Monodominance in tropical forests: modelling reveals emerging clusters and phase transitions

Abstract

Tropical forests are highly diverse ecosystems, but within such forests there can be large patches dominated by a single tree species. The myriad presumed mechanisms that lead to the emergence of such monodominant areas is currently the subject of intensive research. We used the most generic of these mechanisms, large seed mass and low dispersal ability of the monodominant species, in a spatially explicit model. The model represents seven identical species with long-distance dispersal of small seeds, competing with one potentially monodominant species with short-distance dispersal of large seeds. Monodominant patches emerged and persisted only for a narrow range of species traits; these results have the characteristic features of phase transitions. Additional mechanisms may explain monodominance in different ecological contexts, but our results suggest that percolation-like phenomena and phase transitions might be pervasive in this type of system.

1. Background

Tropical forests are a prime example of a highly diverse ecosystem [1]. Now, however, forests are declining worldwide owing to logging and other major land-use changes [2]. Understanding both the natural and human-induced mechanisms that generate, erode or stabilize plant diversity is thus of paramount scientific and societal importance [1,3].

Paradoxically, in some regions of the tropics, large areas within highly diverse forests are dominated by only a single tree species, with more than 60% (often approx. 90%) of canopy trees belonging to the same species [4]. These areas are said to be ‘monodominant’ and extend over hundreds of square kilometres [5] in Africa, the Americas and Asia [6]. The mechanisms responsible for such divergent patterns within highly diverse ecosystems are still not fully understood [7], but understanding them might help us understand the emergence and maintenance of biodiversity.

In a recent review, Peh et al. [7] presented a conceptual framework that compiles currently discussed mechanisms potentially leading to monodominance. These mechanisms are related to either intrinsic plant traits (e.g. low dispersal ability and large seeds) or extrinsic factors (e.g. disturbances or interactions with mycorrhizal fungi). However, this framework by itself cannot provide a quantitative assessment of the relative contributions of the suggested mechanisms to the emergence of monodominance. This is critical as several mechanisms might synergistically lead to monodominance [7], and the relative importance of the mechanisms might vary among sites [8].

To explore the mechanisms leading to monodominance, modelling is a promising approach because it allows the generation of simplified and generic settings [9,10]. We used the framework provided by Peh et al. [7] as the starting point of a model, focusing on the simplest scenario with only the two traits: ‘large seeds’ and ‘poor dispersal’. This scenario was motivated by features of a real species, Gilbertiodendron dewevrei, which is one of the best studied of the 22 known monodominant species [4,5,11,12]. It grows in large monodominant patches in the Ituri Forest of Africa as well as in southeast Nigeria, eastern Cameroon and the Guineo–Congolian rainforest. Monodominant G. dewevrei patches are embedded in species-rich forests, and the transition from mixed to monodominant areas is rather abrupt and without apparent change in topography [4]. G. dewevrei produces large seeds of 20 g dry weight, whereas most tropical tree seeds have masses between 0.1 and 0.5 g [13,14]. Moreover, the seeds of G. dewevrei have a limited dispersal range (DR), i.e. they are rarely dispersed beyond 6 m from the mother tree [4]. Once established, G. dewevrei dominates all size classes and occupies more or less circular patches. These patches slowly invade the surrounding mixed forests via tree-by-tree replacement [15].

Large seeds and poor dispersal are the only traits of the framework of Peh et al. [7] which are purely intrinsic, i.e. do not depend on other factors of the ecological context. Furthermore, both traits are pervasive in tropical vegetation and might therefore contribute to monodominance in general. For further simplification, our model builds on neutral theory [16], which means that all other tree species in the model are completely identical (e.g. in reproduction success, mortality and dispersal rates). Only the candidate monodominant species is distinguished by its seed weight and dispersal distance.

We believe our model is the first to explore the emergence of monodominant patches in otherwise highly diverse forests. It can be extended by adding and testing other traits and processes from the framework of Peh et al. [7]. In particular, an important negative local feedback, the Janzen–Connell effect [17,18], or a positive local feedback, for example deep leaf-litter that suppresses establishment of competing, non-monodominant species [5,19], could be implemented. However, understanding the effect of additional mechanisms (e.g. inter- and intraspecific competition) or traits will only be possible after the basic scenario is well understood.

