Landscape erosion is generally considered to take place vertically downward. Here, by writing gradient-driven erosion in Hamiltonian form, we show this is not true. Instead, we find it takes place in two directions simultaneously: (i) normal to the surface and (ii) along rays pointing upstream and either up or down depending on how erosion rate scales with slope. The rays follow the shortest time paths that determine how long it takes for a landscape to respond to changes in external conditions.

Arid ecosystem health is a complex interaction between vegetation and climate. Coupled with impacts from grazing, it can result in quick changes in vegetation cover. We present a wind erosion and vegetation health model with active grazers over 100-year tests to find the limits of arid environments for different levels of vegetation, rainfall, wind speed, and grazing. The model shows the resilience of grass landscapes to grazing and its role as an improved tool for managing arid landscapes.