A Savanna Dynamics Model

Savannas are mixed woodland-grassland ecosystems that cover fifth of Earth’s land surface. They are characterised by a continuous grass layer and a discontinuous layer of woody plants. In understanding complex savanna dynamics of main interest is the question how do trees and grasses co-exist without one dominating the other? Amongst possible explanations there has been recognized that disturbances like fires or browsing and grazing may play fundamental role. There is quite rich literature on models of tree-grass coexistence in savannas with such non-equilibrium approach. From the mathematical point of view models containing many different factors often lack formal stochasticity and their analysis is usually based on numerical simulations. Moreover, another important determinant of savanna structure - water availability - is of less unpredictable nature following seasonality. Most of the plants growth happens during wet seasons while large amounts of dry grasses additionally support fires during dry seasons. These effects usually are not directly included in existing models of savanna dynamics. We introduced two minimalistic models of tree-grass coexistence driven by fire disturbances. We provided careful mathematical analysis of appropriate piecewise deterministic Markov process and showed the existence of a unique stationary distribution of tree and grass biomasses using the tools of stochastic semigroup theory. Continuing this approach we work on analytic results for more realistic setting including the seasonality via its effects on plants growth rate as well as a probability of occurrence and intensity of fires. Moreover, we want to include in the final model also grazing and browsing effects (similarly to the one provided in (6) but along with seasonality).