Propagation dynamics of a parabolic-elliptic chemotaxis model in a time-periodic shifting environment

The current paper is devoted to the study of the propagation dynamics of a parabolic-elliptic chemotaxis model in a shifting environment. We investigate the scenario in which the favorable and unfavorable habitats are spatially separated, with the favorable habitat shifting at a time-periodic speed. We first establish the vanishing, persistence, and spreading properties of solutions with compactly supported and front-like initial data. In particular, we show that if the self-limitation rate of the species is large relative to the chemotactic sensitivity and the species is initially confined to a bounded region, then the species eventually becomes extinct when the habitat shifts rightward with a large average speed. In contrast, when the average shifting speed of the habitat is moderate, the species can persist and spread along the shifting habitat into an increasingly large spatial region. Finally, we establish the existence and nonexistence of time-periodic forced wave solutions whose wave speed coincides with the average shifting speed of the habitat.