High multiplicity and global structure of coexistence states in a predator-prey model with saturation

The paper’s Theorem 4.1 is supported by a standard IFT argument for the ε=0 non-degenerate coexistence states and a priori bounds plus global-continuation reasoning (citing [23]) to obtain at least one additional solution, yielding ≥2n+2 solutions for λ∈A_n when ε>0 is small. These steps and the required non-degeneracy set A and its sub-sets A_n are developed rigorously in the text, with the operator G, its linearization, and the eigenvalue structure made explicit. By contrast, the model’s phase-2 proof of the extra solution via the singular rescaling W=εw contains two substantial gaps: (i) the direct-method functional J(W) is not coercive on H^1 without a nonnegativity constraint (and constant supersolutions fail if a(x) vanishes on a set of positive measure), so existence of a positive W_* is not established as written; and (ii) the asserted non-degeneracy of W_* is deduced from an integral identity that does not imply φ≡0. Hence the model’s additional-branch construction is unproven. The IFT-based part matches the paper, but the crucial +1 solution is unsupported in the model’s argument.