Bifurcation of stable equilibria under nonlinear flux boundary condition with null average weight

The bifurcation and stability structures of equilibria of a parabolic problem motivated by a population genetics model are completely described. In a previous work the authors considered weights having nonzero average and drew the bifurcation and stability diagrams of equilibria. Herein that study is completed by considering weights with null average over the boundary of the domain and other approach is employed to overcome a degeneration which occurs in such case. For each value of the parameter we prove existence of an unique nonconstant equilibrium solution. It belongs to a globally parametrized bifurcation curve constructed using a Lyapunov–Schmidt type reduction combined with the Morse lemma. That solution is showed to be a global minimizer of the corresponding energy functional which attracts all nontrivial semi-orbits. The behavior of its trace, when the parameter is large, is established and complete diagrams containing all bifurcation and stability information are also provided.