Cross-diffusion effects in extended Lotka-Volterra system describing interspecies competition

The classical Lotka-Volterra o.d.e. system describing interspecies competition with bilinear nonlinearities is fully understood. We propose an extension of the model which accounts for the random dispersal of individuals and for their capability to avoid encounters with competitors by means of a chemo-sensory reaction to the "smell" of rivals (chemorepulsion). The cases of diffusing and non-diffusing repellent are studied as well as  the existence of non-constant steady states and long time-behaviour of solutions. The context of the seminal work of  Shigesada-Kawasaki-Teramoto (1979) in which the classical L-V model was extended to take into account self-and cross-diffusion effects is also considered.