GLOBAL DYNAMICS OF PLANAR DISCRETE TYPE-K COMPETITIVE SYSTEMS

The candidate solution reproduces the main statements of Hou’s Theorem 3.1—axis saddles, existence of interior fixed point(s), and the monotone-curve structure of the global attractor—almost verbatim. However, it omits a crucial hypothesis used throughout the paper: the dissipativity/eventual boundedness condition ensuring that every orbit has ω(x) ⊂ [0,r], which is explicitly assumed in Theorem 3.1 (existence of r ≫ 0 with ω(x) ⊂ [0,r] for all x, T([0,r]) ⊂ [0,r), and ρ(M)<1) . The model nevertheless claims global conclusions for “every interior orbit,” and even asserts ω(x) ⊂ A for all x solely from T([0,r]) ⊂ [0,r), which is not justified. The paper’s proof supplies additional machinery—type-K retrotone/weakly retrotone maps (Proposition 2) and the preimage geometry of monotone line segments (Lemma 5.1)—culminating in Proposition 5 and the full proof of Theorem 3.1(c), including ΣH = W^u(Q1) ∪ {Q1}, ΣV = W^u(Q2) ∪ {Q2}, and the monotone-curve structure of Σ0, none of which are rigorously established in the model solution . Conclusion: the paper is correct; the model’s proof is incomplete/incorrect due to missing assumptions and reliance on heuristic steps that the paper proves via retrotone arguments.