STABILITY, BIFURCATION, AND CHAOS CONTROL IN A DISCRETE-TIME PHYTOPLANKTON-ZOOPLANKTON MODEL WITH HOLLING TYPE II AND TYPE III FUNCTIONAL RESPONSES

The paper’s existence proof for a Neimark–Sacker (NS) bifurcation uses a “frozen-equilibrium” Jacobian and concludes the transversality derivative d|λ|/dγ is strictly positive, hence fixing the direction (γ>γ0 if L<0). That derivative is not taken along the branch of positive equilibria E(γ) and can have either sign (in particular, it can be negative for h=2). The candidate solution correctly differentiates det J along E(γ), obtains an explicit formula, proves Δ>0 for h=1, and shows Δ may change sign for h=2, which reverses the direction. Aside from this sign error and missing hypotheses (e.g., L≠0), the paper’s setup, Jacobian, characteristic polynomial, and Lyapunov-coefficient formula match standard theory.