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This series of two papers is devoted to the study of the principal spectraltheory of nonlocal dispersal operators with almost periodic dependence and thestudy of the asymptotic dynamics of nonlinear nonlocal dispersal equations withalmost periodic dependence. In the first part of the series, we investigatedthe principal spectral theory of nonlocal dispersal operators from two aspects:top Lyapunov exponents and generalized principal eigenvalues. Among others, weprovided various characterizations of the top Lyapunov exponents andgeneralized principal eigenvalues, established the relations between them, andstudied the effect of time and space variations on them. In this second part ofthe series, we study the asymptotic dynamics of nonlinear nonlocal dispersalequations with almost periodic dependence applying the principal spectraltheory developed in the first part. In particular, we study the existence,uniqueness, and stability of almost periodic solutions of Fisher KPP equationswith nonlocal dispersal and almost periodic dependence. By the properties ofthe asymptotic dynamics of nonlocal dispersal Fisher-KPP equations, we alsoestablish a new property of the generalized principal eigenvalues of nonlocaldispersal operators in this second part of the series.

Non-local dispersal equations with almost periodic dependence. II. Asymptotic dynamics of Fisher-KPP equations