It is of major interest to estimate the feedback of arctic ecosystems to the global warming we expect in upcoming decades. The speed of this response is driven by the potential of species to migrate, tracking their climate optimum. For this, sessile plants have to produce and disperse seeds to newly available habitats, and pollination of ovules is needed for the seeds to be viable. These two processes are also the vectors that pass genetic information through a population. A restricted exchange among subpopulations might lead to a maladapted population due to diversity losses. Hence, a realistic implementation of these dispersal processes into a simulation model would allow an assessment of the importance of diversity for the migration of plant species in various environments worldwide. To date, dynamic global vegetation models have been optimized for a global application and overestimate the migration of biome shifts in currently warming temperatures. We hypothesize that this is caused by neglecting important fine-scale processes, which are necessary to estimate realistic vegetation trajectories. Recently, we built and parameterized a simulation model LAVESI for larches that dominate the latitudinal treelines in the northernmost areas of Siberia. In this study, we updated the vegetation model by including seed and pollen dispersal driven by wind speed and direction. The seed dispersal is modelled as a ballistic flight, and for the pollination of ovules of seeds produced, we implemented a wind-determined and distance-dependent probability distribution function using a von Mises distribution to select the pollen donor. A local sensitivity analysis of both processes supported the robustness of the model's results to the parameterization, although it highlighted the importance of recruitment and seed dispersal traits for migration rates. This individual-based and spatially explicit implementation of both dispersal processes makes it easily feasible to inherit plant traits and genetic information to assess the impact of migration processes on the genetics. Finally, we suggest how the final model can be applied to substantially help in unveiling the important drivers of migration dynamics and, with this, guide the improvement of recent global vegetation models.
How fast vegetation communities can follow their shifting climate envelope in a changing environment is determined by their ability to migrate. This is exceptionally challenging under current global change and plants might strongly lag behind their moving climate envelope (Harsch et al., 2009; Loarie et al., 2009; Moran and Clark, 2012). Temperatures are increasing most strongly in the Arctic. Accordingly, forests in the tundra–taiga transition zone are expected to respond by migration into the tundra (Bader, 2014; Holtmeier and Broll, 2005; MacDonald et al., 2008). However, empirical studies show diverse responses to the warming, including treelines being stable, advancing or even retreating (Harsch et al., 2009). A taiga range expansion though, might positively feedback to a global temperature increase due to albedo reduction (Bonan, 2008; Piao et al., 2007; Shuman et al., 2011).
To predict forest responses to climate, computer models were designed with different scopes of complexity, between highly general to very specific (Grimm and Railsback, 2005; Thuiller et al., 2008). Among these, simulation studies with dynamic global vegetation models (DGVMs) tend to overestimate the turnover of treeless tundra into forests (Brazhnik and Shugart, 2015, 2016; Frost and Epstein, 2014; Kaplan and New, 2006; Roberts and Hamann, 2016; Sitch et al., 2008; Snell, 2014; Yu et al., 2009; Zhang et al., 2013). On the other hand, forest landscape models (e.g. Snell et al., 2014; Shifley et al., 2017; Epstein et al., 2007) and small-scale models (forest-gap or individual-based) provide sufficient detail to realistically represent the responses at a stand level, but need a lot of effort for parameterization, have higher computational expenses, and are therefore typically not applied over large areas (Martínez et al., 2011; Pacala et al., 1996; Pacala and Deutschman, 1995; Zhang et al., 2011) or lack the implementation of wind-driven seed and pollen dispersal (e.g. Epstein et al., 2007). Further problems of DGVMs arise from the use of plant functional types as they consist of species with a wide variety of traits (e.g. Lee, 2011; Snell et al., 2014; Svenning et al., 2014). Nonetheless, the ability to form a closed canopy forest depends mainly on species traits acting at a fine-scale level such as (1) time needed to mature (life cycle, high generation time) and produce viable seeds, (2) dispersal distance and the chance for long-distance seed dispersal, and (3) germination and establishment of new individuals (Svenning et al., 2014). One source of the overestimation of migration rates of DGVMs is the unconstrained seed availability when climate variables allow a vegetation type to establish, which was recently pointed out by using a dispersal function between the grid points in simulations with a DGVM (Snell, 2014; Snell and Cowling, 2015). However, connecting grid cells to allow dispersal among them increases the computational complexity of such models (e.g. Nabel, 2015), but would be necessary to simulate realistic large-scale vegetation responses. In addition, the structure of a tree stand, and its response to changes in external forcing, is determined by further local processes, such as spatially explicit competition among individuals of all ages and their interactions. Of special interest is the adaptation of the traits of individuals of local populations, which are influenced by gene flow through seed or pollen distributions across populations. High exchange can lead to outcrossing that hinders local adaptation, but also prevents negative consequences from diversity losses caused by inbreeding within isolated populations due to founder effects in the process of colonization over large distances (Austerlitz et al., 1997; Burczyk et al., 2004; Fayard et al., 2009; Nishimura and Setoguchi, 2011; Ray and Excoffier, 2010). These processes have, so far, not been implemented continuously over a large scale in simulation models.
