ON EXISTENCE AND UNIQUENESS OF A MODIFIED CARRYING SIMPLEX FOR DISCRETE KOLMOGOROV SYSTEMS

The paper’s Theorem 2.4 establishes, under (i) axis fixed points, (ii) nonpositive cross-partials with strict decrease in the own variable, and (iii) a spectral-radius condition on M or M̃, that 0 is a repellor, there is a unique modified carrying simplex Σ, and Σ coincides with the boundary of the basin of repulsion of 0 inside [0,r]; it also describes the α-limit behavior below Σ. These statements and their proof route (via the weaker “weakly retrotone” Theorem 2.3, the identity A0 = B(0), and the construction of Σ as the boundary) are explicit in the PDF and are further complemented by the homeomorphism of Σ to Δ via radial projection and its global-attractor property in [0,r] (and even C under an additional tightness assumption) . The candidate solution reproduces the core logic with a different emphasis: it derives 0’s repulsion from axis monotonicity; uses the M-matrix factorization DT=diag(f)(I−M) to get local diffeomorphism and monotonicity of the projectivized dynamics; appeals to strict sublinearity to invoke a carrying-simplex theorem; and identifies Σ with the boundary of B(0) while proving the α-limit statement. Two caveats: (a) the solution asserts “unordered” for Σ and “strict retrotone,” whereas the paper’s modified carrying simplex allows ordered points in the sense of < and only requires weak retrotone under its weaker Jacobian hypotheses (with an explicit example) ; and (b) the paper repeatedly typesets “Σ = B(0) \ ({0} ∪ B(0)),” which is clearly a typographical omission of the closure—in context the intended identity is Σ = cl B(0) \ ({0} ∪ B(0)), as used in the construction via A0 = B(0) and the radial map . With these clarifications, both are essentially correct and reach the same conclusions by different routes.