Analysis of a one-dimensional forager-exploiter model

\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