  In this article we consider a cascaded taxis model for two proliferating and
degrading species which thrive on the same nutrient but orient their movement
according to different schemes. In particular, we assume the first group, the
foragers, to orient their movement directly along an increasing gradient of the
food density, while the second group, the exploiters, instead track higher
densities of the forager group. Specifically, we will investigate an initial
boundary-value problem for a prototypical forager-exploiter model of the form
\begin{align*} \left\{ \begin{array}{r@{\,}c@{\,}l@{\quad
}l@{\quad}l@{\quad}l@{\,}c} u_{t}&=&\Delta u-\nabla\cdot\big(u\nabla
w\big)+f(u),\ &x\in\Omega,& t>0,\\ v_{t}&=&\Delta v-\nabla\cdot\big(v\nabla
u\big)+g(v),\ &x\in\Omega,& t>0,\\ w_{t}&=&\Delta w-(u+v)w-\mu w+r(x,t),\
&x\in\Omega,& t>0, \end{array}\right. \end{align*}%} in a smoothly bounded
domain $\Omega\subset\mathbb{R}^2$, where $\mu\geq 0$, $r\in\
C^1(\bar{\Omega}\times[0,\infty))\cap
L^\infty\!\left(\Omega\times(0,\infty)\right)$ is nonnegative and the functions
$f,g\in C^1\!\left([0,\infty)\right)$ are assumed to satisfy $f(0)\geq0$,
$g(0)\geq0$ as well as $$-k_f s^{\alpha} -l_f\leq f(s)\leq -K_f s^\alpha+L_f\
\text{ and }\ -k_g s^\beta -l_g\leq g(s)\leq -K_g s^\beta+L_g\quad\text{for
}s\geq0,$$ respectively, with constants $\alpha,\beta>1$, $k_f,K_f,k_g,K_g>0$
and $l_f,L_f,l_g,L_g\geq0$ and $\alpha,\beta>1$.
  Assuming that $\alpha>1+\sqrt{2}$,
$\min\{\alpha,\beta\}>\frac{\alpha+1}{\alpha-1}$ and that $r$ satisfies certain
structural conditions we establish the global solvability of this system with
respect to a suitable generalized solution concept and then, for the more
restrictive case of $\alpha,\beta>1+\sqrt{2}$ and $\mu>0$, investigate an
eventual regularity effect driven by the decay of the nutrient density $w$.