Interestingly, our first simulation runs produced output resembling phenomena known from percolation theory. Percolation [20,21] originally described the trickling movement of liquid through a porous medium and was first applied in physics [22,23]. It was soon adopted by other natural sciences, for instance to simulate epidemic processes [24,25]. Percolation theory is used in ecology to calculate connectivity between fragmented and isolated habitats [26,27], to understand the equilibrium of trees and grassland in savannahs [28] and to model forest fires [29].

Our model has features that can be directly linked to percolation. Each grid cell can include one tree, so grid cells can be categorized either as monodominant or as belonging to one of the seven identical species, so the ‘medium’ (forest) characteristics are binary [30]. In each time step, vacant grid cells are populated following a stochastic process [31], and populated grid cells become vacant with a certain probability (i.e. mortality rate) [31]. Finally, monodominant trees can percolate through the whole simulation area, so that a cohesive cluster of monodominant trees spans the simulated forest.

These similarities to classical percolation settings enabled us to use percolation theory to analyse our model. We applied the technique of finite-size scaling [21,32,33] (see the electronic supplementary material for further explanation) because we hypothesized that seed mass (SM) is the control parameter of a phase transition between mixed and completely monodominant forests. Clusters should only appear when SM is close to a critical value separating the two phases. Finite-size scaling allowed accurate estimation of the critical point. The results obtained by using percolation theory are universal and thus do not depend on idiosyncratic properties of a system. This makes them more easily comparable with other ecological systems.

2. Methods

2.1. Model description

A full and detailed model description following the overview, design concepts, details (ODD) protocol [34,35] is provided in the electronic supplementary material. Here, we present a brief and general description. The tropical forest was represented by a grid of square cells. Every grid cell (typical size 20 × 20 m, approximately the area of the largest single adult tree) represented either a canopy tree or a gap, which occurs when a tree died in the previous time step. The simulated forest area is approximately 10 000 ha (512 × 512 grid cells), and we used periodic boundary conditions, i.e. the model world was wrapped into a torus.

The trees belonged to one of eight species, seven of which had identical species traits, as suggested by neutral theory [16], and one of which was the candidate for monodominance. The latter differed from the other seven species in SM, a proxy for the competitive ability of the seedlings and DR.

The monodominant species had a DR of one grid cell (20 m), i.e. the four nearest neighbour cells. Additionally, it had large, highly competitive seeds, whereas the other seven species showed long-distance seed dispersal with a default radius of 20 grid cells (400 m) and small and less competitive seeds. In essence, the seeds of the monodominant species were distributed among four neighbouring grid cells, but those of the other species were distributed on 1256 cells in total. Within a time step, each tree contributes one seed to every grid cell within its dispersal kernel.

All seven of the identical species have the same SM of 20, whereas the monodominant species has a default SM of 9450. These values were chosen because they led, for a certain range of tree mortality, to the emergence of monodominant clusters. The SM as used in the model is a scaled unit and does not refer to the actual SM (mg) of the trees. Rather, it represents the energy and, hence, the competitive strength of the seeds that every tree can put into a single grid cell within its dispersal kernel. Within this kernel, seeds are in principle distributed on all cells but are only considered further on empty cells, i.e. gaps. At the beginning of a simulation, the trees of all eight species were distributed randomly over the entire grid (alternative initial configurations were tested as well; see below and the electronic supplementary material).

For each time step, the following three processes were performed. (i) Disperse seeds into empty grid cells: seed input to empty grid cells was determined. (ii) Recruitment: for all empty grid cells, the species to establish was chosen via lottery. For each species, the establishment probability in the empty grid cells was proportional to the percentage of its total SM on that cell. Thus, for the sake of simplicity, our model assumed that processes such as seed germination, seedling emergence and survival, and tree establishment were positively correlated with the SM [36,37]. (iii) Mortality: the trees that already existed at the beginning of this time step could die randomly according to a constant mortality rate that was identical for all eight species, including the monodominant species.

2.2. Simulations

In all conducted experiments, we used a simulation time of 10 000 time steps and a system size of 512 × 512 grid cells as the default settings. Annual mortality was set to 1.5%. In order to analyse the impact of annual mortality (figure 1), mortality rate was varied between 0.1% and 3% and, in total, 1000 simulation runs were performed. In addition, two parameters were varied for the relative abundance simulations (figure 2): annual mortality between 0.1% and 2.5% and SM of the monodominant species between 9000 and 10 000. We performed 1000 simulation runs.

