Tropical Tree Cover in a Heterogeneous Environment: a Reaction-diffusion Model

TL;DR

This paper analyzes a reaction-diffusion model of tropical tree cover, revealing conditions for stable forest and savanna states, and explaining observed bimodal distributions through analytical and numerical methods.

Contribution

It provides an analytical derivation of the Maxwell point and explores complex dynamics like cycles and bistability in a spatial model of tropical vegetation.

Findings

The Maxwell point depends on rainfall and human impact.

Fronts between forest and nonforest settle at the Maxwell point.

Bistability and cycles explain observed bimodal distributions.

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. 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.