Periodic monostable traveling waves for a nonlocal dispersal weak competition system with seasonal succession

This paper is concerned with the monostable time periodic traveling waves of a nonlocal dispersal weak competition system with seasonal succession in the whole space. Due to the lack of parabolic estimates and compactness, we establish the existence of periodic traveling waves with speed $ c>c^{*} $ by the asymptotic fixed point theorem together with the upper-lower solutions, where $ c^* $ is some positive constant. The asymptotic behavior of periodic traveling waves at positive infinity is given by the comparison argument and the asymptotic spreading properties of single nonlocal dispersal equation with seasonal succession. Moreover, when $ c<c^{*} $, the nonexistence of periodic traveling waves is confirmed by the contradiction argument.