Using Mathematica to Explore the Role of Vegetation-Fire Feedbacks in Structuring Ecological Communities

Brian Beckage and Chris Ellingwood

Department of Plant Biology, University of Vermont, Burlington, VT 05405, United States.

Abstract

We used Mathematica to investigate the potential for vegetation feedbacks on fire to stabilize savanna communities. Savannas consist of widely spaced trees in an  intervening matrix of grasses. Repeated fire disturbances have been shown to result in non-equilibrium savannas with grasses and trees coexisting, as long as fire frequency is maintained within a specified range.  Fire is a function of vegetation, however, as well as climatic conditions, creating the potential for non-linear feedbacks between vegetation and fire with stable savanna states. We use Mathematica to investigate both non-spatial and spatial representations of vegetation-fire feedbacks formulated alternatively as a system of differential equations and a cellular automaton, respectively. We show that vegetation-fire feedbacks can stabilize ecosystems in a savanna state where trees and grasses coexist, with abrupt state transitions between grassland, savanna, and forest states. In our spatial model, prototyped in Mathematica but ported to Java, we specifically allow for the probability of fire spreading to a cell to be a function of the local neighborhood, with the probability maximized where grasses interface with savanna trees. This spatially localized feedback stabilizes the ecosystem in a savanna state with a spatially overdispersed pattern of savanna trees that is consistent with observations from pine savannas. We were able to formulate and analyze our models, visualize results, and call code from other languages using Mathematica.