Figure 1.

The dependence of relative cluster size of the monodominant species on mortality at an intermediate seed mass of the monodominant species (SM = 9500) after 10 000 simulation years. The black horizontal lines in the boxes show the median, the boxes show the first and third quartiles and the vertical line shows the range of relative abundance, except extreme values (=points), in 1000 simulation runs for each combination of parameter values. The curve is hump-shaped, indicating that at mortalities between 1% and 2% the monodominant species can best be established.

Figure 2.

The relative abundance of the monodominant species plotted against its seed mass and representing different annual mortality rates for the trees (mortality rates are the same for all species). The boxes show the range of relative abundance in 1000 simulations of each parameter combination, and the circles mark outliers.

For the finite-size scaling experiments (figure 3; electronic supplementary material, figures S1 and S2), the SM of the monodominant species was varied from 9200 to 9700 for different system sizes (128 × 128, 178 × 178, 256 × 256, 400 × 400 and 512 × 512 grid cells). Two thousand simulation runs were conducted for smaller systems (size < 400) and 1000 runs for larger systems.

Figure 3.

Finite-size scaling of the percolation probability P_L (SM). The intersection of all system sizes indicates a critical point for the seed mass of the monodominant species at SM_c = 9488. The inset shows the same results after rescaling the x-axis; this analysis results in an estimate of 1.4 for the critical exponent v. Two thousand simulation runs were performed for small systems (<400 × 400 grid cells) and 1000 simulation runs for larger systems.

For variation of the dispersal radius (figure 4), finite-size scaling was performed for each dispersal radius of the identical species (1, 3, 5, 7, 10, 15, 20, 30), meaning that, for each dispersal radius, the SM was varied (between 6000 and 10 000) for different system sizes (128 × 128, 178 × 178, 256 × 256, 400 × 400 and 512 × 512 grid cells). Here, for system size below 400 × 400 grid cells, 1000 simulation runs were performed and for bigger system sizes 500 simulation runs. For the comparison of different initialization schemes, for each parameter combination, we ran 1000 simulations.

Figure 4.

Variation of dispersal range DR_i of the seven identical species. The dispersal radius of the seven identical species (default: DR_i = 20, i.e. 1256 grid cells in range) is changed: DR_i ∈ [1 … 30], whereas the dispersal radius of the monodominant species is kept constant with DR m ≡ 1 (four grid cells in range). For every time step, each tree invests a certain amount of energy into seeds, defined here as seed mass × dispersal area (in number of grid cells). The proportion of the energy a tree of the monodominant species and a tree of one of the identical species invests into its propagules is kept constant, independent of the dispersal radius. The critical seed masses SM_c (DR_i) are calculated as shown in figure 3 and electronic supplementary material, figures S1 and S2, and are plotted against the dispersal radii of the non-monodominant species. The value obtained for the case DR_i = DR_m = 1 (as in the neutral model case [16]) is set to 100%. One thousand simulation runs were performed for each parameter combination for small systems (<400 × 400 grid cells) and 500 simulation runs for larger systems.

2.3. Identifying and characterizing monodominant clusters

We refer to a large area dominated by the monodominant tree species as a cluster. We defined a cluster as an aggregation of directly (via one or more of the four nearest neighbours) connected, monodominant grid cells. For visualization, we recorded a movie of a full simulation run where colours marked different clusters (see the electronic supplementary material). To characterize the clusters, we determined the cluster size s (number of grid cells belonging to a cluster), the relative cluster size, i.e. cluster size s divided by the total number of grid cells, and the radius of gyration R_s (a metric of the cluster's spatial extent; see the electronic supplementary material). By using the radius of gyration, one can calculate the fractal dimension d_F of the monodominant clusters [32,33,38]. For this, the logarithm of the cluster size s is plotted against the logarithm of the corresponding R_s (electronic supplementary material, figure S3). The resulting slope of the plot yields the fractal dimension d_F [21].

