Some effects of nonlocal diffusion on the solutions of Fisher-KPP equations in disconnected domains

The question under study is the existence and uniqueness of a non-trivial bounded steady state of a Fisher-KPP equation involving a fractional Laplacian (-Delta)(alpha) in a fragmented domain with exterior Dirichlet conditions. Of particular interest here is the rigidity on the steady states entailed by the nonlocal dispersion. Our results also provide criteria on the domain for the subsistence of a species subject to a nonlocal diffusion in a fragmented area. Further effects, such as the continuity of this principal eigenvalue with respect to the distance between two compact patches in the one dimensional case, are presented.

Automated summary

This paper investigates the existence and uniqueness of a non-trivial bounded steady state of a Fisher-KPP equation involving a fractional Laplacian in a fragmented domain with exterior Dirichlet conditions. The rigidity of the steady states due to nonlocal dispersion is of particular interest. The results also provide criteria for the subsistence of a species subject to nonlocal diffusion in a fragmented area. Furthermore, the paper presents further effects, such as the continuity of the principal eigenvalue with respect to the distance between two compact patches in the one-dimensional case.