Nonlocal grazing in patterned ecosystems.

Many ecosystems exhibit gapped, labyrinthine, striped or spotted patterns. Important examples are vegetation patterns in drylands: these patterns are viewed as precursors of a catastrophic transition to a degraded state. A possible source of degradation is overgrazing, but many current spatially extended models include grazing in a local linear way. In this article nonlocal grazing responses are derived, taking into account (1) how many consumers there are (demographic response) (2) where they are (aggregative response) and (3) how much they forage (functional response). Different assumptions lead to different grazing responses, the type of grazing has a large influence on how ecosystems adapt to changing environmental conditions. In dryland simulations the different types of grazing are shown to alter the desertification process driven by decreasing rainfall. A sufficiently strong aggregative response leads to the suppression of vegetation patterns, nuancing their role as generic early warning signals.

Striped pattern selection by advective reaction-diffusion systems: resilience of banded vegetation on slopes.

For water-limited arid ecosystems, where water distribution and infiltration play a vital role, various models have been set up to explain vegetation patterning. On sloped terrains, vegetation aligned in bands has been observed ubiquitously. In this paper, we consider the appearance, stability, and bifurcations of 2D striped or banded patterns in an arid ecosystem model. We numerically show that the resilience of the vegetation bands is larger on steeper slopes by computing the stability regions (Busse balloons) of striped patterns with respect to 1D and transverse 2D perturbations. This is corroborated by numerical simulations with a slowly decreasing water input parameter. Here, long wavelength striped patterns are unstable against transverse perturbations, which we also rigorously prove on flat ground through an Evans function approach. In addition, we prove a "Squire theorem" for a class of two-component reaction-advection-diffusion systems that includes our model, showing that the onset of pattern formation in 2D is due to 1D instabilities in the direction of advection, which naturally leads to striped patterns.