A negative answer to a conjecture arising in the study of selection–migration models in population genetics

We deal with the study of the evolution of the allelic frequencies, at a single locus, for a population distributed continuously over a bounded habitat. We consider evolution which occurs under the joint action of selection and arbitrary migration, that is independent of genotype, in absence of mutation and random drift. The focus is on a conjecture, that was raised up in literature of population genetics, about the possible uniqueness of polymorphic equilibria, which are known as clines, under particular circumstances. We study the number of these equilibria, making use of topological tools, and we give a negative answer to that question by means of two examples. Indeed, we provide numerical evidence of multiplicity of positive solutions for two different Neumann problems satisfying the requests of the conjecture.