DYNAMICS AND PATTERN FORMATION IN A DIFFUSIVE BEDDINGTON-DEANGELIS PREDATOR-PREY MODEL WITH FEAR EFFECT

The paper proves existence of at least one nonconstant positive steady state for the Neumann problem (2.3) using Leray–Schauder degree and a spectral window defined via H(d1,d2;μ)=d1 d2 μ^2+(Q d1−M d2) μ+PN−MQ, together with a nonexistence anchor when both diffusions are large; see the stationary system (2.3) and the unique positive constant equilibrium (Lemma 3.1) , the determinant/index setup and window limits (4.7)–(4.8) , the explicit index formula and existence theorem (Theorem 4.9) with homotopy (4.11)–(4.14) , and the nonexistence result for large d1≤d2 (Theorem 4.6) providing the anchor . The candidate solution reproduces the same framework: compact resolvent formulation, uniform a priori bounds, mode-by-mode index computation via H, a large-diffusion nonexistence estimate, and a degree homotopy, leading to the same conclusion. The only notable gap in the candidate write-up is the missing explicit justification that rules out fixed points on the boundary of the degree set (uniform positive lower bounds), which the paper secures via a Harnack-based estimate and careful parameter choices along the homotopy; this is a minor completeness issue rather than a contradiction. Overall, the arguments agree in substance and conclusion.