Bifurcations analysis and pattern formation in a plant-water model with nonlocal grazing

To investigate the formation mechanism of vegetation patterns in dry-land ecosystems, this paper delves into the impact of non-local grazing on the stability and spatiotemporal dynamics of a plant-water model. We first establish the conditions for the occurrence of codimension-1 bifurcations: Turing bifurcations, Hopf bifurcations, as well as codimension-2 bifurcations: Turing-Turing bifurcations, Turing-Hopf bifurcations, and determine the stable and unstable regions of the positive equilibrium. Regarding Turing bifurcations, utilizing weakly nonlinear analysis methods to derive amplitude equations, we conclude that under the influence of non-local grazing, the system exhibits complex patch patterns, including spot, mixed, and stripe patterns. The main analytical challenges arise from non-local interactions, which increase the difficulty of deriving the amplitude equations. From a biological perspective, besides water and nutrients, herbivores also play a significant role in the self-organization of patch patterns in dry-land ecosystems.