EVOLUTIONARY BEHAVIOR IN A TWO-LOCUS SYSTEM

The paper defines the same operator W on S3, identifies Fix(W) = {yu − xv = 0}, and proves Theorem 1: either the initial point is fixed when (x0 + y0)(u0 + v0) = 0, or else the orbit converges to the explicit fixed point whose coordinates are determined by A(x0,u0) = (b x0 + a u0)/((u0 + v0)a + (x0 + y0)b) . Their proof reduces W to a 2×2 linear map on each invariant slice Xα = {x+y=α, u+v=1−α}, computes eigenvalues λ1=1 and λ2=1−(1−α)a−αb, and shows convergence using matrix powers/Cayley–Hamilton, yielding the same limit formula (2.5) . The model’s solution uses a different, direct invariant approach: it exhibits integrals X=x+y, U=u+v, I=bx+au, sets D=bX+aU, and proves the exact one-step identity T'=(1−D)T for T=yu−xv, which gives geometric decay and the same limit via p=I/D. Assumptions match the paper (a,b∈[0,1], a+b≠0). No logical gaps found in either argument.