A Novel Nonlinear Fertility Catastrophe Model Based on Thom’s Differential Equations of Morphogenesis

The paper defines −V(ξ;α,β)=αξ+½βξ²−¼ξ⁴ and asserts that equilibria solve α+βξ−ξ³=0, which is standard. But it then claims the “Cardan discriminant” is δ=27α−4β³ and uses δ≤0 vs δ>0 to separate unimodal/bimodal regimes. This is mathematically incorrect: for the depressed cubic t³+pt+q with p=−β, q=−α, the classical discriminant is Δ=−4p³−27q²=4β³−27α². The model solution gives the correct partition (three real equilibria iff 4β³−27α²>0), including the essential α↦−α symmetry. The paper’s linear dependence on α contradicts this symmetry and misclassifies regions; see the paper’s statements around “α+βξ−ξ³=0 … Cardan discriminant δ=27α−4β³ … unimodal (δ≤0) vs bimodal (δ>0)” .