ini}}\geq 0$ and $ G_m <\infty$. Then, for all $t\geq 0$, ${n_1}_\epsilon(t)\geq 0$ and ${n_2}_\epsilon(t)\geq 0$.

To show the nonnegativity we use the Stampaccchia method. We multiply by $\mathbf{1}_{{n_1}_\epsilon<0}$ and denote $|n|_- = \max(0,-n)$ for the negative part, we get $$\mathbf{1}_{{n_1}_\epsilon<0}\partial_t {n_1}_\epsilon - \mathbf{1}_{{n_1}_\epsilon<0}\partial_x ({n_1}_\epsilon \partial_x p_\epsilon) = \mathbf{1}_{{n_1}_\epsilon<0}{n_1}_\epsilon G_1(p_\epsilon).$$ With the above notation, it reads $$\partial_t |{n_1}_\epsilon|_{-} -\partial_x (|{n_1}_\epsilon|_{-} \partial_x p_\epsilon) = |{n_1}_\epsilon|_{-} G_1(p_\epsilon).$$ We integrate in space, using assumption and $\partial_x p_\epsilon (\pm L,t) = p'_{\epsilon}(n_\epsilon) \partial_x n_\epsilon (\pm L,t)=0$, we deduce $$\frac{d}{dt} \int_{-L}^L |{n_1}_{\epsilon}|_{-}dx \leq \int_{-L}^L |{n_1}_{\epsilon}|_{-} G_1(p_{\epsilon})dx \leq G_m\int_{-L}^L |{n_1}_{\epsilon}|_{-}dx.$$ Then we integrate in time, $$\int_{-L}^L |{n_1}_{\epsilon}|_{-}\,dx \leq e^{G_mt}  \int_{-L}^L |{n_1}_{\epsilon}^{\mbox{\scriptsize ini}}|_{-}\,dx.$$ With the initial condition ${n_1}_{\epsilon}^{\mbox{\scriptsize ini}} > 0$ we deduce ${n_1}_{\epsilon} > 0$. With the same method we can show that if ${n_2}_{\epsilon}^{\mbox{\scriptsize ini}} > 0$ we have ${n_2}_{\epsilon}> 0$.

We notice that the positivity gives a formal proof of the segregation of any solution of -. Indeed, defining $r_{\epsilon}={n_1}_{\epsilon} {n_2}_{\epsilon}$ and multiplying by ${n_2}_{\epsilon}$, by ${n_1}_{\epsilon}$ and adding, we obtain the following equation for $r_{\epsilon}$, $$\partial_t r_{\epsilon} - \partial_x r_{\epsilon}\, \partial_x p_{\epsilon} - 2 r_{\epsilon} \partial_{xx} p_{\epsilon} = r_{\epsilon}(G_1(p_{\epsilon})+G_2(p_{\epsilon})).$$ Given that $r^{\mbox{\scriptsize ini}}_\epsilon =0$, we get that $r_\epsilon=0$ at all time.

### A 