energy $$\mathcal{E}(n) = \int_\RR P(n(x))dx.$$ as a gradient flow for the Wasserstein metric. Using this property we derive a model for two species of cells. Let us denote $n_1(x,t)$ and $n_2(x,t)$ the two cell densities depending on the position $x \in \RR^d$ and the time $t>0$. We assume that the pressure depends on the total density $n=n_1 +n_2$. As the pressure depends on a parameter $\epsilon$, we introduce this dependancy in the notation. We define the free energy for the two cell populations by, $$\mathcal{E}(n_{\epsilon}) = \int_\RR P({n_1}_{\epsilon}(x)+{n_2}_{\epsilon}(x))dx.$$ Restricting to the one dimensional case, the system of equation deriving from this free energy is then defined by, $$\begin{aligned}
\partial_t {n_1}_{\epsilon} - \partial_x ({n_1}_{\epsilon} \partial_x p_{\epsilon}) = {n_1}_{\epsilon} G_1(p_{\epsilon}), \label{eq:n1} \\
\partial_t {n_2}_{\epsilon} - \partial_x ({n_2}_{\epsilon} \partial_x p_{\epsilon}) = {n_2}_{\epsilon} G_2(p_{\epsilon}), \label{eq:n2} \\
p_\epsilon = P(n_\epsilon) = \epsilon \frac{n_{\epsilon}}{1-n_{\epsilon}},  \label{eq:p2}  \\
n_\epsilon = {n_1}_\epsilon + {n_2}_\epsilon, \label{eq:n}\end{aligned}$$ with $G_1, G_2$ the growth functions, and $p_{\epsilon}$ the pressure.

The existence of solution for system - has been proven in [@BHIM; @BPM] for a compact domain $(-L,L)$ with $L>0$, with Neumann homogeneous boundary condition. In particular, it is shown that at a fix $\epsilon>0$, given initial conditions ${n_1}^{\mbox{\scriptsize ini}}_{\epsilon}$ and ${n_2}^{\mbox{\scriptsize ini}}_{\epsilon}$ satisfying, 