The generalised principal eigenvalue of time-periodic nonlocal dispersal operators and applications.

This paper is mainly concerned with the generalised principal eigenvalue for time-periodic nonlocal dispersal operators. We first establish the equivalence between two different characterisations of the generalised principal eigenvalue. We further investigate the dependence of the generalised principal eigenvalue on the frequency of the periodic oscillation, the dispersal rate and the dispersal spread. Finally, these qualitative results for time-periodic linear operators are applied to time-periodic nonlinear KPP equations with nonlocal dispersal, focusing on the effects of the frequency of the periodic oscillation, the dispersal rate and the dispersal spread on the existence and stability of positive time-periodic solutions to nonlinear equations. [ABSTRACT FROM AUTHOR]