A Network Approach to Cascading Tipping Points in the Amazon Rainforest

By Arie Staal and Nico Wunderling

The future of the Amazon rainforest is a topic of great societal and scientific interest. The heating climate is intensifying droughts across the region, forested areas are continually cleared for agricultural use, and fires contribute to further loss of forest cover. The effects of these threats on the rainforest are nonlinear — they may trigger positive feedbacks that amplify the initial change [10]. Of particular concern is the possibility of tipping points beyond which forest loss becomes uncontrollable. Remote sensing studies have found that different regions of the Amazon might be bistable, depending on local levels of precipitation [1, 4]. Under the region’s wettest conditions, only rainforest areas are located; under the driest conditions, only natural savannas occur. However, either forest or savanna can subsist under intermediate rainfall levels; this is a strong indication that the landscape is locally bistable, meaning that tipping points could occur [1, 4].

To understand possible tipping dynamics in the Amazon, one must consider the correct spatial scales. Locally (on the order of kilometers), feedback between fire and forest cover is at play. Fire becomes more likely as conditions get increasingly drier. Furthermore, a savanna ecosystem that is dominated by grasses catches fire more easily than a forest; grasses dry out quicker than trees and a tree canopy maintains relatively humid conditions that suppress fires. After a fire, grasses bounce back quickly but forest tree species may not recover. This basic mechanism is responsible for the local-scale bistability under intermediate precipitation levels. If conditions become too dry for a forest to maintain itself, it would theoretically reach a bifurcation point and transition to a savanna-type ecosystem [6]. Similarly, at a critical loss of forest cover, fires may percolate through the landscape and tip the forest to the alternative state [7].

Figure 1. Conceptual representation of the Amazon rainforest as a set of interacting tipping elements. 1a. Bifurcation plot of an Amazon rainforest grid cell, depicted here as \(1^\circ \times 1^\circ\) cells. If precipitation falls below a critical level within a certain grid cell, the rainforest state cannot be sustained in this cell and a nonlinear transformation towards an alternative savanna-like state occurs. 1b. Interdependency of several forest grid cells due to atmospheric moisture recycling. This mechanism creates a network of interacting “tipping elements” (see equation (2)). Figure courtesy of Nico Wunderling.

Although these fire dynamics are local, their effects may not remain so. Forests in the Amazon enhance precipitation up to hundreds and thousands of kilometers away [3], so forest loss causes drier conditions that increase the likelihood of fires elsewhere. Therefore, tipping events could technically cascade through the coupled forest-climate system. Where and to what extent this phenomenon occurs depends on evaporation, atmospheric mixing, winds, and precipitation. We utilize a Lagrangian atmospheric moisture tracking model to simulate the trajectories from evaporation to precipitation [5]. This model is forced with atmospheric data and tracks the locations of many moisture parcels throughout the atmosphere, which we can use to map regional forest-to-forest atmospheric connections.

To analyze the possibility of tipping cascades in the Amazon, we construct a complex network of individually bistable nodes that are connected via atmospheric moisture recycling. Each node in the network is represented by a \(1^\circ\times 1^\circ\) grid cell of the Amazon rainforest (see Figure 1). One can simply represent the nonlinear reaction of a rainforest cell to moisture deprivation with the following differential equation [2]:

\[\frac{dx_i}{dt}=-x^3_i+x_i+p_i, \tag1\]

where \(x_i\) is the state of the respective forest grid cell and \(p_i\) is the precipitation that this cell receives. If a particular rainforest grid cell begins to receive a level of precipitation that falls below a critical threshold, this grid cell shifts to an alternative state that represents the degraded, savanna-like regime. 

Atmospheric moisture transport can convey the adverse impacts of cells that become rainforest-free to further cells, which implies that one cannot regard certain pairs of cells in the rainforest as independent of each other. In mathematical terms, one can retrieve the following system of coupled differential equations:

\[\frac{dx_i}{dt}=-x^3_i+x_i+p_i+\Sigma^N_{j=1, i\neq j}A_{ij} \cdot x_j. \tag2\]

Here, \(A_{ij}\) describes the cells that are linked via atmospheric moisture recycling. More details about the specifics of the mathematical background are available in [2] and [8].

