use Hilbert duality method to establish uniqueness of solution to the limiting system. Finally in Section \[sec:numeric\], we present some numerical simulations of the system - when $\epsilon$ is going to 0 and simulations of a specific application on tumor spheroid growth.

Main results {#sec:mainresult}
============

In this paper we aim to prove the incompressible limit $\epsilon \rightarrow 0$ of the two populations model with non overlapping constraint - in one dimension. We first introduce a list of assumptions on the growth terms and the initial conditions. For the growth, we consider the following set of assumptions: $$\label{hypG}
  \left\{
      \begin{aligned}
         &\exists\, G_m>0, \quad \| G_1 \|_{\infty} \leq G_m, \quad \| G_2 \|_{\infty} \leq G_m,\\
        & G_1', G_2' <0, \quad \mbox{ and }  \exists\, P^1_M, P^2_M>0, \quad G_1(P^1_M)=0 \mbox{ and } G_2(P^2_M)=0,\\
        & \exists\, \gamma>0, \quad \min(\inf_{[0,P^1_M]} |G_1'|,\inf_{[0,P^2_M]} |G_2'| )= \gamma,\\
        & P_M := \max(P_M^1,P_M^2), \quad \exists\, g_m \geq 0, \ \min\left(\inf_{[0,P_M]} G_1,\inf_{[0,P_M]} G_2\right) \geq -g_m. 
      \end{aligned}
    \right.$$ The set of assumptions on the growth rate is standard and similar to the one in [@HV]. The parameters $P_M^1$ and $P_M^2$ are called homeostatic pressures which represent the maximal pressure that the tissue can handle before starting dying. For the initial datas, 