2.4. Analysing phase transitions

By observing the sharp transitions from forests with and without monodominant clusters in response to varying the SM of the potentially monodominant species, we explored this transition with methods from statistical physics, especially percolation theory [21]. Near phase transitions, the relationship between model parameters and most measurable quantities such as cluster size or correlation length (see the electronic supplementary material) can be described by power laws. The exponents of these power laws characterize phase transitions and are called critical exponents [39,40]. By checking the transitions in our model for these characteristic exponents, we could possibly link our specific model to the large class of spatial models and systems that show phase transitions and in which the geometry of interactions plays a decisive role. We applied a suite of existing methods, e.g. finite-size scaling [32,33], to determine both the critical exponents and the critical seed mass, SM_c, at which the phase transition occurs (electronic supplementary material, table S2).

2.5. Establishment probability

To explore whether higher local densities of monodominant trees in the initial distribution could be a predictor of the emergence and location of monodominant clusters, we determined for all grid cells the establishment probability for the monodominant species, PR_mono, which is the sum of SM contributed by monodominant trees divided by the sum of the SMs of all species within the relevant grid cell. The resulting map of PR_mono (electronic supplementary material, figure S4) was used to visually check for hidden patterns in the initial distribution that would allow for the prediction of where monodominant clusters would emerge.

2.6. Different initialization schemes

To explore the effects of non-random initial distributions of the eight tree species, we used four different initial configurations (see the electronic supplementary material) of the monodominant species, whereas the other seven species were randomly placed. We explored these settings for three different SMs of the monodominant species (9300, 9500, 9600).

2.7. Long versus short dispersal strategy

In the default scenarios, the total SM invested per time step by a single tree of the monodominant species, i.e. SM multiplied by the sum of grid cells within the DR (9450 × 4 = 37 800), is larger than that of the other species (20 × 1256 =25 120). To analyse the effect of the relative total SM that is invested by all species, we systematically varied the DR of the non-monodominant species (from 1 to 30 grid cells) while keeping its SM constant (20). For each DR of the non-monodominant species, we determined the critical SM for the monodominant species that leads to the emergence of clusters. For this, the SM of the potentially monodominant species was varied to identify the smallest SM that would enable the long-term coexistence of clusters and mixed forest over 10 000 time steps. These simulation experiments include the neutral case in which all eight species are identical (DR = 1 and SM = 20).

3. Results

For the default parameter set (electronic supplementary material, table S1), clusters of the monodominant species appeared after approximately 3000 time steps within the initially mixed forest and remained stable over the whole run time (figure 5). In the areas where no clusters could be established, the density of the monodominant species decreased until in the final state no monodominant species could be found outside of a cluster.

Figure 5.

Simulations of 10 000 1-year time steps, with a seed mass of the monodominant species of SM = 9425 (close to the critical point SM_c 9488). The forest landscape consists of 512 × 512 grid cells, corresponding to 10 × 10 km. Black points symbolize the monodominant species, blue/grey/white colours show the non-monodominant tree species. The 20 squares show the simulated forest at different times, starting with one simulation year (top left) and proceeding in increments of 500 years to the bottom left. After some time, black clusters of the monodominant species formed and remained stable over the whole run time. In the areas where no clusters established, the density of the monodominant species decreased until in the final state no monodominant trees could be found outside a cluster.

The relative size of the largest observed cluster showed a hump-shaped relationship with mortality (figure 1). For mortalities below 0.5%, the dynamics were too slow to ensure the formation of monodominant clusters during the simulation time (10 000 time steps). For mortalities above 3.5%, the monodominant species' establishment rate is lower than its mortality. Clusters thus emerged only temporarily but then disaggregated faster than they would grow. Furthermore, an increase in overall mortality raised the probability that two or more neighbouring grid cells will become available for colonization. Such spatial aggregation of empty grid cells is a disadvantage for the monodominant species, because each of the two neighbouring and vacant grid cells can have only three monodominant neighbours by definition, which reduces their establishment probability. Because we were particularly interested in cluster dynamics, for further simulations we chose a mortality rate (1.5%) that enabled cluster growth.

Cluster dynamics are clearly driven by the local configuration of monodominant trees within the mixed forest. Cluster growth occurs only at cluster boundaries, mainly by integrating smaller clusters of the neighbourhood (figure 6).

Figure 6.

Aggregations of monodominant trees after 10 000 simulation years in a forest landscape of 512 × 512 grid cells, with monodominant seed mass at the critical point SM_c = 9488. Each cluster is defined by monodominant cells linked via the four direct neighbours and is represented by a different colour; white cells show mixed-forest areas. The centre of each aggregation is occupied by a single cluster; smaller clusters are found around the circumference of the central clusters.

