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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 $a_{\lambda,\mu}$ isa 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 realparameters. We investigate the topological structure of the set of positivesolutions 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 bifurcationdiagram concerning the number of solutions in the $(\lambda,\mu)$-plane isdepicted. The analyzed Neumann problem has an application in the analysis ofstationary solutions to reaction-diffusion equations in population geneticsdriven by migration and selection.

On the number of positive solutions to an indefinite parameter-dependent Neumann problem