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

Abstract

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.

Impulsive Fire Disturbance in a Savanna Model: Tree–Grass Coexistence States, Multiple Stable System States, and Resilience

“…Some, like the model in this paper, focus on biomass, while others model the fraction of coverage for different plants. In contrast, spatially extended models have the potential to capture patterns of trees and grass within the savanna, as well as the boundaries between savanna and forest (Wuyts et al 2019;Yatat et al 2018). For example, in Goel et al (2020), Goel et al create a reaction-diffusion model which reflects natural biome boundaries between savanna and forest and conclude that biome recovery may be easier in a spatially extended model than it is in a non-spatial model.…”

“…Given the ecological, economic and cultural value of savannas as well as their precarious ecological role, savanna ecosystems are a frequent target of modeling investigations (Accatino et al 2010;Batllori et al 2015;Baudena et al 2010;Beckage et al 2009;Djeumen et al 2021;Goel et al 2020;Patterson et al 2020;Ratajczak et al 2017;Schertzer et al 2015;Staver et al 2011a;Tamen et al 2016Tamen et al , 2017Touboul et al 2018;Wuyts et al 2019;Yatat et al 2018). In tree-grass-fire interaction models of savanna ecosystems, the impact of fire on tree and grass biomass is often represented as a continuous mortality in ordinary differential equation (ODE) models (Accatino et al 2010;Beckage et al 2009;Djeumen et al 2021;Staver et al 2011a;Touboul et al 2018) and in spatial partial differential equations (PDE) models (Goel et al 2020;Wuyts et al 2019;Yatat et al 2018). Continuous fire is an obvious simplification, and in this paper, we show that it is not equivalent to similar models with discrete fire.…”

A minimalistic model of vegetation physiognomies in the savanna biome

“…Spatially explicit versions of the present model are desirable, for instance to better address the dynamics of savanna-forest mosaics found under humid climates and investigate the stability of particular landscape features such as localized structures (e.g. groves, Lejeune et al [2002]) or abrupt boundaries (Yatat et al [2018a], Wuyts et al [2019]) that are of particular relevance to understand the dynamics of forest-savanna mosaics in the face of global change.…”

Partial tipping in a spatially heterogeneous world

“…Recently, the bistability between tropical forests and grasslands in a heterogeneous environment has been studied [62]. In a spatially extended (rescaled) version of the model in [63], the local fraction of fire-prone forest trees at is given by F and its evolution is modelled as…”