Global Dynamics and Time-Optimal Control Studies for Additional Food provided Holling Type-III Mutually Interfering Prey-Predator Systems with Applications to Pest Management

The paper explicitly derives the Jacobian at E1=(γ,0) with eigenvalues −1 and [(δ−m)γ^2+δξ−m(1+αξ)]/(1+γ^2+αξ), giving the same stability threshold as the model’s λ2(ξ)=δ(γ^2+ξ)/(1+γ^2+αξ)−m and the critical curve δξ−m(1+αξ)=−(δ−m)γ^2 (equivalently, ξ=[m−(δ−m)γ^2]/(δ−mα)) . The paper also gives the interior-equilibrium cubic and the predator nullcline y*=((δ−m)x^2+δξ−m(1+αξ))/(mϵ), matching the model’s step (1) . However, the paper merely asserts a “pseudo transcritical bifurcation” between E1 and E* along the same curve without proving that x*=γ and y*=0 at threshold or that the prey and predator nullclines cross transversely there; these are only observed in figures and summarized in Table 2 . The candidate solution closes these gaps: it shows x=γ is a root of the interior cubic at ξ=ξc and y*(γ,ξc)=0, establishes transversal intersection of f=0 and F=0 (hence a unique C^1 interior branch) and performs a center-manifold reduction yielding the logistic normal form and a rigorous stability exchange at ξc. The only minor correction is that the 2×2 determinant used for transversality need only be nonzero; its sign can depend on 1+(α−1)ξ, so it is not uniformly positive as stated. With this clarification, the model’s argument is correct and more rigorous than the paper’s qualitative “pseudo transcritical” claim.