we impose homogeneous Dirichlet boundary conditions, modeling in this way an unfavorable region at level $y=0$. The system we consider is therefore $$\label{14}
  \left\{ \begin{array}{lll}
  u_t(x,t)-Du_{xx}(x,t)=\nu v(x,L,t)-\m u(x,t) & \text{for } x\in{{\mathbb{R}}}, & t>0 \\
  v_t(x,y,t)-d\D v(x,y,t)=f(v) & \text{for } (x,y)\in\O, & t>0  \\ 
  dv_y(x,L,t)=\m u(x,t)-\nu v(x,L,t) & \text{for } x\in{{\mathbb{R}}}, & t>0 \\
  v(x,0,t)=0 & \text{for } x\in{{\mathbb{R}}}, & t>0, \end{array}\right.$$ where $D,d,\mu,\nu,L$ are positive constants. Ascertaining the long time behavior of the solutions of the Cauchy problem associated to will be the first step in the study of the problem. The result is the following

\[thltb\] Let $(u,v)$ denote the solution of starting from a nonnegative, not equal to $(0,0)$, bounded and continuous initial datum $(u_0, v_0)$. If $$\label{15}
\frac{f'(0)}{d}\leq\left(\frac{\pi}{2L}\right)^2,$$ then $$\label{16}
\lim_{t\to+\infty}(u(x,t),v(x,y,t))=(0,0)$$ locally uniformly in $\overline{\O}$, while if $$\label{17}
\frac{f'(0)}{d}>\left(\frac{\pi}{2L}\right)^2$$ and, moreover, $$\label{18}
\frac{f(s)}{s} \text{ is nonincreasing},$$ then $$\label{19}
\lim_{t\to+\infty}(u(x,t),v(x,y,t))=\left(\frac{\nu}{\mu}V(L),V(y)\right)$$ locally uniformly in $\overline{\O}$, where $V(y)$ is the unique solution of $$\label{110}
  \left\{ \begin{array}{l}
  -dV''(y)=f(V(y)) \quad \text{for } y\in(0,L)\\
  V'(L)=0, \quad V(0)=0. \end{array}\right.$$

A remarkable difference with respect to is that here the width of the strip $L$ plays a role in the existence of positive steady states. Moreover, condition was not necessary to guarantee uniqueness in 