  This paper studies the following system of differential equations modeling
tumor angiogenesis in a bounded smooth domain $\Omega \subset \mathbb{R}^N$
($N=1,2$): $$\label{0}
  \left\{\begin{array}{ll}
  p_t=\Delta p-\nabla\cdotp p(\displaystyle\frac \alpha {1+c}\nabla
c+\rho\nabla w)+\lambda p(1-p),\,& x\in \Omega, t>0,
  c_t=\Delta c-c-\mu pc,\, &x\in \Omega, t>0,\\ w_t= \gamma p(1-w),\,& x\in
\Omega, t>0,
  \end{array}\right. $$ where $\alpha, \rho, \lambda, \mu$ and $\gamma$ are
positive parameters. For any reasonably regular initial data $(p_0, c_0, w_0)$,
we prove the global boundedness ($L^\infty$-norm) of $p$ via an iterative
method. Furthermore, we investigate the long-time behavior of solutions to the
above system under an additional mild condition,
  and improve previously known results. In particular, in the one-dimensional
case, we show that the solution $(p,c,w)$ converges to
  $(1,0,1)$ with an explicit exponential rate as time tends to infinity.
