We investigate the evolution of population density vector, u=(u1,⋯,uk), of k-species whose diffusion is controlled by its absolute value |u|. More precisely, we study the properties and asymptotic large time behaviour of solution u=(u1,⋯,uk) of degenerate parabolic system(ui)t=∇⋅(|u|m−1∇ui)for m>1 and i=1,⋯,k. Under some regularity assumptions, we prove that the component ui which describes the
... [Show full abstract] population density of i-th species with population Mi converges to Mi|M|B|M| in space with two different approaches where B|M| is the Barenblatt solution of the standard porous medium equation with L1 mass |M|=M12+⋯+Mk2.
As an application of the asymptotic behaviour, we establish a suitable Harnack type inequality which makes the spatial average of ui under control by the value of ui at one point. We also find 1-directional travelling wave type solutions and the properties of solutions which has travelling wave behaviour at infinity.