\text{ such that } \un{u}(x, t) > 0, \\
\un{u}(x, t) &\geq u_1(x, t) \qquad \text{ for all } (x,t) \text{ such that } \un{v}(x, L, t) > 0,
\end{split}$$ then, for any supersolution $(\ov{u},\ov{v})$ of bounded from below and such that $\un{u}\leq\ov{u}$ and $\un{v}\leq\ov{v}$ at $t = 0$, we have $\un{u}\leq\ov{u}$ and $\un{v}\leq\ov{v}$ for all $t > 0$.

\[recp2\] The same result of Proposition \[prcp2\] holds for problems like with an additional drift term in the differential operator.

As a consequence of the previous analysis, we will consider continuous nonnegative initial data throughout the rest of this work, since we are interested in nonnegative solutions of .

Liouville-type result and long time behavior {#sec3}
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In order to determine the long time behavior of the solutions of we need to study the solutions of the elliptic system associated to it, precisely $$\label{31}
  \left\{ \begin{array}{ll}
  -D U_{xx}(x)=\nu V(x,L)-\m U(x) & \text{for } x\in{{\mathbb{R}}}, \\
  -d\D V(x,y)=f(V) & \text{for } (x,y)\in\O,   \\ 
  d V_y(x,L)=\m U(x)-\nu V(x,L) & \text{for } x\in{{\mathbb{R}}}, \\
  V(x,0)=0 & \text{for } x\in{{\mathbb{R}}}. \end{array}\right.$$ Actually, Propositions \[pr31\] and \[pr34\] below suggest that we have to focus on solutions of which are $x-$independent. They are of the form $(U,V(y))$, 