and $$\begin{aligned}
\label{segre}
{n_1}_0{n_2}_0=0,\end{aligned}$$ and the complementary relation $$\begin{aligned}
\label{relcompbis}
p_0\left(\partial_{xx} P_0 + \int_0^t (n_{10}G_1(p_0) + n_{20} G_2(p_0))\,ds + n_0^{\mbox{\scriptsize ini}} -1\right) = 0.\end{aligned}$$ where $P_0$ is defined by $P_0(t,x) =\int_0^t p_0(s,x)ds$.

Introducing the set $\Omega_0=\{p_0>0\}$, we deduce that on $\Omega_0$ we have $$-\partial_{xx} P_0 = \int_0^t (n_{10}G_1(p_0) + n_{20} G_2(p_0))\,ds + n_0^{\mbox{\scriptsize ini}} -1.$$ Deriving with respect to $t$, we find formally $$-\partial_{xx} p_0 = n_{10}G_1(p_0) + n_{20} G_2(p_0).$$ We recognise the Hele-Shaw model. Noticing also that taking $t=0$ into the relation , we recover the expected relation $p_0^{\mbox{\scriptsize ini}} n_0^{\mbox{\scriptsize ini}} = p_0^{\mbox{\scriptsize ini}}$.

The proof of this convergence result is given in Section 3. It is straightforward to observe that adding and provides an equation on the total density similar to the one found in the one species case [@HV; @PQV]. Then we use a similar strategy for the proof relying on a compatness method. However the presence of the two populations generate some technical difficulties. To overcome them, we use the segregation property. Notice that this paper is written in the specific case where the two species are separated by one interface, but could be generalised to many interfaces. Using the segregation of the species we 