Exact model of aerotactic band: From Fokker-Planck equation to band structure and fluid flow

F. Detcheverry

REVIEW 3 major objections 5 minor 149 references

Exact model of aerotactic band: From Fokker-Planck equation to band structure and fluid flow

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read An exact model yields the aerotactic band shape and the flow it drives.

desk verdict Worth refereeing, but the central 'exact' Laplace-band derivation has a factor-k scaling error that needs fixing before the exactness claim holds. read the letter →

arxiv 2507.16314 v1 pith:CB6VYSZP submitted 2025-07-22 cond-mat.soft physics.bio-ph

F. Detcheverry This is my paper · ORCID

classification cond-mat.softphysics.bio-ph

keywords aerotaxisFokker-PlanckequationactivestressmagnetotacticbacteriaLaplacebandsingularperturbationStokesflowself-organization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to show that the aerotactic band—a stationary layer of bacteria that forms at a fixed distance from an air-water interface—can be described analytically from the microscopic equations of swimming and tumbling, not just through mesoscopic or numerical models. Combining the Fokker-Planck equation for bacterial position and orientation with oxygen diffusion and consumption reduces the steady band to one third-order nonlinear differential equation linking the oxygen profile to the aerotactic tumbling response. For the simplest relevant response, binary sensitivity to a preferred oxygen level, the bacterial density is exactly a two-sided exponential (Laplace) distribution whose width is fixed by the mean run length and the strength of tumbling modulation. Applying the same description to magnetotactic bacteria in a weak magnetic field yields a closed-form expression for the fluid flow generated by active stresses, and that expression reproduces the experimental flow profile without fitting parameters. If correct, this turns a phenomenon known since 1881 into a solvable model system for aerotaxis-driven self-organization.

What carries the argument

The load-bearing object is the Fokker-Planck equation for the bacterial distribution $p(x,\theta,t)$, with self-propulsion, rotational diffusion, and a tumbling rate $\lambda(x,\theta)=\lambda_o(1+k(x)\cos\theta)$. In steady state this equation admits the solution $p\propto e^{-K(x)/\kappa}$, and combining it with oxygen diffusion and consumption gives the single equation $-\kappa c'''=c''k(c,c')$, where $k(c,c')$ is the aerotactic kernel. For binary sensitivity $k=k\,\mathrm{sign}(c_*-c)$, matched asymptotic expansions in the small parameter $\epsilon$ solve this equation and produce the Laplace band. For the flow problem, the second ingredient is the active stress tensor $T_a=\sigma_o\rho L\,p(x)(\langle ee\rangle-I/2)$, whose divergence provides the force density driving the Stokes flow; the capillary geometry enters through the Green function $G(x)$ of $\Delta G=-\delta(x)$, approximated by $G_{\mathrm{app}}(x)=\frac{\Lambda}{2}e^{-|x|/\Lambda}$, $\Lambda=8Ch/\pi^2$. Convolving this Green function with the force density yields the closed-form velocity profile.

What would settle it

Measure the orientation distribution $p(\theta)$ at the band center under the same capillary and field conditions as the flow measurement. The low-field prediction contains a $\sin\theta$ modulation but no $\sin2\theta$ term, whereas flow-induced Jeffery rotation would add a $\sin2\theta$ component proportional to the shear rate; detecting that component at the predicted flow speeds would rule out the quasi-spherical assumption on which the velocity formula rests.

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Extended reading notes

Core claim

The central claim is that a band formed by bacteria with binary aerotactic sensitivity has the exact density profile $p(x)=\frac{1}{2l}e^{-|x|/l}$, with width $l=v_o\lambda_o^{-1}\bar\alpha_o/k$, while the oxygen concentration is piecewise linear away from the band and crosses over exponentially inside it. For magnetotactic bacteria at low magnetic field, the orientation distribution can be computed to second order in the field, and from it the active stress tensor follows; the divergence of that stress acts as a force density on the fluid. Solving the Stokes equation with the no-slip capillary walls through an approximate Green function gives the antisymmetric flow profile $v(x)=V\sin(2\beta)\frac{b^2}{8}\frac{\mathrm{sign}(x)}{l^2/\Lambda^2-1}\left(e^{-|x|/l}-e^{-|x|/\Lambda}\right)$, with velocity scale $V=\pi^2\sigma_o\rho L/\eta$ and hydrodynamic length $\Lambda=8Ch/\pi^2$. This profile, rescaled by its maximum, collapses onto the experimental master curve with no free parameters, and the inferred bacterial force-dipole moment and characteristic magnetic field fall in the biologically expected range. The paper presents these results as an exact, microscopic model of the band and of the flow it induces.

Load-bearing premise

The most fragile premise is that the swimmers are quasi-spherical, so the self-generated flow does not rotate their bodies; if real elongated magnetotactic bacteria rotate in the shear flow, the orientation distribution, active stress, and predicted velocity profile would all change.

