we assume that there exists $\epsilon_0>0$ such that, for all $\epsilon\in (0,\epsilon_0)$, for all $x\in(-L,L)$, $$\label{hypini}
  \left\{
      \begin{aligned}
      & 0 \leq {n_1}^{\mbox{\scriptsize ini}}_{\epsilon}, \quad 0 \leq {n_2}^{\mbox{\scriptsize ini}}_{\epsilon}, \quad {n}^{\mbox{\scriptsize ini}}_{\epsilon}={n_1}^{\mbox{\scriptsize ini}}_{\epsilon}+{n_2}^{\mbox{\scriptsize ini}}_{\epsilon}, \quad 0 < A_0 \leq {n}^{\mbox{\scriptsize ini}}_{\epsilon} \leq B_0, \quad \partial_x n_\epsilon^{\mbox{\scriptsize ini}} (\pm L) = 0, \\
      & \exists\, \zeta^0 \in (-L,L) \mbox{ such that } {n_1}^{\mbox{\scriptsize ini}}_{\epsilon}= {n}^{\mbox{\scriptsize ini}}_{\epsilon} \mathbf{1}_{x \leq  \zeta^0} \mbox{ and } {n_2}^{\mbox{\scriptsize ini}}_{\epsilon}= {n}^{\mbox{\scriptsize ini}}_{\epsilon} \mathbf{1}_{x \geq  \zeta^0}, \\
       & p^{\mbox{\scriptsize ini}}_{\epsilon}:= \epsilon \frac{n_\epsilon^{\mbox{\scriptsize ini}}}{1-n_\epsilon^{ini}} \leq P_M := \max(P_M^1,P_M^2), \\
       &  \max (\| \partial_{x} {n_1}^{\mbox{\scriptsize ini}}_{\epsilon} \|_{L^1(-L,L)}, \| \partial_{x} {n_2}^{\mbox{\scriptsize ini}}_{\epsilon} \|_{L^1(-L,L)}) \leq C,   \\
       & \exists\, n_1^{\mbox{\scriptsize ini}}, n_2^{\mbox{\scriptsize ini}} \in L_+^1(-L,L), 
\mbox{ such that } \|{n_1}^{\mbox{\scriptsize ini}}_\epsilon - n_1^{\mbox{\scriptsize ini}}\|_{L^1(-L,L)} \to 0  \\
& \qquad \mbox{ and } \|{n_2}^{\mbox{\scriptsize ini}}_\epsilon - n_2^{\mbox{\scriptsize ini}}\|_{L^1(-L,L)} \to 0, \mbox{ as } \epsilon \to 0.
      \end{aligned}
    \right.$$ These initial conditions imply that ${n_1}^{ini}_{\epsilon}$ and ${n_2}^{ini}_{\epsilon}$ are uniformly bounded in $W^{1,1}(-L,L)$. Notice also that the existence of $\zeta^0$ being the interface between the two species implies that the two populations are initially segregated.

From [@BPM], we recover that at a fix $\epsilon>0$ under assumption , given initial conditions ${n_1}^{\mbox{\scriptsize ini}}_{\epsilon}$ and ${n_2}^{\mbox{\scriptsize ini}}_{\epsilon}$ satisfying 