During the past decades treeline stands in the Siberian Arctic were
densifying, but only rather slowly colonizing the tundra (Frost et al., 2014;
Kharuk et al., 2006; Montesano et al., 2016), which could be attributed to
seed limitation (Wieczorek et al., 2017). We developed the
The new spatially explicit pollination function tracks the full genealogy of a simulated tree stand and furthermore allows the inheritance of individually varying traits of each tree, rather than randomly drawing the actual trait value from the pool of available traits (cf. Scheiter et al., 2013). Additionally, the implementation of spatially explicit seed dispersal and pollination would enable us to align the model to detailed biogeographical knowledge gained from molecular methods (e.g. Navascués et al., 2010; Polezhaeva et al., 2010; Semerikov et al., 2007, 2013; Sjögren et al., 2017). We started with a very detailed small-scale model that can later be used to inform large-scale models especially about plot connectivity through seed dispersal and pollination and subsequent gene flow in landscapes.
We aim with this study to enable the simulation of spatially explicit and wind-dependent seed dispersal and pollination in the individual-based model LAVESI. After the coupling and verification of the seed dispersal kernel to prevailing winds and the incorporation of the pollination we test the model's sensitivity to its parameterization in local sensitivity analyses and the influence on stand development, migration rates, and pollination distances.
LAVESI is an individual-based spatially explicit model that currently
simulates the life cycle of larch species as completely as possible from
seeds to mature trees (Kruse et al., 2016). It was set up to improve our
understanding of past and future treeline displacements under changing
climates, focusing on the open larch forest ecosystem in northern Siberia,
which is underlain by permafrost. The relevant processes (growth, seed
production and dispersal, establishment and mortality) are incorporated as
submodules, which were parameterized on the basis of field evidence and
complemented with data from literature. Simulation runs proceed in yearly
time steps and are forced by monthly temperature and precipitation time
series. The area simulated represents spatially homogeneous forest plots of
variable size with the use of an environment grid (e.g. competition) with
20 cm tiles and where the handling of seeds dispersed beyond the plot
borders can be set to deletion or reintroduction from the other side to
simulate a forest patch. The model is programmed in C
The model was successfully applied to conduct temperature-forcing experiments, where simulations revealed that the responses of the larch tree stands in Siberia – densification and northwards migration – could lag the applied hypothetical warming by several decades, until the end of the 21st century (Kruse et al., 2016; Wieczorek et al., 2017).
Here we present the implementation of wind-dependent seed dispersal as well
as the newly introduced pollination. The absorbing boundary condition had to
be revised to allow the simulation of larger areas. Hence, we introduce a new
mode of periodic boundary conditions that allows seeds leaving the simulated
area (
Schematic representation of wind pollination as newly implemented in the LAVESI-WIND model. Based on actual winds, a distance-dependent pollination probability of ovules is estimated for each adult tree (potential pollen source) and for each seed source in the simulated area. The shaded areas on the ground represent the pollination probability for the labelled seed source for winds from the upper-right corner. These are generally higher for adult trees in upwind direction of the central seed source.
