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.
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Figure 1.
Qualitative bifurcation digram of the solutions to (1.1) in the
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