Evolutionary Dynamics on a Regular Networked Structured and Unstructured Multi-population

Existence: Both the paper and the model correctly derive the consensus equilibria by imposing (γ + r d ξ)ζ = ξ(α + σ d μ) and (γ + r d μ)ζ = μ(α + σ d ξ), yielding either ξ = μ (symmetric case) or ζ = α/(r d) . However, in Case 2 the paper’s printed quadratic and closed form appear dimensionally inconsistent (using α r − d instead of r d − α and 4αγ/(rσ) without matching linear terms), and do not reproduce the paper’s own numerical example; the model’s formula does, and matches the equilibrium found in the paper’s simulation (γ = 0.2, α = 0.4, r = 0.3, σ = 0.4, d = 3 ⇒ ξ = 2/9, μ = 1/3) . Stability: The paper’s Theorem 3 claims that two lower bounds on σ suffice for local exponential stability , but its Jacobian used for Gershgorin omits the state factors (−σξ, −σμ) on the off-neighbour blocks (it prints −σA instead), which changes the stability margins materially . The model block-diagonalizes the exact Jacobian via the adjacency eigenmodes and shows that in the symmetric case the consensus is stable iff α > r d ζ (independent of σ), and otherwise provides a correct uniform Gershgorin-based sufficient condition α − r d ζ > σ d |ξ − μ|. A concrete symmetric counterexample has α < r d ζ while the paper’s σ-inequalities hold trivially, and the linearized eigenvalue at λ = d is positive; hence the paper’s Theorem 3 is false as stated.