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

Guglielmo Feltrin; Paolo Gidoni

doi:10.1016/j.nonrwa.2020.103108

paper

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

arXiv:1909.05737 · doi:10.1016/j.nonrwa.2020.103108

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}''+λ_{1} w_{1}(x,p_{2}) f_{1}(p_{1}) = 0, &\text{in $Ω$ ,} \, p_{2}''+λ_{2} w_{2}(x,p_{1}) f_{2}(p_{2}) = 0, &\text{in $Ω$ ,} \, p_{1}'=p_{2}'=0, &\text{on $\partial Ω$ ,} \end{cases} \end{equation*} where the coupling-weights $w_{i}$ are sign-changing in the first variable, and the nonlinearities $f_{i} : [0, 1] \to [0, + \infty [$ satisfy $f_{i} (0) = f_{i} (1) = 0$ , $f_{i} (s) > 0$ for all $s \in] 0, 1 [$ , 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 $λ_{1}$ and $λ_{2}$ are sufficiently large.

23 pages