Through the analysis of its heterogeneity, the Drape concept provides rich insights about a particular ecosystem or a biome shift. Using multiscale textural analysis (e.g. Gaucherel, 2007) on a 2D projection of the Drape (Figure 4A, similar to Figure 3A) produces a Drape heterogeneity map (Figure 4B) that can help detect different behaviors of the system. The heterogeneity index used is the evenness diversity index (see Supplementary information S1.2 for details of the computation). The higher the evenness diversity, the more heterogeneous the neighbor pixels on the map. Heterogeneous areas are therefore defined as having a high degree of evenness diversity amongst neighboring pixels (Figure 4B) highlighting statistically significant differences in the averaged biomass between neighboring pixels on the Drape (Figure 4A). Homogeneous areas (plateaus and valleys where neighboring pixels have low relative evenness diversity values) imply stable states and potential basins of attraction to explore through additional tests.

This heterogeneity analysis may seem to be in the same spirit as methods applied for the case of potential landscapes. An increased variance in Z associated with a small change in X or Y, measured as Drape heterogeneity, has utility to distinguish among stable (homogeneous zones), transient (gradients) and unstable (narrow ridges or peaks) behaviors (Briske et al., 2010). Such state characteristics result from the Drape’s topography, and not from the dataset value distribution as in the potential landscape concept (Hirota et al., 2011; Scheffer et al., 2012).

Beyond the presentation of the Drape, the objectives of this study were threefold. The first objective was to test whether the 600–1000 mm.yr⁻¹ rainfall range, first proposed to discriminate closed-canopy forests from savannas (Sankaran et al., 2005), could be characterized on the Drape as a specific area such as a ridge or gradient). The second objective was to see whether several savanna types could be highlighted on the Drape due to climate–fire interactions (Hirota et al., 2011; Staver et al., 2011a; Favier et al., 2012) and whether these different types of savanna qualified as different stable states. Finally, the third objective was to test whether savanna and forest stable states would be found close enough to each other on the Drape to be considered as likely, adjacent alternative system states.

We used the dataset from Ruesh and Gibbs (2008) and their reporting of observed ranges of biomass based on climatic zones (IPCC, 2006), Global Ecofloristic Zones (FAO, 2001), and the Global Land Cover 2000 ecosystem classification (Bartholomé and Belward, 2005) to compare with the Drape value ranges.

The African Drape analysis using the Multiscale Heterogeneity Map method and the computation of the evenness diversity index revealed three homogeneous areas (namely S1, S2, S3 in Figure 4B) that we interpret as stable states, and two heterogeneous areas (the G1 gradient area interpreted as a transient state and the R1 ridge area interpreted as an unstable state in Figure 4B). Based on their African geographic locations (Supplementary Figure S3), we compared the characteristics of each revealed Drape area with those from the compiled observations (Ruesh and Gibbs, 2008) for precipitation, biomass, and ecosystem types (Table 1).

The S1 stable state (Figure 4B) represents the most arid regions with the lowest biomass range (Figure 4A; Table 1). Such characteristics and their associated geographic locations extracted from the LPJ-GUESS simulation (Supplementary Figure S3A), are in agreement with ecosystems from sparse grasslands to shrub-dominated savannas. The S1 biomass range likely matches the lower third of the observed range, because the S1 woody components represent less than 30% (not shown) of the potential C biomass simulated. S1 represents a stable state (a valley) on the Drape constituted by arid and semi-arid grassy biomes. They range from desert to tropical steppes to savannas with low woody cover.

The S2 stable state (Figure 4B) represents a plateau composed of regions with 1300–1800 mm annual rainfall, intermediate fire frequencies, and C biomass that is at least ten times heavier than that found in S1. These characteristics and their geographic locations (Supplementary Figure S3B) match perfectly with the tropical moist deciduous forest and rainforest (Table 1).

