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This paper is concerned with the stability of transition waves and strictlypositive entire solutions of random and nonlocal dispersal evolution equationsof Fisher-KPP type with general time and space dependence, including time andspace periodic or almost periodic dependence as special cases. We first showthe existence, uniqueness, and stability of strictly positive entire solutionsof such equations. Next, we show the stability of uniformly continuoustransition waves connecting the unique strictly positive entire solution andthe trivial solution zero and satisfying certain decay property at the endclose to the trivial solution zero (if it exists). The existence of transitionwaves has been studied in [34, 39, 45, 46, 61] for random dispersal Fisher-KPPequations with time and space periodic dependence, in [41, 42, 43, 51, 52, 53,58, 63] for random dispersal Fisher-KPP equations with quite general timeand/or space dependence, and in [17, 48, 56] for nonlocal dispersal Fisher-KPPequations with time and/or space periodic dependence. The stability resultestablished in this paper implies that the transition waves obtained in many ofthe above mentioned papers are asymptotically stable. Up to the author'sknowledge, it is the first time that the stability of transition waves ofFisher-KPP equations with both time and space dependence is studied.

Stability of transition waves and positive entire solutions of Fisher-KPP equations with time and space dependence