  In this paper, we are concerned with the instability and stability of a
quasi-linear hyperbolic-parabolic system modeling vascular networks. Under the
assumption that the pressure satisfies $\frac{\nu P'(\bar\rho)}{\gamma
\bar\rho} < \beta$, we first show that the steady-state is linear unstable
(i.e., the linear solution grows in time in $L^2$) by constructing an unstable
solution. Then based on the lower grow estimates on the solution to the linear
system, we prove that the steady-state is nonlinear unstable in the sense of
Hadamard. On the contrary, if the pressure satisfies $\frac{\nu
P'(\bar\rho)}{\gamma \bar\rho} > \beta$, we establish the global existence for
small perturbations and the optimal convergent rates for all-order derivatives
of the solution by slightly getting rid of the condition proposed in
[Liu-Peng-Wang, SIAM J. MATH. ANAL 54:1313--1346, 2022].