The S3 stable state (Figure 4B) having biomass range similar to that of S2 and with more than 2300 mm annual rainfall, is a plateau typified by rainforests (Supplementary Figure S3C). Simulated and observed characteristics of these locations conform with the African tropical wet climate regions (Table 1). While simulated biomasses are slightly lower than the mean observed, their range (Table 1) is in agreement with the full range (8.5–33 kgC.m⁻²) of observed rainforest biomasses. Note that vegetation models predict potential vegetation, so that slightly higher biomass than observed biomass is expected – even from reserves such as national parks in which low intensity management activities still exist.

Among the heterogeneous areas revealed by the multiscale map, R1 represents the most heterogeneous area even after discarding outlier pixels (Figure 4B). We describe the R1 area as a ridge because over such small extent area on the multiscale map of the Drape, characterized by a narrow range of quasi inexistent fires and a 700–1200 mm.yr⁻¹ rainfall range, the simulated biomass range is the widest recorded (Figure 4A; Table 1, and Supplementary Figure S3D). This may be partly explained by the rainfall range that crosses the 1000 mm.yr⁻¹ threshold discriminating tropical dry from tropical moist climates (Table 1). Consequently, these geographic areas include several ecosystems from sparse grasslands to tropical moist deciduous forests and rainforests whose observed biomasses, ranging from 0.2 to 15.2 kgC.m⁻², agree well with the wide range extracted from the Drape (Table 1). The last large heterogeneous area is the G1 area, squeezed between S1 and S2 areas on the multiscale map (Figure 4B). G1 is a gradient representative of regions with a slightly wider rainfall range (500–1300 mm.yr⁻¹) than that of R1, a wide range of fire occurrences, and as a consequence with a quite wide range of biomasses (Figure 4A; Table 1) but narrower than that of R1. Simulated characteristics and locations of G1 conditions (Table 1; Supplementary Figure S3E) include regions that are similar to those of R1 and mountain regions. Actually, the R1 ridge could also be considered as the most extreme conditions of the G1 gradient (see Supplementary Figure S3D), where heterogeneity is so high that the system switches from transient states along G1 to unstable states in the R1 area. It is worth noting that African regions composing the G1 transient states (alone or even associated with the R1 regions) are very similar to the regions classified as the projected worst-case biome changes expected in Africa over the 2071–2100 period as compared to the 1961–1990 reference (Niang et al., 2014). Finally, one could also see on the multiscale map a second gradient (dashed line on Figure 4), squeezed between S2 and S3 areas, but narrower and weaker than G1.

Based on these results, we confirm the first hypothesis since the 600–1000 mm.yr⁻¹ rainfall range actually belongs to the G1 gradient that we consider as a transient state area, and to the R1 ridge that we consider as an unstable state occurring in the quasi absence of fires. However, we must reject the second and the third hypotheses because there is no stable state (homogeneous area) within this 600–1000 mm.yr⁻¹ gradient that could be representative of any savanna stable state (second hypothesis), and in turns, the savanna-forest dynamics cannot be considered as alternative stable states. Conversely, we showed that S2 and S3 stable state areas represent different forest stable states and that part of the S1 stable state pixels may represent savannas with low woody cover (less than 30% of the very light biomass (<1.5 kgC.m⁻²)). However, these stable savannas grow with less than 500 mm.yr⁻¹ in rainfall, which is at least 2.6 and 4.6 times drier than annual rainfall for S2 and S3, respectively.

Several paleoecological studies have shown that during the Holocene African Humid Period (de Menocal et al., 2000) and the so-called “Green Sahara” (Chikira et al., 2006; Watrin et al., 2009), several paleolakes were present in the current Saharan and Sahelian regions (Lézine et al., 2011). This was a response to intensified monsoonal rainfall that penetrated inland more than 10° northwards. The rainbelt core migrated up to 5° northwards during the humidity optimum (Texier et al., 2000). Such moister conditions over the 15–20°N latitudinal region, as compared to its present-day aridity, created good environmental conditions for the Guinean-Congolian pollen taxa (Hély et al., 2014) that are exclusively representative of modern tropical rainforest. This suggests that some geographic locations have therefore experienced ecosystem or even biome shifts over the last millennia that may have been similar to those from the S1 to S2 and/or to S3 stable states. Further research is needed to assess the speed of these past changes, to plot geographic-site trajectory on the Drape (as conceptually illustrated on Figure 3C), and to apprehend paths and likely direction shifts.

