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

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 w_i are sign-changing in the first variable, and the nonlinearities f_i:[0,1]→[0,+∞[ satisfy f_i(0)=f_i(1)=0, f_i(s)>0 for all s∈]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 λ1 and λ2 are sufficiently large.