Front propagation in hybrid reaction-diffusion epidemic models with spatial heterogeneity. Part I: Spreading speed and asymptotic behavior

Quentin Griette, Hiroshi Matano

correctmedium confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper’s Theorem 2.6 states the right/left spreading speeds for front-like data and gives the variational formulas c*_R = min_{λ>0} k(λ)/λ and c*_L = min_{λ<0} k(λ)/(-λ), proved via an auxiliary cooperative truncation and a monotone-operator framework linked to Weinberger’s theory, together with properties of the λ-periodic principal eigenvalue k(λ) (analyticity/convexity and bounds) . The candidate model solution reaches the same result but by a different construction: a weighted semigroup upper bound and a refined subsolution with a quadratic correction in e^{-2λ(x-ct)}, plus a cooperative truncation. The logical steps are consistent with the paper’s hypotheses, so both are correct though they use different proof strategies.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The paper establishes spreading speeds for a biologically motivated two-species KPP-type system with spatially periodic coefficients, identifying the right/left minimal speeds via a λ-periodic principal eigenvalue framework and relating them to Dirichlet eigenvalues on large boxes. The mix of spectral, comparison, and abstract monotone-dynamics tools is well-chosen and technically sound. Some arguments (e.g., convexity of k(λ) and parts of the comparison in noncooperative regimes) are succinct and could use minor clarification or pointers, but overall the results are solid, novel for this hybrid (cooperative near zero, competitive at large values) setting, and of interest to the PDE and mathematical biology communities.