The multiscale map analysis suggests that most savannas (i.e. GLC2000 forest mosaic and shrub cover classes with 5.0–10.0 kgC.m⁻² biomass range (Ruesh and Gibbs, 2008)) lie on the G1 gradient area and are therefore transient states but without delimited stable states. These savannas could shift to R1 unstable states if only fire (or a similar disturbance such as grazing (Bond et al., 2005) would almost disappear, while they could shift to S2 (S1) stable states if only rainfall would increase (decrease), and only to S1 if both rainfall would decrease and fire would change. Past savanna dynamics applied on the Drape could provide insights on past realized-trajectories and speed. The overall Drape information could be useful for savanna-ecosystem sustainable management, for instance to avoid bush-encroachment (Ringrose et al., 1998) that one could consider as a shift to R1 conditions.

Once valleys (S1) and plateaus (S2 and S3) are identified on the Drape, their location, width and depth could quantify the importance of these basins of attraction, such characterization corresponding well with the system resilience as defined by Walker et al. (2004) and discussed by Beisner et al. (2003) for ecosystem management perspectives. Similarly, ridges could represent the boundaries of contiguous basins of attraction. As previously noted, ridges could be interpreted as bifurcation states (i.e. tipping points) from which small changes in the state variables would induce an ecosystem shift or even a biome shift. From the location (state) of the system on these Drape areas, one could also estimate objectively the latitude (i.e. width) and the resistance (i.e. depth) of the basin of current ecosystems, as well as its precariousness (i.e. its distance to the ridge, which is the boundary of the basin of attraction) and its likely resilience as suggested by Beisner et al. (2003) and Walker et al. (2004).

Observations from current conditions, from past reconstructions or from future projections based on different socio-economic and climatic scenarios can provide the realized states found in different areas of the Drape. Analysis of an ecosystem at a given geographical location and its trajectory on the Drape could provide expected temporal changes of the state variables. One could therefore analyze and validate the Drape variations (i.e. its topography) from embedded system trajectories reconstructed from observations and/or paleoecological studies as conceptually illustrated in Figure 3C.

Understanding ecosystem dynamics as a scientific objective and predicting future responses as a more practical objective are difficult tasks. We propose departing from the current methods based on differential equation systems and/or statistical approaches, as well as from simplifications of the abiotic and disturbance components of ecosystems. Because our method only relies on observed data for validation, one avoids misinterpretation. Debates concerning possible extrapolation in cases of slow dynamics (as suggested above) should move toward synthesis (e.g. Hirota et al., 2011; Ratajczak and Nippert, 2012). The proposed Drape concept is an improvement in ecosystem understanding for the following seven reasons:

i) The Drape presents in a single multidimensional space all ecosystems governed by the same first-order state variables and allows to position and characterize the different states of these ecosystems in order to later study past, present, or future trajectories. (Figure 2 and Supplementary information S1.1). This process-based approach will allow including ecosystem responses over long periods of time and through several spatial (from local to continental) scales. This satisfies the need to retrieve slow but integrative changes during regime shifts that may appear incremental at human time scales (Ratajczak and Nippert, 2012; Hughes et al., 2013b). For example, the response to global warming by terrestrial ecosystems at the end of the last ice age (Minckley et al., 2012) or at the end of the African Humid Period (Lézine et al., 2011; Hély et al., 2014) took millennia, long after the ice sheets had melted or after the African monsoon intensification had terminated.

ii) The Drape, built from the outputs of the process-based model, includes disturbances as a state variable as important as vegetation or precipitation and thus allows to better characterize each ecosystem, its states, and its functional neighbors. Indeed, disturbances were previously considered as parameters or external forces in the representation of system dynamics and manifested an effect on the ecosystem through the biotic component only (e.g. Staver et al., 2011b; Touboul et al., 2018). By definition, ecosystems are simultaneously biotic and abiotic, with several tightly interacting components (e.g. flora, fauna, soils, atmosphere and humans). There is no conceptual reason to focus on a specific component over the others (Gignoux et al., 2011; Schwartz and Jax, 2011; Gaucherel, 2014). This matches the observation that ecosystems are simultaneously conditioned by the laws of thermodynamics and the by natural selection.

