MODELLING AND ANALYSIS OF A MODIFIED MAY–HOLLING–TANNER PREDATOR-PREY MODEL WITH ALLEE EFFECT IN THE PREY AND AN ALTERNATIVE FOOD SOURCE FOR THE PREDATOR

The paper’s Appendix B states, for the combined Allee + alternative-food system (their system (13)), that det J(u,u+C) has the sign of Q−g′(u) and tr J(u,u+C) has the sign of u g′(u)/(u+A)−S, then proves: (i) when Δ>0 the lower interior equilibrium P1 is a saddle (det<0), and the upper interior equilibrium P2 has det>0 and stability flips at S* = D / [4(1+A+M+G+√Δ)] with D as given; (ii) when Δ=0, the double-root equilibrium (E,E+C) is a saddle-node with stability threshold H / [4(1 + A + M + G)] (Lemmas B.2–B.4, and equations (13), (22)–(23) of the Appendix) . The candidate solution reaches precisely the same conclusions and thresholds. It uses a slightly different argument for the sign of Q−g′(ui) via the derivative of the cubic S3 at its roots to show P1 is a saddle and P2 has det>0, and then re-expresses the trace threshold S* in closed form, also matching the paper’s D and H formulas. Aside from a minor notational slip in the paper text (“Evaluating Q−g′(u) at u2 gives: Q−g′(u1)=…”) and a sign-expression that is easy to reconcile with the final inequality claimed, the two proofs agree on all substantive points .