  We study the local H\"older continuity and the asymptotic behaviour of
solution, $\mathbf{u}=(u^1,\cdots, u^k)$, of the degenerate system
\begin{equation*} u^i_t=\nabla\cdot\left(m\,U^{m-1}\nabla u^i\right) \qquad
\text{for $m>1$ and $i=1,\cdots,k$ } \end{equation*} which describes the
populations density of $k$-species whose diffusion is determined by their total
population density $U=u^1+\cdots+u^k$. For the local H\"older continuity, we
adopt the intrinsic scaling and iteration arguments of DeGiorgi, Moser, and
Dibenedetto. Under some regularity conditions, we also prove that the
population density function of $i$-th species with the population $M_i$
converges in $C_s^{\infty}$ to $\frac{M_i}{M}\mathcal{B}_M(x,t)$ as $t\to
\infty$ where $\mathcal{B}_M$ is the Barenblatt profile of the standard porous
medium equation with $L^1$ mass $M=M_1+\cdots+M_k$. As a consequence of
asymptotic behaviour, it is shown that each density function becomes a concave
function after a finite time.
