A NOTE ON THE PERIODIC ORBITS OF WOLBACHIA SPREAD DYNAMICS IN MOSQUITO POPULATIONS IN PERIODIC ENVIRONMENTS

The paper proves (via unimodality and negative Schwarzian) that under Conjecture 1 the T-fold composition has at most two non-zero fixed points in (0,1], with T=2 handled separately and T≥3 stated as Theorem 12. The candidate’s solution proves the same bound by a different, clean argument: the change of variables y=1/x turns each step into a strictly increasing, strictly convex map on (1,∞); their composition remains strictly increasing and strictly convex; hence G(y)−y is strictly convex and has at most two zeros, bijectively yielding at most two fixed points of F. Aside from a minor inequality slip when justifying µ*_n≤1 (which is easily corrected and not essential to the core convexity argument), the model’s proof is sound and matches the paper’s main claim.