A class of solutions of the asymmetric May-Leonard model

The paper proves that, under the algebraic initial constraints x1(0)+a12 x2(0)+a13 x3(0)=a21 x1(0)+x2(0)+a23 x3(0)=a31 x1(0)+a32 x2(0)+x3(0)=:z, the asymmetric May–Leonard system admits the explicit solution xn(t)=xn(0)/{e^{−η t} + (z/η)[1−e^{−η t}]} (its Proposition, eqs. (8)–(10); system (2); reduction (4a)–(5) and the homogeneous-polynomial argument leading to (6)–(7)) . The candidate solution reaches the same formula via the standard scaling-ansatz xn(t)=xn(0) y(t), yielding a one-dimensional logistic ODE y'=y(η−z y). This is the same scaling mechanism the paper invokes (via homogeneity) and transforms back with (4a). The model’s derivation also covers the η=0 limit explicitly, which the paper leaves implicit. Overall, both are correct and rely on essentially the same scaling invariance.