  This work considers a model for oncolytic virotherapy, as given by the
reaction-diffusion-taxis system $$ \left\{ \begin{array}{l} u_t = \Delta u -
\nabla \cdot (u\nabla v)-\rho uz, \\[1mm] v_t = - (u+w)v, \\[1mm] w_t = D_w
\Delta w - w + uz, \\[1mm] z_t = D_z \Delta z - z - uz + \beta w, \end{array}
\right.
  $$ in a smoothly bounded domain $\Omega\subset\mathbb{R}^2$, with parameters
$D_w>0, D_z>0, \beta>0$ and $\rho\ge 0$.\\ % Previous analysis has asserted
that for all reasonably regular initial data, an associated no-flux type
initial-boundary value problem admits a global classical solution, and that
this solution is bounded if $\beta<1$, whereas whenever $\beta>1$ and
$\frac{1}{|\Omega|}\int_{\Omega} u(\cdot,0)>\frac{1}{\beta-1}$, infinite-time
blow-up occurs at least in the particular case when $\rho=0$.\abs %
  In order to provide an appropriate complement to this, the present work
reveals that for any $\rho\ge 0$ and arbitrary $\beta>0$, at each prescribed
level $\gamma\in (0,\frac{1}{(\beta-1)_+})$ one can identify an
$L^\infty$-neighborhood of the homogeneous distribution $(u,v,w,z)\equiv
(\gamma,0,0,0)$ within which all initial data lead to globally bounded
solutions that stabilize toward the constant equilibrium $(u_\infty,0,0,0)$
with some $u_\infty>0$.
