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

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.