A New Perspective On Modeling Forest Fires

A New Perspective on Modeling Forest Fires
Jos´ Leonel Larios-Ferrer1 , Justin Peterson2 , Arturo Vargas3 , e Leon Arriola4 , Michael Golinski5 , and Benjamin Morin5 September 7, 2010
Universidad Aut´noma del Estado de Hidalgo1 , o Northwestern College2 , University of California Irvine3 , University of Wisconsin Whitewater4 , Arizona State University5 ,
Abstract In this study, we usenumerical simulations to heuristically explore the spread of forest fires. Our numerical studies are based on a “bottom-up” framework in which we start with a model with no spatial information on how forest vegetation is distributed (the Mean Field model (MF)). The MF is then replaced by a more detailed model which explores the effects of local, spatial interactions between vegetation and fire (thePair-Approximation model (PA)). In this detailed study, the MF model serves as our “null” model because of its disconnection from actual biological processes (i.e. the absence of spatial interactions between vegetation and fire and how it affects fire spread). The most developed model in our framework is a Cellular Automata (CA) model. The stochastic and spatially explicit features of the CA model make itideal for exploring the effects of distance and random behavior on the spread of fire. With the CA model, we gain insight that is directly applicable to actual forest fire management. For each model, we compare and contrast the dynamics of fire spread using a single and two layered (connected) lattice to measure the effect of including differential behavior of fire between the understory and canopy. Fromeach of the models we observe thresholds (when available) for the stability of the fire-free equilibrium (FFE). We also

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utilize sensitivity analysis to determine the relationships between parameters in the MF and PA model and the basic ignition number, a measure for the average number of new trees that should catch on fire when a single source of fire is introduced into a forest. Resultsindicate that for all three models, the rate of fire spread (α), the rate at which an occupied burning state returns to a non-burning occupied state (β), and the rate at which a burning occupied state becomes an empty site (γ) determines the stability or instability of a forest fire. In the case of the two-layered lattice versions of the models, we find that fire controls are best focused on theunderstory level.

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Introduction

Empirical and theoretical management strategies for minimizing the spread of potentially harmful forest fires have important consequences for the environment and its inhabitants. For example, strategies should be implemented to reduce potentially devastating effects caused by the spread of a forest fire into nearby residential areas. Recently, a number of mathematicalmodels have been developed to predict the spread of forest fires and the environmental consequences that may ensue (Schueller, 2003; Mandel, Beezley, Bennethum, Chakraborty, Coen, Douglas, Hatcher, Kim, and Vodacek, 2007; Michelis and Consolini, 2002; D’Ambrosio, Spataro, and Trunfio, 2006). Many of these models germinated from Rothermel’s fire model (1983), which predicted the spread of a forestfire by taking into account biologically relevant parameters such as meteorological trends, fuel characteristics, and forest topography. In the present study, we use previous theoretical research to develop a multi-layered mathematical modeling approach based within a bottom-up framework to understand the spread of fire and to determine strategies for minimizing its spread and impact.

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AMulti-Model Approach

Given the complex nature of forest fires, theoretical models that capture their dynamics in a satisfactory manner are sparse. In addition, it is not always clear what the advantages and disadvantages are in using different models (Figure 1). As a result, our study uses several models based within a bottom-up framework to capture the effects of the explicit inclusion of space 2...