Multiplicity of clines for systems of indefinite differential equations arising from a multilocus population genetics model

TL;DR

This paper studies the existence of multiple coexistence states in a complex population genetics model using nonlinear differential equations, revealing conditions under which many stable genetic configurations can occur.

Contribution

It introduces a topological degree method to prove the existence of multiple positive solutions in a multilocus population genetics model with sign-changing coupling weights.

Findings

Existence of 2^N positive solutions for large parameters

Conditions for coexistence states in heterogeneous habitats

Application of topological degree to nonlinear differential systems

Abstract

We investigate sufficient conditions for the presence of coexistence states for different genotypes in a diploid diallelic population with dominance distributed on a heterogeneous habitat, considering also the interaction between genes at multiple loci. In mathematical terms, this corresponds to the study of the Neumann boundary value problem \begin{equation*} \begin{cases} \, p_{1}''+\lambda_{1} w_{1}(x,p_{2}) f_{1}(p_{1}) = 0, &\text{in $\Omega$,} \, p_{2}''+\lambda_{2} w_{2}(x,p_{1}) f_{2}(p_{2}) = 0, &\text{in $\Omega$,} \, p_{1}'=p_{2}'=0, &\text{on $\partial\Omega$,} \end{cases} \end{equation*} where the coupling-weights $w_{i}$ are sign-changing in the first variable, and the nonlinearities $f_{i}\colon\mathopen{[}0,1\mathclose{]}\to\mathopen{[}0,+\infty\mathclose{[}$ satisfy $f_{i}(0)=f_{i}(1)=0$, $f_{i}(s)>0$ for all $s\in\mathopen{]}0,1\mathclose{[}$, and a superlinear growth condition at zero. Using a topological degree approach, we prove existence of $2^{N}$ positive fully nontrivial solutions when the real positive parameters $\lambda_{1}$ and $\lambda_{2}$ are sufficiently large.