Stability and bifurcation in a reaction-diffusion-advection predator-prey model

Yihuan Sun, Shanshan Chen

correcthigh confidence

Category: math.DS
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper rigorously establishes existence/uniqueness of the positive steady state for small λ, develops an eigenvalue reduction, and proves stability/Hopf alternatives depending on the sign of T(α), culminating in Theorem 3.10. The candidate solution reaches the same high-level claims but its proof contains critical internal inconsistencies: it incorrectly asserts v(λ,l) → 0 as λ → 0 (the paper shows v → q0l > 0), mis-evaluates M11 at λ = 0 by omitting the q0l-term, and contradicts itself about M21’s behavior at λ → 0. These errors undermine the model’s derivations, even though its final conclusions match the paper.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The work provides a careful, rigorous treatment of stability and Hopf bifurcation for a heterogeneous predator–prey PDE with advection. The analysis combines an existence result for positive steady states with a well-designed reduction of the eigenvalue problem and a clear transversality computation. The results are significant for the modeling community. Minor clarifications would further strengthen readability and rigor in places where operator-theoretic assertions are condensed.