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

## TL;DR

This paper investigates the uniqueness of polymorphic equilibria in population genetics models with selection and migration, providing counterexamples that demonstrate multiple equilibria can exist under certain conditions.

## Contribution

It offers the first negative answer to a longstanding conjecture about the uniqueness of clines, using topological methods and numerical evidence to show multiple solutions can occur.

## Key findings

- Counterexamples with multiple equilibria are constructed.
- Numerical evidence supports the existence of multiple solutions.
- The conjecture about unique polymorphic equilibria is disproven.

## Abstract

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.