Role of Stochastic Fluctuations and Climate Heterogeneity in Vegetation Patterns of Mesic Savannas

In this talk, I will discuss a few mathematical problems arising in the study of the competition between savannas and forests, with a particular focus on where the boundary between these two vegetation types arises and its stability in the face of climate change. Savanna trees and forest trees compete for grassy spaces to grow, and this competition is mediated by fires carried mostly by grass. Climate and precipitation govern birth and death rates. In addition to competing for space, fires play a key role in mediating this competition. Forest trees do not carry fires efficiently, but when exposed to a fire, they are killed. Savanna trees are less sensitive to fires: while they may carry it more efficiently, adult savanna trees resist fires, and savanna saplings subject to fires are not killed but delay their maturation to adult trees (they are top-killed, their roots remain healthy). Simon Levin and Carla Staver introduced an age-based mathematical model describing this competition. From the mathematical viewpoint, these relatively simple equations are of an astounding dynamical richness. I will present a few studies around the Staver-Levin model, first exhibiting their subtle dynamics and rich bifurcation diagram, then showing how one can derive a spatially-extended integro-differential model from a system of stochastic equations representing more finely the individual behaviors of trees, grassy patches, seed transport and fire propagation. I will then explore the dynamics of these spatially-extended systems in the presence of resource limitations or in the presence of gradients of precipitation. Most of this material is joint work with Simon Levin, Denis Patterson and Carla Staver.