Dynamics of the one-dimensional self-organized forest-fire model

We examine the dynamical evolution of the one-dimensional self-organized forest-fire model (FFM), when the system is far from its statistically steady state. In particular, we investigate situations in which conditions change on a time scale that is faster than, or of the order of the typical time needed for relaxation. An analytical approach is introduced based on a hierarchy of first-order nonlinear differential equations. This hierarchy can be closed at any level, yielding a sequence of successively more accurate descriptions of the dynamics. It is found that our approximate description can yield a faithful description of the FFM dynamics, even when a low order truncation is used. Employing both full simulations of the FFM and our approximate descriptions, we examine the time scales and cluster-size-dependent dynamics of relaxation to the statistical equilibrium. As an example of changing external conditions in a natural forest, the effects of a time-dependent lightning frequency are considered.