  \begin{abstract} \noindent % We consider the one-dimensional parabolic system
  The system
  \bas
  \left\{ \begin{array}{l}
  u_t= u_{xx} - \chi_1 (uw_x)_x, \\[1mm]
  v_t = v_{xx} - \chi_2 (vu_x)_x, \\[1mm]
  w_t = dw_{xx} - \lambda (u+v)w - \mu w + r,
  \end{array} \right.
  \eas % that has been proposed as a model to describe social interactions
within mixed forager-exploiter groups. is considered in a bounded real
interval, with positive parameters $\chi_1,\chi_2,d,\lambda$ and $\mu$, and
with $r \ge 0$. Proposed to describe social interactions within mixed
forager-exploiter groups, this model extends classical one-species
chemotaxis-consumption systems by additionally accounting for a second axis
mechanism coupled to the first in a consecutive manner. \abs % It is firstly
shown that for all suitably regular initial data $(u_0, v_0, w_0)$, an
associated Neumann-type initial-boundary value problem possesses a globally
defined bounded classical solution. Moreover, it is asserted that this solution
stabilizes to a spatially homogeneous equilibrium at an exponential rate under
a smallness condition on $\min\{\io u_0, \io v_0\}$ that appears to be
consistent with predictions obtained from formal stability analysis.\abs
