Emergent structure and dynamics of tropical forest-grassland landscapes.

Previous work indicates that tropical forest can exist as an alternative stable state to savanna. Therefore, perturbation by climate change or human impact may lead to crossing of a tipping point beyond which there is rapid forest dieback that is not easily reversed. A hypothesized mechanism for such bistability is feedback between fire and vegetation, where fire spreads as a contagion process on grass patches. Theoretical models have largely implemented this mechanism implicitly, by assuming a threshold dependence of fire spread on flammable vegetation. Here, we show how the nonlinear dynamics and bistability emerge spontaneously, without assuming equations or thresholds for fire spread. We find that the forest geometry causes the nonlinearity that induces bistability. We demonstrate this in three steps. First, we model forest and fire as interacting contagion processes on grass patches, showing that spatial structure emerges due to two counteracting effects on the forest perimeter: forest expansion by dispersal and forest erosion by fires originating in adjacent grassland. Then, we derive a landscape-scale balance equation in which these two effects link forest geometry and dynamics: Forest expands proportionally to its perimeter, while it shrinks proportionally to its perimeter weighted by adjacent grassland area. Finally, we show that these perimeter quantities introduce nonlinearity in our balance equation and lead to bistability. Relying on the link between structure and dynamics, we propose a forest resilience indicator that could be used for targeted conservation or restoration.

Mean-field models of dynamics on networks via moment closure: An automated procedure.

In the study of dynamics on networks, moment closure is a commonly used method to obtain low-dimensional evolution equations amenable to analysis. The variables in the evolution equations are mean counts of subgraph states and are referred to as moments. Due to interaction between neighbors, each moment equation is a function of higher-order moments, such that an infinite hierarchy of equations arises. Hence, the derivation requires truncation at a given order and an approximation of the highest-order moments in terms of lower-order ones, known as a closure formula. Recent systematic approximations have either restricted focus to closed moment equations for SIR epidemic spreading or to unclosed moment equations for arbitrary dynamics. In this paper, we develop a general procedure that automates both derivation and closure of arbitrary order moment equations for dynamics with nearest-neighbor interactions on undirected networks. Automation of the closure step was made possible by our generalized closure scheme, which systematically decomposes the largest subgraphs into their smaller components. We show that this decomposition is exact if these components form a tree, there is independence at distances beyond their graph diameter, and there is spatial homogeneity. Testing our method for SIS epidemic spreading on lattices and random networks confirms that biases are larger for networks with many short cycles in regimes with long-range dependence. A Mathematica package that automates the moment closure is available for download.

Tropical tree cover in a heterogeneous environment: A reaction-diffusion model.

Observed bimodal tree cover distributions at particular environmental conditions and theoretical models indicate that some areas in the tropics can be in either of the alternative stable vegetation states forest or savanna. However, when including spatial interaction in nonspatial differential equation models of a bistable quantity, only the state with the lowest potential energy remains stable. Our recent reaction-diffusion model of Amazonian tree cover confirmed this and was able to reproduce the observed spatial distribution of forest versus savanna satisfactorily when forced by heterogeneous environmental and anthropogenic variables, even though bistability was underestimated. These conclusions were solely based on simulation results for one set of parameters. Here, we perform an analytical and numerical analysis of the model. We derive the Maxwell point (MP) of the homogeneous reaction-diffusion equation without savanna trees as a function of rainfall and human impact and show that the front between forest and nonforest settles at this point as long as savanna tree cover near the front remains sufficiently low. For parameters resulting in higher savanna tree cover near the front, we also find irregular forest-savanna cycles and woodland-savanna bistability, which can both explain the remaining observed bimodality.

Author Correction: Amazonian forest-savanna bistability and human impact.

This corrects the article DOI: 10.1038/ncomms15519.

Amazonian forest-savanna bistability and human impact.

A bimodal distribution of tropical tree cover at intermediate precipitation levels has been presented as evidence of fire-induced bistability. Here we subdivide satellite vegetation data into those from human-unaffected areas and those from regions close to human-cultivated zones. Bimodality is found to be almost absent in the unaffected regions, whereas it is significantly enhanced close to cultivated zones. Assuming higher logging rates closer to cultivated zones and spatial diffusion of fire, our spatiotemporal mathematical model reproduces these patterns. Given a gradient of climatic and edaphic factors, rather than bistability there is a predictable spatial boundary, a Maxwell point, that separates regions where forest and savanna states are naturally selected. While bimodality can hence be explained by anthropogenic edge effects and natural spatial heterogeneity, a narrow range of bimodality remaining in the human-unaffected data indicates that there is still bistability, although on smaller scales than claimed previously.