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

<p style='text-indent:20px;'>We study the second-order boundary value problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ a_{\lambda,\mu} $\end{document}</tex-math></inline-formula> is a step-wise indefinite weight function, precisely id="M2">\begin{document}$ a_{\lambda,\mu}\equiv\lambda in id="M3">\begin{document}$ [0,\sigma]\cup[1-\sigma,1] and id="M4">\begin{document}$ a_{\lambda,\mu}\equiv-\mu id="M5">\begin{document}$ (\sigma,1-\sigma) $\end{document}</tex-math></inline-formula>, for some id="M6">\begin{document}$ \sigma\in\left(0,\frac{1}{2}\right) with id="M7">\begin{document}$ \lambda id="M8">\begin{document}$ \mu positive real parameters. We investigate topological structure of set solutions which lie id="M9">\begin{document}$ (0,1) as id="M10">\begin{document}$ id="M11">\begin{document}$ vary. Depending on id="M12">\begin{document}$ based phase-plane analysis time-mapping estimates, our findings lead to three different (from point view) global bifurcation diagrams terms parameter id="M13">\begin{document}$ $\end{document}</tex-math></inline-formula>. Finally, first time literature, qualitative diagram concerning number id="M14">\begin{document}$ (\lambda,\mu) $\end{document}</tex-math></inline-formula>-plane depicted. The analyzed Neumann problem has an application stationary reaction-diffusion equations population genetics driven by migration selection.</p>