Steady State Classification of Allee Effect System

The paper proves two key ingredients: (i) any steady state has at most three distinct coordinate values via the cubic g(z) reduction (Theorem 2), and (ii) a border polynomial for the full n-dimensional steady-state system factors as the product of border polynomials of the 2- and 3-block reduced systems (Theorem 3, formula (26)). Both are explicitly stated in the paper and justified by reducing to subsystems G1 and G2 and then multiplying their border polynomials, ensuring constancy of solution counts off the zero set of that product . The candidate solution follows the same structure: it proves the “≤3 values” lemma using the same g(·) argument, reduces to 2- and 3-block prototypes, and then constructs the full-system border polynomial as the product of all bp1 and bp2 factors. It adds an explicit inclusion–exclusion step to remove degeneracies (y=z and related equalities) when counting prototypes, which the paper handles more implicitly via the “distinct numbers” phrasing in Theorem 4 and a separate +3 term for uniform solutions in the final count formula (29) . Minor notational imprecision in the paper (when c1, c2 are introduced as “the numbers of positive solutions” without restating the distinctness constraint) is clarified by the model’s explicit identities. Overall, the proofs are the same in substance: cubic fiber argument → block reductions → product border polynomial, with the model giving slightly more explicit combinatorial bookkeeping. Hence, both are correct and substantially the same proof.