for the growth functions which will be defined latter in this paper.

The aim of this paper is is to study the incompressible limit $\epsilon \rightarrow 0$ for the two populations systems. When the two species are not in contact, the system is equivalent to the one population model [@HV], this is why we limit ourself in this paper to the case where the two populations are initially in contact. To use the solutions defined in [@BPM], we restrict the space to a compact domain $(-L,L)$ with $L>0$ and assume and are verified. Outside the domain $(-L,L)$, the system will be equivalent to the one population model.

We firstly remark that by adding and , we get, $$\begin{aligned}
\partial_t n_{\epsilon} - \partial_x (n_{\epsilon} \partial_x {p}_{\epsilon}) = {n_1}_{\epsilon} G_1({p}_{\epsilon})+{n_2}_{\epsilon}G_2({p}_{\epsilon}) \mbox{ in } (-L,L). 
\label{eq:nn}\end{aligned}$$ Multiplying by $P'(n_\epsilon)$ we find an equation for the pressure, $$\label{eq:pp}
 \partial_t p_{\epsilon} - (\frac{p_{\epsilon}^2}{\epsilon}+p_{\epsilon})\partial_{xx} p_{\epsilon} -  |\partial_x p_{\epsilon}|^2 = \frac{1}{\epsilon}(p_{\epsilon}+\epsilon)^2({n_1}_{\epsilon} G_1(p_{\epsilon})+{n_2}_{\epsilon} G_2(p_{\epsilon})) \mbox{ in } (-L,L).$$ Formally, passing at the limit $\epsilon \rightarrow 0$, we expect the relation, $$-p_0^2 \partial_{xx} p_0 = p_0^2 ({n_1}_{0} G_1(p_{0})+ {n_2}_{0} G_2(p_{0})) \mbox{ in } (-L,L).$$ In addition, passing formally to the limit $\epsilon\to 0$ into , it appears clearly that 