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

We study the second-order boundary value problem - u'' = a_{λ,μ}(t) u^2(1-u), t ∈ (0,1), u'(0) = 0, u'(1) = 0, where a_{λ,μ} is a step-wise indefinite weight function, precisely a_{λ,μ}=λ in [0,σ] ∪ [1-σ,1] and a_{λ,μ}=μ in (σ,1-σ), for some σ∈(0,1/2), 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-diffiusion equations in population genetics driven by migration and selection.