[@BRR1], while here it is (see Remark \[re33\] below). From a biological point of view, Theorem \[thltb\] says that, if the strip is not sufficiently large, the influence of the unfavorable region $y=0$ drives the species to extinction. On the contrary, the species will persist if the strip is sufficiently large. In this latter case, a natural question is to study more deeply how the convergence to the steady state occurs. To this end we consider the following concept

\[deasp\] We say that $c^*\in{{\mathbb{R}}}_+$ is an *asymptotic speed of propagation* (in the $x-$direction) for if, denoting by $(u, v)$ the solution of with a continuous, nonnegative, compactly supported initial datum $(u_0, v_0)\neq(0, 0)$, we have

-   for all $c > c^*$, $$\label{111}
    \lim_{t\to +\infty} \sup_{\substack{|x|\geq ct \\ y\in[0,L]}} |(u(x, t), v(x, y, t))| = 0,$$

-   for all $0<c < c^*$, $$\label{112}
    \lim_{t\to +\infty}\sup_{\substack{|x|\leq ct \\ y\in[0,L]}} \left|(u(x, t), v(x, y, t)) - \left(\frac{\nu}{\mu}V(L), V(y)\right)\right|=0,$$ where $V(y)$ is the unique solution of .

In this sense, the main result of the paper is the following

\[thmain\] Problem admits an asymptotic speed of propagation, denoted by $c^*=c^*_L(D,d,\mu,\nu)$, such that:

-   $D\mapsto c^*(D)$ is increasing,

-   the following limits exist and are 