Pollen was not represented in the former LAVESI version, but is needed to independently track gene flow by seeds and pollen through time. Accordingly, Figure 1 illustrates how we implemented an individual based pollination for each seed's ovule using a wind-determined and distance-dependent probability distribution function for pollen dispersal (similar to Gregory, 1961). It makes use of the von Mises distribution, which is an angular equivalent to the Gaussian normal distribution, for the two-dimensional representation (Abramowitz and Stegun, 2012).
A pollen dispersal function was newly implemented as a distance-dependent
probability function for pollination of each individual seed's ovule, rather
than simulating the large amount of pollen released by each tree (Gregory,
1961; Kuparinen et al., 2007). For each seed-bearing tree, the probability
of pollen donating trees is calculated and out of the list of potential
fathers for each seed one tree is randomly determined according to this
probability. The pollination probability of each seed's ovule on a tree is
proportional to the amount of pollen in the air column around it, which is,
for simplification in the current implementation, not additionally dependent
on the performance of the tree so that every tree that bears cones is taken
into account. This aspect might be included in future versions. The
following function is used here as the distance-dependent probability
distribution of arriving pollen:
The probability distribution
Consequently, following Gregory (1961) the pollination probability of a
seed's ovule is as follows:
In the initial version of LAVESI, seeds are dispersed in random directions
and at a distance
The wind-dependent distance estimation was implemented as a ballistic flight
following the assumptions of Matlack (1987). Accordingly, seed dispersal
distances depend on the height of the releasing tree top
Overview of model parameters and processes for
The model's parameters had to be revised after implementing the model
extensions to achieve simulated tree densities comparable to field data.
Forest inventory data were recorded for each larch individual with explicit
positions on plots of a minimum area of
Simulations are forced with monthly mean temperature and precipitation sum
series from the CRU TS 3.22 database (Harris et al., 2014). These are used to
estimate long-term responses and derive the auxiliary climate variables
active air temperature (sum of temperatures above 10
Parameter values evaluated in the sensitivity analysis for seed dispersal, migration patterns, and pollination.
The model is driven with pairs of wind speed in m s
To test the influence of the parameterization of the variables from the newly
introduced functions on the model's results, we ran local sensitivity
analyses (Grimm and Railsback, 2005; Cariboni et al., 2007). For each
simulation repeat, the input parameters (Table 2) were changed by 5 % and
50 % and a sensitivity value was calculated by comparing the results with the
reference run:
The simulations were carried out on hypothetical north–south transects with a
width of 100 m and length of 1000 m using the new model version and
allowing seeds to be dispersed along the meridional borders. Populations were
initiated on empty areas only in the lowermost 100 m wide and 100 m long
area by randomly distributing 1000 seeds during the first 10 years of a
1000 year long stabilization period. During this phase, seeds exceeding the
lowermost
For the evaluation of migration rates we selected three target output
variables for the area ahead of the 100 m initialization area:
(1)
The memory load was estimated by adding up the size of all data types within each handled structure simulating a plot of one hectare (Table S1). These were multiplied by the actual number of elements in each of the structures. We calculated mean values of the number of handled items of the final 80 years of the simulations for the evaluation of dispersal processes to estimate the total memory needed for the arrays of trees and seeds and the grid representing the environment (Kruse et al., 2016).
To reduce the computation time, we parallelized the code for estimating
pollination probabilities, seed dispersal, and tree density computation of
the model using the OMP-library and conducted simulations using 1, 4, 8, and
16 CPUs. The performance of the model was evaluated by recording the
computation time of each single simulation year for complete simulation runs
(1080 years). We conducted four different runs, one with only wind dispersal
of seeds (SEED), one with seed and pollination (
Dispersal distances of seeds are wind dependent and positively correlated with the height of the releasing tree. The simulated and hypothetically calculated dispersals were compared across evenly distributed height classes; the results are similar for north and south winds, and here the results with north winds are presented.
