Stability and Bifurcation in a Discrete Phytoplankton–Zooplankton Model with Holling-Type Toxic Effects

The paper proves a Neimark–Sacker bifurcation at the positive fixed point Ẽ by: (i) reducing the Jacobian at Ẽ to F(λ)=λ^2−p(ũ)λ+q(ũ) with p(ũ)=2−ũ and q(ũ) as in (2.7) (so q(ũ)=1 defines θ0), (ii) verifying nonresonance since a(0)=p(ũ)∈(1,2), b(0)=1, hence no m=1..4 resonances, and (iii) establishing transversality via d|λ|/dθ*|_{0}<0 (eq. (3.5)) and then computing the first Lyapunov coefficient L from third-order normal-form coefficients L20,L11,L02,L21; if L<0 (resp. >0) an attracting (resp. repelling) invariant closed curve arises for θ<θ0 (resp. θ>θ0) (Theorem 3). These steps appear in Section 3 and equations (2.6)–(2.7), (3.3)–(3.6), and (3.13)–(3.14) of the paper . The candidate solution follows the same roadmap: defines θ0 by q(ũ)=1, shows nonresonance using p∈(1,2), proves transversality via ∂q/∂θ<0, and uses the standard L-formula for maps to conclude super/subcritical cases. The only small discrepancy is that the model measures transversality using ∂q/∂θ at fixed ũ, whereas the paper’s precise crossing speed derives from det(J(θ)) and includes an extra positive factor (1+h+cũ^h), but both give the same sign; thus the conclusions coincide.