  Vast landscapes extending from southern Angola, Namibia, and South Africa
exhibit localised barren patches of vegetation called fairy circles. They
consist of isolated or randomly distributed circular areas devoid of any
vegetation. They are surrounded by fringes of tall grasses and are embedded in
a sparse grassland on sandy soils. When the aridity increases, the size of the
fairy circles increases and can reach diameters up to 10m. Although several
hypotheses have been proposed, the origin of this phenomenon remains unsolved.
We show that a simple non-local model of plant ecology based on realistic
assumptions provides a quantitative explanation of this phenomenon. Fairy
circles result from strong interaction between interfaces connecting two
uniform covers: the uniform grassland and the bare states, and their
stabilisation is attributed to the Lorentzian-like non-local coupling that
models the competition between plants. The cause of their formation in thus
rather inherent to the vegetation dynamics. Our analysis explain how a circular
shape and fringes are formed, and how the aridity level influences the size of
the fairy circles. In agreement with field observation, these theoretical
findings, provide a link between a strong non-local interaction of plants and
the formation of stable fairy circles. Finally, we show that the proposed
mechanism is model-independent: indeed, it also applies to the
reaction-diffusion type of model that emphasises the influence of water
transport on the vegetation dynamics.
