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The question studied here is the existence and uniqueness of a non-trivialbounded steady state of a Fisher-KPP equation involving a fractional Laplacian(--$\Delta$)^$\alpha$ in a domain with Dirichlet conditions outside of thedomain. More specifically, we investigate such questions in the case of generalfragmented unbounded domains. Indeed, we take advantage of the non-localdispersion in order to provide analytic bounds (which depend only on thedomain) on the steady states. Such results are relevant in biology. Forinstance, our results provide criteria on the domain for the subsistence of aspecies subject to a non-local diffusion in a fragmented area. These criteriaprimarily involve the sign of the first eigenvalue of the operator(--$\Delta$)^$\alpha$ -- Id in a domain with Dirichlet conditions outside ofthe domain. To this end, we exhibit a result of continuity of this principaleigenvalue with respect to the distance between two compact patchs in the onedimensional case. The main novelty of this last result is the continuity up tothe distance 0.

Properties of steady states for a class of non-local Fisher-KPP equations in disconnected domains