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arxiv: 2604.15954 · v1 · submitted 2026-04-17 · 🧮 math.AP

Global stability in a negative chemotaxis system with chemically induced lethality

Federico Herrero-Herv\'as , Mihaela Negreanu This is my paper

Pith reviewed 2026-05-10 08:15 UTC · model grok-4.3

classification 🧮 math.AP
keywords negative chemotaxisKeller-Segel systemglobal stabilitylogistic growthchemically induced deathlong-time dynamicsrepulsive chemotaxishomogeneous steady states
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The pith

Repellent supply strength determines whether cell populations go extinct or equilibrate to a uniform positive density in a negative chemotaxis model with lethality.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the long-time dynamics of a repulsive chemotaxis system that incorporates logistic growth and a death term triggered by the concentration of the chemorepellent, which cells both produce and receive from external sources. It proves that when the external supply is held constant, solutions converge uniformly in space to either complete extinction or a positive constant equilibrium, with the outcome controlled by whether the supply magnitude exceeds or stays below a threshold set by the logistic growth rate. A sympathetic reader would care because the result gives explicit conditions under which the model predicts total population loss or stable uniform survival, without dependence on initial spatial arrangements. The analysis shows how the constant supply simplifies the system enough for comparison arguments to decide the global attractor.

Core claim

In the negative chemotaxis Keller-Segel system with logistic growth and chemically induced lethality, constant external chemorepellent supplies cause solutions to converge in the L^∞ norm to extinction when the supply is large relative to the logistic growth rate, or to a positive spatially homogeneous steady state when the supply is small.

What carries the argument

The threshold comparison between the magnitude of the constant external chemorepellent supply and the logistic growth rate, which determines the sign that controls long-time attraction to the zero state or the positive constant.

If this is right

  • When the constant supply lies below the critical value, every solution approaches the same positive uniform density regardless of initial data.
  • When the supply lies above the critical value, the population density converges uniformly to zero everywhere.
  • The repulsive chemotactic flux does not produce lasting spatial inhomogeneities under constant supply.
  • Boundedness of solutions holds globally in time, with the long-term behavior fully classified by the parameter comparison.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The stabilizing role of a constant supply suggests that time-varying supplies could allow transient patterns before eventual homogenization or extinction.
  • In applications, maintaining an external chemical level above threshold could achieve spatially uniform eradication of a cell population.
  • Related models with nonlinear death rates or additional advection terms might admit analogous thresholds that can be tested numerically.

Load-bearing premise

The external supply of the chemorepellent is constant in both time and space.

What would settle it

An explicit solution or numerical trajectory that develops persistent spatial patterns or fails to reach extinction when the supply exceeds the logistic threshold.

Figures

Figures reproduced from arXiv: 2604.15954 by Federico Herrero-Herv\'as, Mihaela Negreanu.

Figure 1
Figure 1. Figure 1: Two different possibilities for kmin(r) and k2(r) for (18) to hold. The feasible region is shaded. As the leading coefficient is 16Da2χ 2 > 0, and P(0) = − [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
read the original abstract

In this paper, we investigate the long-time dynamics of a repulsive Keller-Segel chemotaxis system. The model features negative chemotaxis, logistic growth and a cell death term, accounting for a lethal chemorepellent that is self-produced by the cells and externally supplied. We prove that, for constant chemorepellent supplies, depending on their magnitude with respect to the logistic growth rate, solutions converge in $L^\infty$ norm toward extinction of the population, or equilibrate toward a nontrivial spatially homogeneous steady state.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper investigates the long-time dynamics of a repulsive Keller-Segel chemotaxis system with negative chemotaxis, logistic growth, and a cell death term induced by a self-produced and externally supplied lethal chemorepellent. The central result establishes that, for constant-in-time-and-space chemorepellent supplies, solutions converge in the L^∞ norm either to extinction or to a unique nontrivial spatially homogeneous steady state, with the outcome determined by comparing the supply magnitude to the logistic growth rate.

Significance. If the proofs hold, the result gives a clean, parameter-driven dichotomy for global stability in this class of models, exploiting the time-independence of the supply to reduce the asymptotics to a scalar comparison. This is a useful contribution to the analysis of repulsive chemotaxis systems with lethality, as it identifies explicit thresholds separating extinction from survival at a homogeneous equilibrium. The approach appears to rely on comparison principles or energy methods that become tractable precisely because the supply is constant, which is a genuine technical advantage.

minor comments (2)
  1. The abstract states L^∞ convergence but does not indicate whether the proof first obtains L^1 or L^2 bounds before upgrading; a brief outline of the bootstrap or comparison argument in the introduction would improve readability.
  2. Notation for the death term and chemotactic sensitivity function should be fixed consistently between the model statement and the steady-state analysis to avoid any ambiguity in the parameter threshold.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript. The referee's summary accurately captures the main contribution: the L^∞ convergence dichotomy for solutions of the repulsive chemotaxis system with constant chemorepellent supply, determined by comparison with the logistic growth rate. We are pleased that the technical advantage of time-independent supply is recognized. No major comments were listed in the report, so we have no specific points requiring response or revision at this stage. We remain available to address any minor suggestions should they be provided.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes global stability for the repulsive Keller-Segel system with logistic growth and lethal death term by applying standard parabolic comparison principles and energy dissipation estimates to the given PDEs under the assumption of constant-in-time-and-space chemorepellent supply. The long-time behavior is controlled by a direct scalar comparison between supply magnitude and logistic growth rate, which follows from the model equations without any fitted parameters, self-referential definitions, or load-bearing self-citations. No step reduces a claimed result to an input quantity by construction, and the proof chain remains independent of the target stability statement.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the well-posedness of the modified Keller-Segel PDE system and the assumption that the death term is linear in the repellent concentration; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The system is governed by the standard parabolic chemotaxis equations augmented with logistic growth and a lethality term proportional to the chemorepellent density.
    This is the foundational model whose long-time behavior is analyzed.

pith-pipeline@v0.9.0 · 5378 in / 1252 out tokens · 58521 ms · 2026-05-10T08:15:05.133969+00:00 · methodology

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Reference graph

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