||We investigate the evolution of population density vector, u = (u(1),..., u(k)), of k-species whose diffusion is controlled by its absolute value vertical bar u vertical bar. More precisely, we study the properties and asymptotic large time behaviour of solution (u(1),..., u(k)) of degenerate parabolic system (u(i))(t) = del . (vertical bar u vertical bar(m-1) del u(i)) for m > 1 and i = 1,..., k. Under some regularity assumptions, we prove that the component u(i) which describes the population density of i-th species with population M-i converges to Mi/vertical bar M vertical bar B-vertical bar M vertical bar in space with two different approaches where B-vertical bar M vertical bar is the Barenblatt solution of the standard porous medium equation with L-1 mass vertical bar M vertical bar = root M-1(2) +....+ M-k(2). As an application of the asymptotic behaviour, we establish a suitable Harnack type inequality which makes the spatial average of u(i) under control by the value of u(i) at one point. We also find 1-directional travelling wave type solutions and the properties of solutions which has travelling wave behaviour at infinity. (C) 2020 Elsevier Inc. All rights reserved.