  In this paper, we are concerned with a quasi-linear hyperbolic-parabolic
system of persistence and endogenous chemotaxis modelling vasculogenesis in
$\mathbb{R}$. Under some suitable structural assumption on the pressure
function, we first predict the system admits a nonlinear diffusion wave in
$\mathbb{R}$ based on the empirical results in the literature. Then we show
that the solution of the concerned system will locally and asymptotically
converges to this nonlinear diffusion wave if the wave strength is small. By
using the time-weighted energy estimates, we further prove that the convergence
rate of the nonlinear diffusion wave is algebraic.
