Ecological Disturbance as a Renewal Process: Theory and Application to Fire History

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Abstract
Failure-time analysis an renewal theory were used to evaluate two assumptions implicit in most studies involving calculations of disturbances frequency. These calculations assume that the disturbance process is stationary (intervals between disturbances are drawn from the same distribution) and that the probability of disturbance does not change with time since the last disturbance. I present quantitative methods and apply them to long-term fire-occurrence data (fire scars on red pine trees and stratigraphic charcoal data) and climate data from northwestern Minnesota. Results show that past decade- and century-scale fluctuations in climate correspond to changes in the disturbance regime. Probability of fire occurrence increases with time since the last fire, albeit at different rates during the various climatic settings that have prevailed over the last 750 yr. These results suggest two reasons to question disturbance regimes calculated from spatial dispersion events. First, expected fire intervals derived as the inverse of the spatial proportion of area disturbed requires a stationary process. The space-time analogy assumed by this method is importantly wrong when the expected interval between disturbances changes over time. Second, because fire hazard is an increasing function of time since the last fire, the number of disturbances predicted to occur over short time intervals will be overestimated by the usual constant-hazard assumption. Probability densities of events having age-specific probabilities that increase over time (e.g. fire, windthrow) exhibit modes at time t>0 compared with exponential decrease with time.