  We consider the system \[
  \left\{
  \begin{aligned}
  u_t &= \Delta u - \chi \nabla \cdot ( \tfrac{u}{v} \nabla v) - uv + \rho u -
\mu u^2, \\
  v_t &= \Delta v - v + u v
  \end{aligned}
  \right.
  \tag{$\star$} \] with $\rho \in \mathbb{R}, \mu > 0, \chi > 0$ in a bounded
domain $\Omega \subseteq \mathbb{R}^2$ with smooth boundary. While very similar
to chemotaxis models from biology, this system is in fact inspired by recent
modeling approaches in criminology to analyze the formation of crime hot spots
in cities. The key addition here in comparison to similar models is the
logistic source term.
  The central complication this system then presents us with, apart from us
allowing for arbitrary $\chi > 0$, is the nonlinear growth term $uv$ in the
second equation as it makes obtaining a priori information for $v$ rather
difficult. Fortunately, it is somewhat tempered by its negative counterpart and
the logistic source term in the first equation. It is this interplay that still
gives us enough access to a priori information to achieve the main result of
this paper, namely the construction of certain generalized solutions to
($\star$).
  To illustrate how close the interaction of the $uv$ term in the second
equation and the $-\mu u^2$ term in the first equation is to granting us
classical global solvability, we further give a short argument showing that
strengthening the $-\mu u^2$ term to $-\mu u^{2+\gamma}$ with $\gamma > 0$ in
the first equation directly leads to global classical solutions.
