Dynamics of a linearly-perturbed May–Leonard competition model

The paper derives the circulant Jacobian at the equal–population equilibrium ec, its spectrum λ1 = −1 and λ2,3 with real part [(α+β−2)/(2(1+α+β)) − 3µ] and imaginary part ±[√3(β−α)/(2(1+α+β))], obtains the Hopf line βc(α, µ) = −α + (6µ+2)/(1−6µ), and computes the first Lyapunov coefficient l1(0) = −(α+β−2)(α+β+4)/(6ω) with ω = √3(β−α)/(2(1+α+β)), concluding the Hopf is supercritical and noting a degeneracy at α=β where two zero eigenvalues occur. These are stated explicitly in the paper’s equations (7)–(8), (10)–(11), and (24)–(26) . The candidate solution reproduces the same Jacobian, spectrum, Hopf locus, and the same l1(0), using a Fourier/circulant decomposition and normal-form calculation. The only minor discrepancy is a notational choice for the left eigenvector p versus the pairing used in Kuznetsov’s formula, but this does not affect the computed sign of l1(0). Overall, both proofs coincide in substance.