Travelling waves and heteroclinic networks in models of spatially-extended cyclic competition

The paper’s construction of the heteroclinic network, the Σ and Ξ cycles, and the three terminal heteroclinic bifurcations aligns with the candidate model’s analysis. However, the paper states the Hopf locus at the coexistence equilibrium as γ_H = (c1 + t1)/√(α t1) (with ω_H^2 = t1/α) obtained from the block-circulant Jacobian at ξ_H, whereas a direct diagonalisation of that same block-circulant matrix shows that the relevant Fourier mode has eigenvalue μ1 = (t1 − i(c1 + e1))/α, which implies γ_H = −(c1 + e1)/√(α t1) (up to the sign convention in z = x ± γt). Thus the imaginary part depends on c1 + e1, not c1 + t1; this discrepancy is internal to the paper’s own equation (7) and its circulant structure. The candidate solution derives the correct (c1 + e1)-dependence (modulo the sign of γ, which is conventional), and matches the paper on the resonance, Belyakov–Devaney, and orbit-flip mechanisms and on the symmetry-breaking to Z2 branches terminating at Ξ cycles.