positive real numbers $$\lim_{D\to 0}c^*(D)=\ell_0,
    \qquad
    \lim_{D\to+\infty}\frac{c^*(D)}{\sqrt{D}}=\ell_\infty,$$

-   for fixed $D,d,\mu,\nu$ we have $$\lim_{L\to+\infty}c^*_{L}(D,d,\mu,\nu)=c^*_{\infty}(D,d,\mu,\nu),$$ where $c^*_{\infty}(D,d,\mu,\nu)$ is the asymptotic speed of propagation of Problem .

The paper is organized like follows: in Section \[sec2\] we recall some tools from [@BRR1; @BRR2] which will be indispensable throughout the rest of the paper; Section \[sec3\] provides the proof of Theorem \[thltb\]; in Sections \[sec4\] and \[sec5\] we construct $c^*$ and derive some properties that will allow us, in Section \[sec6\], to show that it satisfies Definition \[deasp\] and relation $(i)$ of Theorem \[thmain\]. In Section \[sec7\] we prove relation $(ii)$ of Theorem \[thmain\] and finally in Section \[sec8\] we study the influence of the road on the asymptotic speed of propagation, comparing with the case in which no road is present, and give the proof of Theorem \[thmain\]$(iii)$.

Preliminary results {#sec2}
===================

In this section we present some fundamental results that are contained in or follow easily from [@BRR1; @BRR2]. The existence of a solution for the Cauchy problem associated to with a continuous initial datum $(u_0,v_0)$ follows from an easy modification of [@BRR1 Appendix A] and uniqueness follows from a comparison principle which will be diffusely used throughout this paper and 