Global Dynamical Behaviours and Periodicity of a Certain Quadratic-Rational Difference Equation with Delay

The paper’s global stability theorem (Theorem 11) asserts that the unique positive equilibrium is globally asymptotically stable for 0<p<1/2, which is true. However, the provided proof contains a faulty algebraic step when converting the limsup/liminf bounds S ≤ 1 + pS/I^2 and I ≥ 1 + pI/S^2 into the product inequalities; the fractions are swapped, yielding an incorrect chain and the concluding factor (S−I)(1−p(1/S+1/I))≤0 is not justified by the preceding lines. See the paper’s own statement and derivation around Theorem 11 for details . By contrast, the candidate model’s solution proves boundedness and then uses a monotone squeezing via H=F∘F with a uniqueness-of-fixed-point argument, yielding a complete and correct proof of global convergence for 0<p<1/2.