Spreading properties for a time-space periodic Lotka-Volterra cooperative system in multi-dimensional media

This work is devoted to the study of spreading properties for a time-space periodic Lotka-Volterra cooperative system posed in multi-dimensional media. Under some appropriate conditions, we prove that the system admits a single spreading speed as well as an asymptotic spreading speed and an asymptotic spreading set, in the sense of some spreading properties of solutions to the Cauchy problem with certain initial values. Moreover, we give an upper bound for the fastest spreading speed and a lower bound for the slowest spreading speed in the general case. We also give the existence and nonexistence of periodic traveling waves.