Spreading speeds of invasive species in a periodic patchy environment: effects of dispersal based on local information and gradient-based taxis

We consider a periodic environment with favorable and unfavorable patches alternately arranged in one dimension. In such heterogeneous environments, invasive animals may undergo not only random diffusion but also directed movement toward favorable patches. Here we propose reaction–diffusion–advection models in which the diffusion term is of either Fickian or Fokker–Planck type, the advection term is given by a gradient-based taxis, and the growth rate depends on both the environmental conditions and the population density. We first present hypotheses for the existence of a periodic traveling wave and a method to derive its spreading speed. Then this method is applied to a case in which the environment is homogeneous within each patch, so that taxis occurs at interfaces between different patches, causing accumulated distributions in favorable patches with density jumps at the interfaces. We show how the Fickian or Fokker–Planck diffusion and the gradient-based taxis interplay to determine the spreading speed.