Abstract
The role of fire in expanding the global distribution of savanna is well recognized. Empirical observations and modeling suggest that fire spread has a threshold response to fuel-layer continuity, which sets up a positive feedback that maintains savanna–forest bistability. However, modeling has so far failed to examine fire spread as a spatial process that interacts with vegetation. Here, we use simple, well-supported assumptions about fire spread as an infection process and its effects on trees to ask whether spatial dynamics qualitatively change the potential for savanna–forest bistability. We show that the spatial effects of fire spread are the fundamental reason that bistability is possible: because fire spread is an infection process, it exhibits a threshold response to fuel continuity followed by a rapid increase in fire size. Other ecological processes affecting fire spread may also contribute including temporal variability in demography or fire spread. Finally, including the potential for spatial aggregation increases the potential both for savanna–forest bistability and for savanna and forest to coexist in a landscape mosaic.
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Notes
Note that this constitutes a slight departure from Staver et al. (2011a), but one that more accurately reflects the reality of fire spread.
The grass cluster of \(z\) is defined as the set of Grass cells \(z'\), each of which is connected to \(z\) by some continuous Grass path, i.e. by a sequence \(\pi = (z_1, \ldots , z_n)\) with \(z_1 = z\) and \(z' = z_n\), such that for each \(i < n\), the sites \(z_i\) and \(z_{i+1}\) are two neighboring Grass patches. Finally, the density of a cluster \(F(z)\) is defined as the number of sites in the cluster renormalized by the number of sites in the landscape.
More precisley, \(T_\sigma (z) := \frac{1}{Z}\sum _{z' \ \text {is a Tree}} \exp (-\frac{(z-z')^2}{2 \sigma ^2})\) where \(Z\) is the renormalization constant such that \(\sum _{z'} \exp (-\frac{(z-z')^2}{2 \sigma ^2})=Z\).
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Acknowledgments
We acknowledge S. Archibald for helpful discussions. Funding for this work was provided by “The Emergence And Evolution of Ecosystem Functioning” from the Andrew W. Mellon Foundation.
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Schertzer, E., Staver, A.C. & Levin, S.A. Implications of the spatial dynamics of fire spread for the bistability of savanna and forest. J. Math. Biol. 70, 329–341 (2015). https://doi.org/10.1007/s00285-014-0757-z
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DOI: https://doi.org/10.1007/s00285-014-0757-z