Transient and asymptotic distributions of vegetation under recurrent stochastic fires

Fire is a major ecological driver in forests worldwide, strongly influencing vegetation dynamics and ecosystem resilience. This work analyses the stochastic dynamics of tree biomass under recurrent fires with randomly distributed inter-fire times. We consider a model with logistic growth between fire events, each reducing biomass instantaneously by a fixed proportion, and we derive explicit expressions for the deviation of tree biomass from equilibrium after any number of fires. These expressions allow recursive computation of the moments of the transient distributions, including their means and variances, as well as the corresponding asymptotic moments under appropriate convergence conditions. To complete the probabilistic description of the fire dynamics, we apply the Random Variable Transformation method to compute the transient and asymptotic probability density functions of tree biomass of the pre-fire and post-fire events. The theoretical results are illustrated and discussed through examples considering exponentially distributed inter-fire times.