The simulated seeds were solely dispersed in a north or south direction in
coherence to the forcing winds (Fig. 2, Tables S2 and S3). The median seed
dispersal distances were
The pollination events were mainly coming from the direction of the forcing
winds: however,
Wind forcing
Sensitivity values for varied model parameters influencing the seed dispersal process.
In northern central Siberia, the main wind directions observed during the vegetation period are a combination of both west and east (Fig. 3a). In some years, one of these directions predominates, and is also characterized by stronger wind events. Accordingly, simulated seeds are dispersed into the general direction of the forcing wind data (Fig. 3b). Dispersal distances can reach up to a maximum of several thousand kilometres, yet the majority of seeds fall within a few hundreds of metres, and these are dispersed over distances depicting the wind speeds as well.
The median pollen flight distances are generally larger than the seed's, with a technically fixed maximum of about the distance from the central plot to the borders (Fig. 3c). Similar to seed dispersal, pollination follows the wind directions and fathers are positioned in the upwind direction of the main occurring winds.
The sensitivity analyses for the implemented seed dispersal function was
extended for further model parameters that have an influence on the migration
rate. In general, the four target variables have the same response direction
towards changes in the parameters (Table 3). The stronger the changes, the
more apparent becomes the change in the result so that the significance
increases strongly from only 25 % to 79 %. The sensitivity values
were of the same order of magnitude with the extreme values of
The sensitivity values for resulting pollination distances for varied
parameters were an absolute mean of change of 0.11 for 5 % and 0.02 for
50 % with extremes of
Simulation consumption time in relation to the number of trees
present, the number of CPUs used, and for different types of parallelization
of the code. The time increases exponentially with the number of trees and
more quickly when simulating the additional pollination (
Sensitivity values of the model's results assessed by mean distance
per pollination event into an area of
The dynamic arrays need 120 bytes for each tree and 98 bytes for each seed. A
further 54 bytes are needed for each of the environmental map tiles and
another 117 bytes for the storage of output variables for each simulated year
(Table S1). The constant containers use 390 bytes for the weather list and
the parameter structures contain 642 bytes. On the basis of a simulated
typical dense forest with
The simulation time increased with the number of trees in the simulation and
for the contrasting simulation setups – either only wind-dependent seed
dispersal SEED, or also with the calculation of pollination
Without including the pollination events, the computation takes
The assumption of unlimited seedbeds – allowing species in models to grow as
soon as climate space allows them – causes high uncertainty in future
predictions with dynamic global vegetation models (e.g. Midgley et al., 2007;
Neilson et al., 2005; Sato and Ise, 2012). Implementing time-lagged responses
in such models highlighted the need for a proper understanding and
implementation of processes that limit species' migrations (Snell, 2014;
Snell and Cowling, 2015). To reveal and understand the underlying processes
that cause time lags, we designed the model LAVESI that represents all
life-cycle stages of larches in high detail from seeds to mature trees,
producing seeds themselves, which are then distributed in the environment
(Kruse et al., 2016). We built this model to simulate responses of the
Siberian treeline ecotone, which is solely covered over vast areas by a
single tree species of the genus
The simulated seed dispersal strictly followed the wind forcing and seeds
settled in a downwind direction as expected, and not, as in the original
model, in a purely ballistic manner (Kruse et al., 2016). We tested, in a
local sensitivity analysis, the influence of different parameters on the
stand level and the migration process. Sensitivity values were generally low
with mean values between
The use of winds from only the vegetation period might have introduced a bias, but it is based on the observation that this is the time when seeds are primarily dispersed (Abaimov, 2010). However, secondary dispersal by winds, due to uplift in strong winds, travel in winter on frozen surfaces over long distances (Nathan et al., 2011a; cf. Pluess, 2011), or due to wind-independent animal-mediated zoochory (Evstigneev et al., 2017), is currently not represented but could further facilitate the migration into tundra. When applying this model over historical periods, which are not covered by observations, one must be careful as the wind regimes could have shifted their main wind direction from the past to the current setting and might even change in the future (Lisitzin, 2012; Trenberth, 1990). A change, for example, from north–south to the current east–west wind directions could have limited the recent potential migration rate. This could explain the slow response of the treeline in northern Siberia to global warming, in addition to the long life cycle of larches, as well as prevailing seed limitation in the north (Kruse et al., 2016; Wieczorek et al., 2017).
