2009.03221
A Run-and-Tumble Model with Autochemotaxis
Nicholas J. Russell, Louis F. Rossi
correcthigh confidence
- Category
- math.DS
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:55 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper linearizes (10a)–(10c), derives the cubic dispersion relation (17), and proves a two-sided sufficient condition (19) for instability by analyzing the curvature R''(0) of the spectral abscissa near k=0. The candidate solution derives the same dispersion relation and then gives a simpler sufficiency test: if d3 > −(c d2)/δ, one can choose a small nonzero k with p_k(0) < 0, forcing a positive real root by continuity and hence instability on a suitably chosen periodic domain. Thus both establish linear instability; the model’s argument is shorter and shows the paper’s upper bound is not needed for sufficiency, but the paper never claimed necessity. Proofs are different; both are correct.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The dispersion relation and linear instability analysis are correct and well presented. Proposition 2 provides a clear sufficient instability window; the candidate solution shows that the lower bound alone already implies instability, which is a useful sharpening. A small typographical correction to the Taylor expansion and a remark about sufficiency would improve clarity. The work is solidly executed and relevant to chemotactic pattern formation.