iii) The state variables used to build the space and the Drape are symmetric in the system functioning. Each axis has the same weight and equal role in constructing the domain (Figure 2). With this property in mind, the 3D space is the smallest multidimensional space aiming at synthetizing any ecosystem state. More complicated domains can take into account additional extra state variables (e.g. human activities or fauna), if required (Cincotta et al., 2000). The symmetry between state variables also allows the study of the system and its states from different points of view (Figures 2A, B), while maintaining the mechanistic relationships among the state variables.

iv) The Drape itself provides an instantaneous and convenient visualization of the different states encountered, as well as their characterization in terms of associated state variable values. This is interpreted in the example case as an averaged and rasterized (grid-based) surface (Figures 3, 4) of the realized domain (Figure 2).

v) The textural analysis explores the Drape variations to identify basins of attraction, characteristic tipping points and their relationships in terms of distance (Figure 4; Appendix 2). Using a statistical moment such as the mean to build the Drape surface prevents from finding “folded backward” regions previously considered as tipping-points or induced catastrophic shifts (Scheffer et al., 2001). However, tipping-points on the Drape are materialized as ridges (i.e. most heterogeneous areas, Figure 3A). Once identified and delineated, the analysis informs the possible system behavior further away. Methods other than the multiscale textural analysis shown here may be used to analyze the Drape, such as a spectral analysis (e.g. Keitt, 2000).

vi) The Drape concept is generic. It allows featuring all states and associated state variables values extracted from a process-based model. The Drape concept could therefore be used to study other terrestrial, marine, or fresh water ecosystems as long as the chosen state variables are the best representative ones and that values are extracted from (state-of-the-art) process-based model. The Drape can be used for different periods than nowadays, which may point to states being located in different areas of the Drape as compared to the present-day state.

vii) By reporting onto the Drape the trajectory a given geographical location would record through time (observed from time-series such as remote-sensing datasets, simulated from process-based models, or reconstructed from paleoecological datasets, see Figure 3C) we would visualize and analyze its dynamics and clearly characterize the states it went through. Despite its resemblance with real topography or a Waddington’s landscape (Goldberg et al., 2007), it is worth emphasizing that the third axis (Z) of the domain does not carry the meaning of a vertical and oriented direction such as the one constrained by gravity (e.g. Scheffer et al., 2001; Beisner et al., 2003). On this Drape, a ball representing the ecosystem state at a specific location and time could easily move up to higher Z values due to any ecological process. As such, the Drape surface appears much more similar to a chaotic manifold (further discussed below) than the stability landscape of the potential mentioned in previous studies. The Drape surface also seems to combine the ecosystem and the community perspectives from Beisner et al. (2003). Indeed, no process directly acts on this ball (i.e. the ball is not pushed by an external force), but rather intrinsic and stochastic processes modify one or several state variable values which, in turn, “push” the system toward another area of the Drape (Abarbanel, 1996; Strogatz, 2001). Moreover, basins of attraction are not systematically the lowest Drape areas.

In practice, a different domain (and its associated Drape) should be recomputed whenever a state variable should be replaced by another more important or representative variable. This allows different neighboring ecosystems (or even biomes) to occur within the same domain, and approximated by a common Drape. It seems reasonable to find tropical forest and savanna biomes in the same domain, but boreal biomes (e.g. coniferous forests and tundra) would be in another distinct domain, as they would require different ecosystem state variables (e.g. temperature instead of precipitation). This should also allow identifying past ecosystem states for which there are no modern analogues (e.g. the “green Sahara” during the mid-Holocene) in specific areas of the Drape (e.g. Figure 4B, lower right region, which is currently empty). Every biome observed during a given period will likely not occupy all of its potential Drape area. So, the Drape concept does not fully solve the problem of defining the ecosystem or biome boundaries (Gignoux et al., 2011; Gaucherel, 2014). The limits, here defined by the extreme values of each state variable, are physically constrained, while a priori the Drape extent is not. Such boundaries would simultaneously depend on the maximum system variability and by the addressed question.