Pollen dispersal functions are frequently used to reconstruct vegetation
composition from palaeo-archives, for example in the Landscape Reconstruction
Algorithm by Sugita et al. (2010), whereas other models have been used to
track pollen clouds in tree stands (review in Jackson and Lyford, 1999;
Prentice, 1985). Calculating every pollen dispersal event for each tree and
seed is computationally challenging, but it can be simplified following the
assumptions of Kuparinen et al. (2007). Hence, we implemented a
density-dependent probability function and found in the sensitivity analysis
that the pollination process was less affected by changing the input
parameters than by the seed dispersal process. Values only reached a mean of
The individual-based approach of the model LAVESI-WIND, with the extension of wind-dependent seed dispersal and pollination, bears a high potential of knowledge gain, but this comes with some challenges: (1) repeated calculations for millions of individuals (seeds and trees) are computationally intense (e.g. Snell et al., 2014; Svenning et al., 2014; Nabel, 2015), and (2) they require a certain amount of memory during the simulation runs. Whereas the memory during each simulation run could be minimized to the needs of the simulation setup, the computational power was historically the limiting factor (Grimm and Railsback, 2005). But with the development of recent computer clusters with hundreds of CPUs, it seems very likely that one can overcome this, allowing us to use detailed and spatially explicit models at a regional scale (e.g. Paik et al., 2006; Zhao et al., 2013).
We estimated the requirements for a hectare of a dense simulated forest as
15 MB of RAM. This means, on typical computer servers, even broad-scale
simulation runs are easily feasible for
The computational effort of pollination for each seed's ovule increases with the number of mature trees present on a simulated plot. Therefore, to allow simulations to be run on standard computers in manageable time, it was a major goal to minimize the time needed for each simulated year. To meet this requirement, we parallelized parts of the program code that are computationally intensive, namely the processes of pollination and seed dispersal. With our approach, we have been able to decrease the time so far by a factor of 2 when using 8 CPUs, in comparison to using only one. Still, overheads from using a standard template library (STL)-list container lead to a negative exponential progression of the computation time needed per year rather than linear improvements (Fig. 4). Additional gains for other not yet parallelized processes are much smaller than these, but there is further potential to reduce the computation time by using different implementations of the parallelization.
