Planar Kolmogorov systems with infinitely many singular points at infinity

The paper rigorously proves that the family ẏ = y(b0 + b1yz + b2y + b3z), ż = z(c0 + b1yz + b2y + b3z) has exactly 13 topologically distinct global phase portraits on the Poincaré disc: it reduces parameters via H (in particular c0 − b0 ≠ 0), analyzes all local portraits at finite and infinite singularities (including semi-hyperbolic and nilpotent cases at the four special points at infinity), constructs 36 global candidates G1–G36, and then merges them into 13 equivalence classes R1–R13 using separatrix-based invariants and symmetries (Theorem 1.1, Lemma 4.1, Theorems 4.2–4.4, Table 7, Table 8, Theorem 5.1) . The candidate’s solution reaches the same classification and uses a clean, different argument to exclude periodic orbits (monotonicity of ln(y/z)), whereas the paper uses an axis-invariance argument for the absence of limit cycles . Minor issues in the model outline: it (i) does not state the key reduction H (esp. c0 − b0 ≠ 0, b1 ≠ 0) used by the paper , and (ii) treats the four special points at infinity as saddle-nodes only, omitting the nilpotent alternatives that the paper handles explicitly (Theorems 4.2 and 4.4) . Despite these omissions, the model’s classification count and separatrix-based distinctions agree with the paper’s results.