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 p1′′+λ1w1(x,p2)f1(p1)=0,in Ω,p2′′+λ2w2(x,p1)f2(p2)=0,in Ω,p1′=p2′=0,on ∂Ω,where the coupling-weights wi are sign-changing in the first variable, and the nonlinearities fi:[0,1]→[0,+∞[ satisfy fi(0)=fi(1)=0, fi(s)>0 for all s∈]0,1[, and a superlinear growth condition at zero. Using a topological degree approach, we prove existence of 2N positive fully nontrivial solutions when the real positive parameters λ1 and λ2 are sufficiently large.