Permanence for continuous-time competitive Kolmogorov systems via the carrying simplex

The paper’s Theorem 4.2 establishes a permanence/impermanence criterion on the carrying simplex using the geometric-mean Lyapunov function V and g(x)=∑ν_i f_i(x), proved via general repellor/attractor theorems and a compact neighborhood construction M=γ+(Oε(Σ)) (and S=M∩∂Rn_+) . The candidate solution reaches the same conclusions with a direct time-window argument producing uniform multiplicative increases/decreases of V near Λ(∂Σ). While largely correct, the model implicitly treats Λ(∂Σ) as compact and asserts uniform attraction without proof; both issues can be fixed by adopting the paper’s M,S framework or standard limit-set constructions (as used in the proof of Theorem 4.2 and Theorem 4.1) .