The Drape concept applied to ecosystems can play an important role in the tipping point debate (Ratajczak and Nippert, 2012; Brook et al., 2013; Gaucherel and Moron, 2015). With tipping points defined as sharp changes in ecosystem dynamics, such changes would fit with all sharp gradients (clines) and sharp peaks observed in the Drape variations. Indeed, to shift from a relatively smooth or flat (i.e. homogeneous) area of the Drape to another one by a sharp gradient is a clue of abrupt variations in states (Lenton et al., 2008; Kriegler et al., 2009). Such tipping points could easily be identified and quantified on the Drape based on their sharpness (relatively to other zones) and from their direction.

In our African example, the G1 gradient appeared approximately twice as heterogeneous as its neighboring stable states (S1 and S2) assimilated to steppes-grassy savannas and forest types, respectively, based on state variable values (Figure 4A). We stress that such transitions are not necessary “state transitions” such as those observed in chaotic systems (Abarbanel, 1996; Ghil et al., 2002), and they would definitely need a detailed and rigorous mathematical analysis to be demonstrated as attempted in some recent studies (Accatino et al., 2010; Zaliapin and Ghil, 2010).

The limitations of the Drape concept presented here are of two sorts. First, there are methodological limitations, which hopefully can be improved upon and progressively removed. For instance, the variability arising from similar geographic areas (i.e. close points in the domain) probably contributes to the volume or “thickness” of the realized domain (Figure 3B). This variability may be retrieved by the calculation and representation of another statistical moment (e.g. variance) of the domain, and be analyzed in a similar manner than the averaged Drape (Figure 4). Moreover, using different moments or statistics such as the mode would allow representation of bimodal distributions in each (X, Y) interval bin if they exist, and thus to the plotting of more complicated (e.g. folded) Drape variations. Such complicated responses have been studied in detail in the catastrophe theory (e.g. the cusp geometry (Thom, 1972)), and ecological studies would gain in applying such a rigorous analysis to ecosystems (Gaucherel et al., 2020).

For clarity, we did not explore the specific role of human beings as one of the ecosystem components. As with other unmentioned ecosystem components (e.g. soils), humans could partly be involved as part of one of the already included components (e.g. climate and/or fires), or as an additional component. The resultant four-dimensional space and the more complicated 3D-Drape (called a hypersurface) could be treated without technical difficulties. This is obviously useful for applied questions related to ecosystem management and environmental policies, and it would be relevant to explore this possibility in human-perturbed systems (Cincotta et al., 2000; Gaucherel et al., 2012).

The second category of limitations of the Drape concept involves the effort needed ultimately to represent a more functional concept such as ecosystem chaos. The Drape concept carries on the ecosystem trajectory that follows the Drape response surface from the spatial and temporal autocorrelations of system components (Figure 3C). With the exception of ridge areas and other possible singularities, each ecosystem state globally shows a high correlation with the next (and nearby) state (Figure 4), due to the processes involved and function of model equations (e.g. Cramer et al., 1999; Smith et al., 2001; Sitch et al., 2008), as well as due to the spatial autocorrelation in the abiotic state variables.

It would be convenient and parsimonious to build the Drape with the system trajectories themselves, which is the approach that provides the dynamical system theory and chaotic system representations in previous studies (Lorenz, 1963; Takens, 1981). To write and solve the equations for a dynamical system would directly lead to a trajectory in the phase-space. This approach assumes that ecosystems are dynamical and possibly chaotic, i.e. fully deterministic, ergodic and sensitive to small variations in initial conditions (Lorenz, 1963; Strogatz, 2001). To our knowledge, this has never been demonstrated for an ecosystem as a whole, although it is documented for some components such as prey–predator subsystems (May, 1977), vegetation dynamics (Solé and Bascompte, 2006) or climate dynamics (e.g. Ghil et al., 2002). However, a focus on purely biotic (or abiotic) components omits the necessary symmetry between biotic, abiotic and disturbance components (e.g. Gaucherel et al., 2020). This is the spirit of the present study.