  We study the second-order boundary value problem \begin{equation*}
\begin{cases} \, -u''=a_{\lambda,\mu}(t) \, u^{2}(1-u), & t\in(0,1), \\ \,
u'(0)=0, \quad u'(1)=0, \end{cases} \end{equation*} where $a_{\lambda,\mu}$ is
a step-wise indefinite weight function, precisely
$a_{\lambda,\mu}\equiv\lambda$ in $[0,\sigma]\cup[1-\sigma,1]$ and
$a_{\lambda,\mu}\equiv-\mu$ in $(\sigma,1-\sigma)$, for some
$\sigma\in\left(0,\frac{1}{2}\right)$, with $\lambda$ and $\mu$ positive real
parameters. We investigate the topological structure of the set of positive
solutions which lie in $(0,1)$ as $\lambda$ and $\mu$ vary. Depending on
$\lambda$ and based on a phase-plane analysis and on time-mapping estimates,
our findings lead to three different (from the topological point of view)
global bifurcation diagrams of the solutions in terms of the parameter $\mu$.
Finally, for the first time in the literature, a qualitative bifurcation
diagram concerning the number of solutions in the $(\lambda,\mu)$-plane is
depicted. The analyzed Neumann problem has an application in the analysis of
stationary solutions to reaction-diffusion equations in population genetics
driven by migration and selection.