The new model version LAVESI-WIND allows for the evaluation of the importance of driving processes, which determine the response speed of tree stands growing at the treeline in Siberia. It can therefore be used for a very detailed evaluation of intra-stand processes determining migration speeds and help to improve abstract dynamic global vegetation models (e.g. Sato et al., 2007; Sitch et al., 2003), forest landscape models (e.g. Seidl et al., 2012), or regional forest gap models (e.g. Brazhnik and Shugart, 2015, 2016). Such a detailed representation of forest stands, as in the model presented here, is unlikely to be able to simulate forest dynamics on a continental to global scale (cf. Neilson et al., 2005). Nonetheless, the model can be used to parameterize dispersal kernels constraining inter-grid cell migration in DGVMs (Snell, 2014; Snell and Cowling, 2015). This could be achieved by comparing the migration rate in a continuous landscape in LAVESI-WIND, which covers grid cells of the DGVM to achieve a better representation of processes constraining or enhancing the spread of a plant species (cf. Lehsten et al., 2018). With this new model version, we can approach novel research questions, such as “do wind regime shifts explain faster or slower migration rates in past climate changes?” Furthermore, one could test how different treeline types determine the migration behaviour in changing environments. These can vary widely, based on the treeline type, being abrupt or with stand densities decreasing with the abiotic gradient and might further be influenced by shrubs that respond faster to current climate warming (e.g. Frost and Epstein, 2014), but which are not represented in the model yet. In addition, this may be influenced by single-tree stands growing ahead of the migration front (Holtmeier and Broll, 2005). Further interesting questions could be addressed, such as the role of refugia during past glacial periods and their influence on present-day tundra colonization by trees (Wagner et al., 2015), with a simplified and thus computationally effective approach. This is a necessary step because the current model version is computationally to demanding to track the full genealogy over simulated areas and time periods. Upscaling approaches could decrease generally the computation time and allow to expand the simulation over larger areas (e.g. Nabel, 2015; Epstein et al., 2007), however, the individual genetic information that passes thorough the landscape would be lost, which might be of interest. By connecting the borders of a simulation plot along the meridional borders we already implemented boundary conditions that allow the simulation of south-to-north transects, which are representative of the treeline area where highest tree densities occur in the south and treeless areas in the north. Thus, with this model, past migration corridors and timings can be revealed by a landscape-scale simulation, potentially answering important questions of the past biogeography of larch species in Siberia.
Before applying this new model version, however, a proper parameterization is necessary. Because pollen productivity and pollination distances as well as seed dispersal distances are not yet available for forests of the northernmost treeline area, the next important step would be to evaluate the modelled seed dispersal and pollination processes with field-based data, and finally, to apply this model to achieve realistic predictions of a future treeline. Molecular methods can help to improve the seed dispersal function, especially microsatellite markers, which can uncover connections among subpopulations and even kinships by parentage analyses at the stand level, which would make the effective seed dispersal distances directly inferable (Ashley, 2010; Dow and Ashley, 1996; Piotti et al., 2009; Pluess, 2011). Additionally, these methods can be used to estimate the fat tail of the dispersal function indirectly (Piotti et al., 2009).
Another interesting application would be to use this model to estimate the pollen influx in lakes (cf. Sugita, 2007). Pollen influx rates are widely used for vegetation reconstructions at the tundra–taiga transition zone (e.g. Klemm et al., 2016) and could now be used either to tune the dispersal parameters for a more precise population dynamics prediction, or inversely, to reconstruct ancient tree stands by simulations. Before the genetics or the influx rates are included in the model, however, a revision of boundary conditions for pollen in the model is necessary. This must include a relevant source area for the pollen (cf. Sugita, 2007) to determine to what extent genetic traits are delivered by pollen from beyond the borders of the simulated area. If this can be efficiently parameterized, the model could further be used to track genetic lineages in time.
We conclude that it is feasible to implement wind-driven seed dispersal and pollination in an individual-based model, which is then able to run across broader areas. However, the simulated area and duration of the simulation are constrained by available computer power and memory, and thus further effort is needed to minimize the computational load of this model in order to allow landscape-scale simulations on a standard computer. With the new model setup, further applications in combination with the genetics of the represented species are now feasible and can bring us detailed knowledge about the behaviour of the treeline and the biogeography of larch species through time.
The source code of the host model is available from GitHub,
The supplement related to this article is available online at:
SK and AG planned the study, NK, AG, and SK updated the model and implemented new functions. AG and SK performed the simulations and the statistical analysis. SK and AG wrote the manuscript. UH provided substantial advice in the process of data analysis and paper writing.
The authors declare that they have no conflict of interest.
We acknowledge Sven Willner for valuable advice in the process of parallelizing the program code and Cathy Jenks for proofreading and improving the manuscript. Furthermore, we particularly thank the handling topic editor Hisashi Sato as well as Julia Nabel and one anonymous reviewer for their valuable comments on the previous version of the manuscript. The position of Stefan Kruse is funded by the Helmholtz Initiative Fund.The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.Edited by: Hisashi Sato Reviewed by: Julia Nabel and one anonymous referee