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In this paper, we investigate the asymptotic dynamics of Fisher-KPP equationswith nonlocal dispersal operator and nonlocal reaction term in time periodicand space heterogeneous media. We first show the global existence andboundedness of nonnegative solutions, and then obtain some sufficientconditions ensuring the uniform persistence. In particular, we study theexistence, uniqueness and global stability of positive time periodic solutionsunder several different conditions.

Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media