A Mathematical Model of the Cell Cycle: Exploring the Impact of Zingerone on Cancer Cell Proliferation

The paper correctly identifies that the rectangle Ω=[0,1]^2 is not forward invariant for the smooth Hill system because at x=1 the outward normal condition fails for small y (they restrict the domain using the trajectory through (1,ŷ) to ensure invariance) . It numerically infers a repelling equilibrium and then invokes Poincaré–Bendixson to claim a limit cycle for the smooth model, without a rigorous equilibrium analysis . For the piecewise (Heaviside) system, the paper provides the four regional dynamics and parameter constraints, including the return-map coordinate x̂ and the closure condition K2 ≤ x̂ ≤ 1, but stops short of a formal limit-cycle existence proof via Poincaré–Bendixson in the piecewise-smooth setting . It also presents only a numerical fit for T(α1)=A/α1+B rather than an analytic derivation . By contrast, the candidate solution gives: (A) a precise face-by-face tangent-condition check on ∂Ω confirming non-invariance of Ω; (B) a clean, parameterized, no-sliding transverse-crossing argument that builds a trapping region and invokes Poincaré–Bendixson to produce a periodic orbit; (C) explicit flight times across the four regions yielding T=A/α1+B; and (D) a closed-form cell-viability CV(t,z) that matches the paper’s expression (Eq. 20) under c(t,z)=α1e^{-ρzt} and r=ln2/τ−μ . Minor notational slips (using f where g is meant on x=K2) do not affect correctness; the missing step that g<0 on the R11 side at x=K2 follows from the paper’s own parameter condition (α1+α3)/(α3+β1)<K2. Overall, the model provides the rigorous derivations the paper omits.