Impact of directed movement on invasive spread in periodic patchy environments

To address the effect of taxis of invasive animals on their spreading speed in heterogeneous environments, we deal with an advection-diffusion-reaction equation (ADR) in a periodic patchy environment. Two-types of advection that spatially vary depending on environmental heterogeneity are taken into consideration: a stepwise taxis function and a saw-like taxis function. We first analyze the ADR with the stepwise taxis advection, and derive an invasion criterion. When the invasion criterion holds, an initially localized population evolves to a traveling periodic wave (TPW). The asymptotic speed of the TPW is found to be equal to the minimal speed of the TPW analytically derived. Thus, we examine how the minimal speed is influenced by the taxis. The major results are: (1) As the magnitude of the taxis toward favorable patches increases, invasion becomes more feasible. However, the spreading speed increases at first, and then decreases to show a one-humped curve against the magnitude of the taxis; (2) As the scale of fragmentation in the patchy environment is increased, the spreading speed increases when the magnitude of the taxis is small, while it decreases when the magnitude of the taxis becomes sufficiently large. These characteristic features qualitatively apply to the ADR model with the saw-like taxis function.