We plotted the relative abundance of the monodominant species against its SM for different annual mortality rates. It turns out that both SM and the mortality determined the relative abundance of the monodominant species (figure 2). At low SMs, the relative abundance was low, and either no clusters formed at all or they were too small to stabilize. For high SMs, the monodominant clusters became percolating, which means that a single cluster spanned the whole forest either in one dimension or both (electronic supplementary material, figure S5). Neither case corresponds to observations of monodominant areas embedded in species-rich tropical forests. However, these extremes determine the range within which a realistic representation of monodominance is possible.

Thus, clusters emerged and grew further to a certain size and then stabilized in only a narrow range of values of SM. We thus assumed that SM was the control parameter for a phase transition separating both phases and we confirmed this by applying the method of finite-size scaling. Therefore, we plotted the percolation probability P_L against the SM for different system sizes.

As a result, we found an intersection of all curves that indicates a critical point (figure 5). We determined the critical SM, where the transition between the non-monodominant and monodominant phase of forest structure occurred, independently via three different methods (figure 3 and electronic supplementary material, figures S1 and S2). The average value and range of the three values obtained was SM_c = 9488±10.

The critical exponents and the fractal dimension characterizing the transition are summarized in the electronic supplementary material, table S4. For the default parameter set, the fractal dimension of the monodominant clusters was d_F = 1.89 ± 0.3 (electronic supplementary material, figure S3), which is exactly the value that characterizes percolation in two dimensions [21]. The distribution of cluster sizes followed a power law, which is characteristic of phase transitions; the exponent was −1.986 (electronic supplementary material, figure S6).

Although all eight species were distributed randomly at the initial stage, some small areas showed random aggregations of the monodominant species. We hypothesized that the emergence of clusters could be a relic of the initial stochastic clumping of the monodominant species that leads to the formation of clusters. Such clumps were indeed visible on a map of establishment probability that was taken at the beginning of the simulation, but we could not visually identify a link between this initial pattern and the location of the finally emerging clusters (electronic supplementary material, figure S4). This finding suggests that clustering is seeded by a highly stochastic process and cannot be predicted from specific configurations. Most potential ‘nuclei’ of monodominant clusters eventually disappeared. This dominant role of stochasticity is well known from systems which are close to a phase transition.

Additionally, we compared different initial spatial distributions of the monodominant species, including two completely regular configurations where no accidental local clumping of the monodominant species was possible. Even for those scenarios, monodominant clustering occurred after some time.

The dispersal radius of the non-monodominant species and critical SM of the monodominant species were positively correlated. This relationship is concave and saturates for dispersal radii larger than 20 grid cells (figure 4) and shows that a dispersal radius larger than 20 does not provide further advantages for the non-monodominant species for which the monodominant species would have to compensate via a higher SM. Under default model settings with a radius DR_i = 20, the monodominant species had to invest more than an additional 50% in total SM than its competitors to obtain a similar fitness. Fitness was measured as the number of occupied grid cells at the end of the simulation.

4. Discussion

The relative importance of the presumed mechanisms underlying the emergence of monodominance in tropical forests [7] is an open important question. Simulation models allow exploration of whether the suggested pathways can explain the emergence of monodominant clusters within a well-mixed, species-rich forest.

Here, we present a generic, extendable model describing the phenomenon of monodominance. We investigated whether a combination of two traits—large SM and limited dispersal ability—is sufficient to explain the existence of monodominant stands in mixed forests. Using seven identical long-distance dispersing species and one locally dispersing species, we found that clusters of monodominance do emerge and persist (figure 5) if the monodominant species invests more in SM than the other species do. The clusters in our model correspond to observations in real tropical forests where monodominant trees dominate circular areas (=clusters) even without prior disturbance [7].

Using a similarly abstract model, Gandhi et al. [41] showed that, for the establishment and growth of a cluster, the curvature of the boundary between the cluster and its environment is critical. For small clusters, the curvature is high, which makes it difficult for the monodominant species to win local competition for empty grid cells because it is unlikely that more than one or two of the neighbour cells of an empty one will be occupied by the monodominant species. However, as the cluster grows, the curvature of its boundary decreases, and the chances of taking over empty grid cells increases.

In our case as well, a cluster's growth increases with its size; however, in contrast to Gandhi's continuous model, we used a grid-based approach with different dispersal distances and therefore cannot compare our outcomes directly with Gandhi's findings.

