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

We study the second-order boundary value problem \begin{equation*} \begin{cases} \, -u''=a_{λ,μ}(t) \, u^{2}(1-u), & t\in(0,1), \\ \, u'(0)=0, \quad u'(1)=0, \end{cases} \end{equation*} where $a_{λ,μ}$ is a step-wise indefinite weight function, precisely $a_{λ,μ}\equivλ$ in $[0,σ]\cup[1-σ,1]$ and $a_{λ,μ}\equiv-μ$ in $(σ,1-σ)$, for some $σ\in\left(0,\frac{1}{2}\right)$, with $λ$ and $μ$ positive real parameters. We investigate the topological structure of the set of positive solutions which lie in $(0,1)$ as $λ$ and $μ$ vary. Depending on $λ$ 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 $μ$. Finally, for the first time in the literature, a qualitative bifurcation diagram concerning the number of solutions in the $(λ,μ)$-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.