Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model

An integro-differential reaction-diffusion equation is proposed as a model for populations where local aggregation is advantageous but intraspecific competition increases as global populations increase. It is claimed that this is inherently more realistic than the usual kind of reaction-diffusion model for mobile populations. Three kinds of bifurcation from the uniform steady-state solution are considered, (i) to steady spatially periodic structures, (ii) to periodic standing wave solutions, and (iii) to periodic travelling wave solutions. These correspond to aggregation and motion of populations.