Figure 2. Structures of vulnerability within the Amazon rainforest. 2a. Motif of the “feed forward loop” structure (representing a triangular structure) in which one rainforest grid cell (lower right) is targeted by two incoming moisture recycling links (from the left and upper right grid cells). Therefore, a potential tipping event in the left cell and subsequent decrease of atmospheric moisture transport would impact the lower right grid cell in two ways: via the direct link from the left cell and the indirect link from the left cell to the upper right grid cell. This occurrence can lead to a negative amplification and additional threat for the lower right cell. 2b. Number of feed forward loop motifs in the Amazon rainforest, exemplary drawn for the year 2014 on a \(2^{\circ} \times 2^{\circ}\) grid. Figure 2a courtesy of Nico Wunderling, 2b adapted from [9].

From a physical point of view, the tipping of one rainforest grid cell can adversely affect other cells and possibly contribute their tipping as well. The strength of this positive feedback mechanism depends upon the structure of the atmospheric moisture recycling network itself. In this sense, triangular structures seem to have a particularly detrimental effect on the stability of complex tipping networks [9]. The structure—called a feed forward loop in more technical language (see Figure 2a)—is particularly prevalent in the Amazon rainforest, especially in northern areas and in the west near the Andes (see Figure 2b).

Our network approach to study cascading tipping points in the Amazon rainforest illustrates how applied system dynamics—when informed by observations—could benefit the conservation of an essential component of Earth’s climate system. By understanding both the potential for and structure of cascading change, we will be better equipped to prevent unwanted nonlinear consequences of global climate heating, deforestation, and fires.

References
[1] Hirota, M., Holmgren, M., van Nes, E.H., & Scheffer, M. (2011). Global resilience of tropical forest and savanna to critical transitions. Science, 334(6053), 232-235.
[2] Krönke, J., Wunderling, N., Winkelmann, R., Staal, A., Stumpf, B., Tuinenburg, O.A., & Donges, J.F. (2020). Dynamics of tipping cascades on complex networks. Phys. Rev. E, 101(4), 042311.
[3] Staal, A., Tuinenburg, O.A., Bosmans, J.H.C., Holmgren, M., van Nes, E.H., Scheffer, M., ..., Dekker, S.C. (2018). Forest-rainfall cascades buffer against drought across the Amazon. Nat. Clim. Change, 8(6), 539-543.
[4] Staver, A.C., Archibald, S., & Levin, S.A. (2011). The global extent and determinants of savanna and forest as alternative biome states. Science, 334(6053), 230-232.
[5] Tuinenburg, O.A., & Staal, A. (2020). Tracking the global flows of atmospheric moisture and associated uncertainties. Hydrol. Earth Syst. Sci., 24(5), 2419-2435.
[6] Van Nes, E.H., Hirota, M., Holmgren, M., & Scheffer, M. (2014). Tipping points in tropical tree cover: Linking theory to data. Glob. Change Biol., 20(3), 1016-1021.
[7] Van Nes, E.H., Staal, A., Hantson, S., Holmgren, M., Pueyo, S., Bernardi, R.E., …, Scheffer, M. (2018). Fire forbids fifty-fifty forest. PLoS ONE, 18(1), e0191027.
[8] Wunderling, N., Krönke, J., Wohlfarth, V., Kohler, J., Heitzig, J., Staal, A., …., Donges, J.F. (2021). Modelling nonlinear dynamics of interacting tipping elements on complex networks: The PyCascades package. Eur. Phys. J. Spec. Top.
[9] Wunderling, N., Stumpf, B., Krönke, J., Staal, A., Tuinenburg, O.A., Winkelmann, R., & Donges, J.F. (2020). How motifs condition critical thresholds for tipping cascades in complex networks: Linking micro- to macro-scales. Chaos, 30(4), 043129.
[10] Zemp, D.C., Schleussner, C.F., Barbosa, H.M.J., Hirota, M., Montade, V., Sampaio, G., …., Rammig, A. (2017). Self-amplified Amazon forest loss due to vegetation-atmosphere feedbacks. Nat. Commun., 8, 14681.

	Arie Staal is assistant professor at the Copernicus Institute of Sustainable Development at Utrecht University. His research focuses on the resilience of tropical forests in the Anthropocene.
	Nico Wunderling is a postdoctoral researcher at the Potsdam-Institute for Climate Impact Research and the Stockholm Resilience Centre, and a guest researcher at the High Meadows Environmental Institute at Princeton University. His research interests focus on the resilience of the Earth system and the nonlinear dynamics of interacting tipping elements in the Earth's climate system.