimate over time (see Supplement Sect. S3.2 for details).

Being a single index for the whole simulation area, the SC is a rough indicator for the similarity between two simulations and could in particular conceal biases in the spatial biomass distribution. Furthermore, depending on the research question, other aspects could be more important than absolute biomass values per grid cell, especially when studying the migration of a species. Therefore, simulations of application scenario A2 were additionally compared focussing on the migration of the tracked species. Comparisons of the spatial spread of O. carpinifolia showed a reasonable approximation of 1L-ORG simulations by 2L simulations with all bioclimate discretisations (Fig.  and Supplement Fig. S3.11).

In summary, SCs comparing 2L and 1L-ORG simulations were within the magnitude of deviations due to interannual bioclimatic variability and deviations where smaller than reported for previous upscalings e.g.. Furthermore, comparisons of the spatial spread of O. carpinifolia between 1L-ORG and 2L simulations did not indicate spatial biases and comparisons with 2L-NDA simulations underlined the necessity to track migrating species.

Simulations with two layers led to considerable reductions in CPU time for both application scenarios (Table ). When considering the ratio between the number of bioclimate types and the number of cells in the simulation areas (Table ), larger CPU time reductions could have been expected for scenario A1 than for A2. Yet, the actual reductions are not systematically larger for A1 (E1: 52.9 %, E2: 56.0 %, E3: 57.8 %) than for A2 (E1: 32.6 %, E2: 65.6 %, E3: 84.7 %). To a small part this is due to increased dynamics in the number of elements on the non-spatial layer in A1 compared to A2 (Fig. ), leading to a comparable magnitude in the peak element-cell ratio for both application scenarios (Table ). However, sensitivity tests for the application scenario A1 showed that changes in the number of tracked species and in the merging interval leading to large changes in the peak element-cell ratio did not lead to notable changes in CPU time reductions (see Supplement Sect. S3.3.1). Moreover, the small difference in CPU time reductions among A1 simulations with the three sets of bioclimate types in combination with notable differences in the peak element-cell ratio (Table ) also indicates that the peak element-cell ratio is not the main cause for the limitation in the CPU time reduction for A1.

The actual main cause is revealed when looking at the percentage of executed instructions for the different processes (Table ): the percentage of instructions spent on seed dispersal was simply much larger for A1 than for A2. 1L-ORG simulations for application scenario A1 spent about 45 % of executed instructions on seed dispersal and only about 50 % on adult dynamics, i.e. on processes simulated on the non-spatial layer. A2 1L-ORG simulations, in contrast, only spent about 6 % of the executed instructions on seed dispersal and about 86 % on adult dynamics (Table ). Differences in the execution ratios were already expected (see Sect. ) because of the increased number of sink cells considered in seed dispersal calculations for the grid with 200 m cell side length in A1 compared to the grid with 1 km cell side length in A2. As a consequence, the base load that cannot be reduced with D2C was more than 7 times larger for A1 than for A2.

For A2 differences in CPU time reductions were strongly driven by the applied set of bioclimate types. Sensitivity tests showed that splitting and merging thresholds as well as the merging interval were far less influential (Supplement Sect. S3.3.2).