Additionally, in our model, monodominant clusters appeared only for narrow ranges of parameter values for SM, dispersal radius and mortality. We found that for low SMs no monodominant clusters were formed, whereas for high SMs the majority of the grid was covered by monodominant trees at the end of the simulations. For intermediate SMs, we found monodominant clusters and mixed forest in coexistence.

We interpreted this sharp change in model behaviour as a phase transition between a mixed-species phase and a monodominant phase. To analyse this, we used methods from statistical physics and percolation theory. Therefore, we plotted different measures, such as the percolation probability or the correlation length against its SM for different system sizes (figure 3 and electronic supplementary material, figures S1 and S2). The bigger the system size, the steeper the curve of the percolation was probability, meaning that the parameter space, in which the monodominant and the identical species can coexist on the grid, becomes narrower. Therefore, we called it a ‘sharp change’: coexistence—a state in which monodominant clusters emerge and persist within a mixed forest—is only possible at the critical point for infinite systems.

The existence of a phase transition indicates that the emergence of monodominance shows universal behaviour and is strongly determined by local interactions. Results obtained by finite-size scaling are independent of the details of interactions, i.e. triangular, square or hexagonal grid, of the specific type of neighbourhood considered (four or eight neighbours; [40]). This finding and the fact that the critical exponents are the same for a wide class of models and settings imply that the mechanisms underlying the emergence of monodominance are related to those underlying other spatial phenomena observed in forests. Canopy gaps in real forests, e.g. the 50 ha Barro Colorado Island plot in Panama, have a fractal dimension (d_F = 1.86 [42]) that is very close to the fractal dimension of the clusters in our model (d_F = 1.89 ± 0.003).

The critical exponents observed in our model correspond to those of two-dimensional standard percolation [21]. This is surprising because two-dimensional percolation grid cells are occupied independently, whereas we use a growth process where the cells influence each other. Usually, the occurrence of such correlations changes the universality class (i.e. the exponents). One consequence of two-dimensional percolation is that the location of clusters cannot be predicted from the initial spatial pattern. The reason for this is that percolation is a second-order phase transition, which means that the correlation length, which describes the average distance of two sites that belong to the same cluster [21], diverges at the critical point. Thus, near this critical point, all regions contribute (over the iterations) to the outcome at any position in the grid. This finding suggests that the clustering process is not only a consequence of specific geometric conditions created by, for example, disturbances, but can also be a consequence of inherent properties of the monodominant species.

This hypothesis is confirmed by the analysis of the different initialization schemes. The occurrence of clustering was not sensitive to the initialization scheme, suggesting that the model's stochastic mechanisms made the simulated forests rather quickly ‘forget’ their initial configuration (see the electronic supplementary material for further information).

The sharp change in model behaviour indicates as well that the processes in our model (large seeds and poor dispersal) may not be sufficient for fully explaining the occurrence of monodominance in Nature. It is unlikely that these traits evolved precisely to values that produce cluster formation and monodominance. Nevertheless, seemingly unstable configurations do occur in Nature, for example the tree–grass coexistence in savannahs that develop into neither woodlands nor grasslands. In savannahs, buffer mechanisms have been hypothesized to stabilize the coexistence [43].

The high sensitivity of our model outcomes to the parameters characterizing the SM/dispersal trade-off is a common feature in competition models and indicates that equalizing trade-offs may not be the sole mechanism explaining monodominance and coexistence [44]. Therefore, local stabilizing feedback mechanisms may contribute to the clumping of the monodominant species [5]. These mechanisms may include changing the understorey environment by casting a significantly deeper shade than measured in mixed-species forests or producing large amounts of slowly decomposing litter which reduces the availability of nitrogen in the soil and drives losses at the ecosystem level by destabilizing plant–nutrient systems of competing tree species [45]. Deep litter poses a mechanical barrier for seeds; therefore, it is well complemented by a large seed size, because larger seeds can penetrate the thick layer of litter more easily to reach the top soil for germination [5]. Moreover, mycorrhiza networks may favour the monodominant species [7]. In short, these processes enable the monodominant species to shape its local surrounding to its needs, thereby giving it an advantage over its competitors. We hypothesize that an extended model that includes one or more of these mechanisms should still reproduce the observed pattern of monodominant clustering, with more rapid cluster formation and with coexistence more robust to parameter values.

