Hunting Co-operation in the Middle Predator in Three Species Food Chain Model

The paper’s Theorem 2.1 claims linear stability of the coexistence equilibrium u[3] under the assumptions J11−J22<0 and J11J22>J23J32, based on a Jacobian with J13=J31=0 and J33≈0 and the characteristic polynomial coefficients A1=−(J11+J22), A2=J11J22−J23J32−J21J12, A3=J11J23J32. However, the proof does not actually establish A3>0 or the decisive Routh–Hurwitz condition A1A2>A3 and even treats the special case A3=0, which is nongeneric at coexistence; moreover, J33 is exactly 0 at coexistence (not merely “almost”), and the paper mixes two model forms ((1) vs (2)), introducing inconsistencies in G1,G2 with and without hunting cooperation H=1+αu2 . The candidate model solution corrects part of this (derives A1,A2,A3 correctly and proves J11<0 using the original u1-equation) and requires the proper trace condition J11+J22<0 for A1>0, but its proof of the key inequality A1A2>A3 is not fully justified without additional quantitative bounds and can fail in admissible sign patterns. Hence both are incomplete.