Date Published: June 27, 2019
Publisher: Public Library of Science
Author(s): Bert Wuyts, Alan R. Champneys, Nicolas Verschueren, Jo I. House, Grant Lythe.
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.
First analyses of the satellite-derived MODIS Vegetation Continuous Fields (VCF) tree cover product  found strong evidence for the bistability hypothesis [2, 3]. They did this by showing that tropical tree cover data are multimodal at intermediate rainfall values, i.e. they have multiple maxima in their empirical probability distribution function. When taking the plausible assumption that more frequently observed tree cover values are more stable, such multimodality implies multistability.  found forest-savanna bistability, from the observation that the tree cover data has a bimodal distribution in a rainfall range of intermediate rainfall, with as modes savanna (about 20% tree cover) and forest (about 80% tree cover). Similarly,  found forest-savanna-treeless tristability, with an extra treeless state (about 0%). The treeless state was not found by , most likely because they excluded areas with bare soil. A scatterplot of tree cover versus rainfall revealed how the stability of the states depends on rainfall. In such a scatterplot, the modes—stable states according to the dynamical interpretation—show up as regions with high point density. With increasing mean annual rainfall, the inferred probability of being in a higher tree cover mode increases. Hence it was concluded that rainfall can be seen as the bifurcation parameter in a dynamical system with a hysteresis loop. From here, we restrict our focus to forest-savanna bistability.
In the first section below, we derive the MP of the homogeneous forest model. In the second section, the front pinning location in the heterogeneous forest model is derived via a numerical continuation. The third section shows simulation results of the heterogeneous forest and forest-savanna models.
In this paper, we have provided a first analytical and numerical analysis of our spatially heterogeneous reaction-diffusion model of tropical tree cover. We have treated this model before with a more realistic set-up  (in 2D, with noise and forced by observed climate, soil and human impact), but we formulated it here in an as simple as possible form (in 1D, deterministic and forced by linear rainfall) for easier mathematical analysis. The heterogeneity was captured with the relation (3), such that low x values represent dry and high x values represent wet areas. From the homogeneous system without savanna trees/saplings [S = T = 0, (5)], a Maxwell point was derived. We showed via a numerical continuation and linear stability analysis of the spatially heterogeneous forest model that this MP is still of use for the spatially heterogeneous case because here, it is the parameter value at which the forest front pins. The MP of the homogeneous forest model and the rainfall value at which the forest model’s front pins as a function of external parameters (the dashed red line and the solid blue line in Fig 1 respectively) are indistinguishable and have the same shape as what was obtained in  by simulation. Existence of a MP in reaction-diffusion equation with a bistable reaction term [13, 14, 20] and pinning under heterogeneity  is consistent with previous work. For parameters that lead to low cover of savanna trees, the MP of (5) is also a good predictor of the forest-savanna model’s forest front [S, T ≠ 0, (1)] (Fig 2C–2F). This is because the effect of savanna trees on forest trees, mediated by burnt area [see (1)], remains negligible when savanna tree cover near the forest front stays below the threshold where fire spread is inhibited, i.e. T < Yc. Choosing parameters such that savanna tree cover near the forest front exceeds this threshold (T ≳ Yc) makes the forest front shift away from the MP of (5), towards drier areas (Fig 3A and 3B). In this regime where savanna tree cover affects forest tree cover, we also found forest-savanna cycles and savanna-woodland bistability, which both can lead to bimodal tree cover distributions under the same external forcings. These cycles are consistent with the existence of Hopf bifurcations in the nonspatial system  above a certain value of the parameters equivalent to P and rs. For an explanation of the physical mechanism behind the cycles, we refer to . We found that the cycles can turn irregular by diffusion. That the irregular cycles are produced endogenously suggests that close to the forest front, sudden and unpredictable loss of forest can occur without climatic or anthropogenic perturbations. We speculate that the irregularity is due to spatiotemporal chaos, which is known to occur in the wake of traveling fronts [24, 25]. To prove this, it would need to be shown additionally that the cycles produced by the deterministic system are truly aperiodic and that there is sensitivity to initial conditions . We further showed via simulation that bistability of a savanna and a woodland state can arise in the savanna model (i.e. the model without forest trees) under a regime of high sapling recruitment and high fire occurrence (Fig 3C and 3F). When introducing forest trees (under the same conditions), the savanna-woodland bistability does not survive at higher rainfall, due to competition between savanna and forest trees (Fig 3B). Instead, the irregular cycle discussed above appears. Where it is too dry for forest, savanna tree cover bistability does survive. To obtain a complete picture of the behavior of the spatial model and how it differs from the nonspatial model, its bifurcation diagrams need to be made. A step towards increased realism is then the consideration of two spatial dimensions instead of one, with a further step towards increased realism being the verification of how this diagram is affected by spatial heterogeneity. Source: http://doi.org/10.1371/journal.pone.0218151