We consider it likely that the explanation of monodominance has both generic and idiosyncratic elements. On the other hand, our findings, particularly the occurrence of phase transitions, may also explain why monodominance is not observed more generally and for more species, as tropical forests can harbour hundreds of tree species. It has been found in spatially explicit models that the persistence of competitively superior but short-dispersing species is strongly affected by spatial aggregation [9,46]. Clustering can increase intra- versus interspecific competition and thus acts as a stabilizing mechanism [10,41].

One might argue that our model may be described by the moment closure approximation presented by Bolker & Pacala [9]. However, we used a simulation model because it is the declared purpose of this model to serve as a reference for future modifications, where other mechanisms are added, in a modular way. We cannot expect that those versions could be approximated by moment closures or by similar methods for space discrete models.

We observed that the monodominant species needs to invest far more (approx. 50%) in SM than its longer-dispersing competitors to reach similar abundances. Thus, a large DR was—assuming other species properties are equal—generally a preferable strategy, a result consistent with previous investigations [47,48]. However, long-distance dispersal also comes at a price that we have not considered in our model, i.e. investments in seed morphology such as wings or in fruits to attract potential dispersal vectors. Many tree species circumvent the trade-off between seed size and dispersal by producing highly rewarding seeds or fruits and relying on large-bodied seed dispersers able to ingest and disperse large seed loads over long distances [49,50]. Indeed, most tropical tree species (70–90%) are animal-dispersed [51,52], including many monodominant species (e.g. Dacryodes excelsa [53], Parashorea malaanonan (Dipterocarpaceae) [54]). For example, seeds and fruits of our model species, Gilbertiodendron dewevrei (Fabaceae), are indeed consumed and probably dispersed by a variety of hunted mammals in different African forests of Zaire and Central African Republic [15,55–57].

However, owing to selective defaunation, with large-sized seed dispersers being decimated in most tropical forests [58], many such tree species are experiencing limitations to seed dispersal, which often leads to clumped spatial patterns [59–62] similar to those produced in our model from a similar mechanism. A loss of dispersal vectors for large seeds due to human impact could be expected to reduce large-seeded species owing to dispersal limitation, and to increase spatial aggregation and, thus, negative density-dependence [51]. However, our results suggest that, at least at restricted spatial scales, the opposite might occur for some tree species which are able to use local feedbacks and therefore take advantage of their strongly clumped distribution. These tree species might unexpectedly stabilize or even increase in abundance and eventually drive a slow transformation of mixed forests into monodominant stands and thus decreased biodiversity [62–64], with important outcomes for ecosystem functioning [65].

We focused on two main mechanisms potentially leading to monodominance and left other mechanisms out, to concentrate on the system behaviour in the light of phase transition and percolation theory. Importantly, however, our novel model provides a solid starting point for further investigations into monodominance. In particular, modelling the relevance of defaunation and both positive (e.g. shade tolerance or slowly decomposing litter) and negative feedbacks loops (e.g. Janzen [17] and Connell [18]) is certainly needed. We expect that positive feedback loops might reduce the time needed for a cluster to form, whereas negative feedback loops might slow down or even stop cluster growth. Furthermore, it would indeed be interesting to explore non-neutral settings where diversity is maintained by stabilizing rather than by equalizing mechanisms [3], but this is beyond the scope of our study.

Our model is meant as a baseline for further model development. On the methods side, it implements approaches to explore cluster formation; on the ecology side, it allows, in a robustness analysis sensu V. Grimm and U. Berger (2016, unpublished data), the addition of mechanisms and investigation of whether and under what conditions cluster formation still occurs.

Although generic, the model already allows deeper understanding of the cluster formation processes of the monodominance species and interspecific geometrical interactions. We suggest that mechanisms relevant for percolation phenomena and spatial self-organization might also be relevant for understanding monodominance in general, a new direction in the discussion of monodominance.

Authors' contribution

M.K. designed the study and developed the model; P.B. extended the model, wrote the source code, did the analysis and wrote the manuscript; F.M., R.F. and F.T. helped to develop the model; M.S.M. supported the development of source code; V.G., J.G. and A.K.H. significantly supported the modelling process and the analysis; J.M.F. contributed intellectually for embedding the model into the ecological context; and all co-authors significantly contributed to improve the manuscript.

Competing interests

We have no competing interests.

Funding

We received no funding for this study.