Critical patch size and insufficient maximal biomass density level in the Klausmeier vegetation model with non-local dispersal on flat landscapes

We study a reaction–diffusion–dispersal model originally proposed by Klausmeier (1999) to describe the interactions between vegetation biomass and water availability. Motivated by ongoing climate change and the resulting decline in rainfall, our primary focus is on identifying conditions under which stable stationary solutions—corresponding to vegetation survival—exist. From a theoretical standpoint, we successfully construct such solutions and derive precise criteria for their existence. Our analysis also reveals the presence of a critical biomass threshold: below this level, vegetation inevitably goes extinct, with no possibility for recovery. Additionally, we identify a critical patch size—i.e., a minimal domain size—sufficient to sustain vegetation. Domains smaller than this threshold cannot support stable vegetation patterns.