Editorial extensions

If this is right

A band with binary sensitivity always has an exponential density profile, so measuring the width of a real band gives the combination of run time, tumble efficiency, and modulation strength.
The oxygen concentration is piecewise linear outside a sharp interior layer, a two-scale structure that could be looked for with oxygen-sensitive dyes.
The induced flow is antisymmetric across the band, scales as $\sin(2\beta)$ with the magnetic-field angle, and its rescaled shape depends only on the ratio of capillary half-thickness to band width.
The maximum flow speed saturates in thick capillaries at $V\sin(2\beta)$ and grows as the square of the capillary thickness in thin ones.
Using the reported experiments, the model infers a pusher force-dipole moment around $1.5\times10^{-19}\,\mathrm{J}$ and a characteristic magnetic field near $1\,\mathrm{mT}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One could invert Eq. (41) on individual experimental profiles to extract the force-dipole moment per experiment, testing whether the inferred $\sigma_o$ depends on bacterial density or field strength; the paper does not perform this test.
The same third-order equation with a sign kernel should produce exponential bands for any chemical taxis with a preferred concentration, not only oxygen; a direct extension would be chemotaxis to an attractant with a steep consumption profile.
Real M. gryphiswaldense cells are elongated, so the $C_B=0$ assumption is the fragile point: a testable signature of its failure would be a $\sin2\theta$ component in the orientation distribution inside the band, which the low-field prediction (28a) does not contain.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper has a real result and a genuinely nice flow prediction, but the central derivation of the Laplace band is not as clean as it looks. The stress-test note is correct: Eq. (14) does not satisfy Eq. (9a) unless k=1. Substituting the proposed inner solution leaves a leftover factor k on the right-hand side; the correct inner scale is epsilon/k, not epsilon. As written, the dimensionless density (14), the band-position shift in (12a), and the derivation of the dimensional width l = vo/(lambda_o alpha_bar_o k) are internally inconsistent. The final dimensional formula (15) has the right k dependence, but it does not follow from the algebra in the paper. This is a load-bearing flaw because the paper is explicitly an "exact model" --- but it is fixable by rescaling the inner layer.

What the paper does well: it derives a third-order ODE for the oxygen profile from a Fokker-Planck description, and then solves it exactly for two idealized sensing kernels. The exponential Laplace band and the algebraic-decay linear-sensing band are new analytical results; prior work was numerical, one-dimensional, or mesoscopic. The second half gives an analytical flow prediction, Eq. (41), that matches the experimental master curve without free parameters. That comparison is the strongest part of the paper and deserves credit.

The other soft spots are more minor. The CB=0 assumption (quasi-spherical swimmers) is explicit and acknowledged; it could change the quantitative flow shape but is a tractability choice, not a hidden error. The low-field expansion in Appendix C relies on a finite Fourier truncation, which is a technical assumption that appears to work to second order but that I could not fully verify. The footnote about the factor-of-five discrepancy with agent-based simulations is honest but unexplained; with the corrected k scaling, that estimate may change.

Who this is for: people working on aerotaxis, bacterial self-organization, and active suspensions. The flow prediction is the reason to read it, and the exact-profile results are useful once the scaling error is corrected.

Recommendation: send to peer review. The error is real but fixable, and the paper has enough new material — the flow prediction alone — to justify a serious referee. The referee should be asked to check the inner-layer scaling carefully.

Referee Report

3 major / 5 minor

Summary. This paper develops a microscopic Fokker-Planck description of a steady aerotactic band and of the fluid flow induced by a magnetotactic band. The author derives a third-order equation for the oxygen concentration (Eq. 8), solves it analytically for binary and linear sensing kernels, and obtains a Laplace density profile for the band. For magnetotactic bacteria in a weak magnetic field, a second-order expansion in field strength yields the orientation distribution and the active stress; convolution with an approximate Green function gives a closed-form velocity profile (Eq. 41). The velocity profile is compared with the experimental master curve of Marmol et al., without fitting the shape.

Significance. The paper is significant because it provides one of the few analytical, microscopic treatments of aerotactic band structure and connects it to a quantitative prediction for flow in magnetotactic bands. The Laplace-band solution and the parameter-free shape comparison in Fig. 8 are valuable and, if correct, would constitute a solid reference model for aerotaxis-driven self-organization. The derivation of the basic ODE, the matched-asymptotic structure, and the hydrodynamic Green-function computation are mostly clean and well documented. However, the central 'exact' solution for the binary-sensitivity band contains a factor-k inconsistency, and the low-field expansion rests on an unproven finite Fourier truncation; both points affect the quantitative claims and need to be fixed.

major comments (3)

[Sec. II B 1, Eqs. (9a), (11), (13)-(15)] The inner solution (13)-(14) does not satisfy the governing equation for k<1. For x>x*, substitution of c_in into Eq. (9a) gives -epsilon c''' = e^{-u}/(2 gamma epsilon) while c'' k(c,c') = k e^{-u}/(2 gamma epsilon); balance requires k=1, and the same problem occurs on x<x*. The correct inner scale is delta = epsilon/k, leading to p(x) = k/(2 epsilon) exp(-k|x-x*|/epsilon), a band-position shift epsilon/(2k) in Eq. (12a), and a dimensional width l = v_o lambda_o^{-1} bar_alpha_o / k. As printed, Eq. (14) is therefore not exact for general k, and the factor-k error propagates into the parameter estimates and the simulation comparison in footnote [79].
[Appendix C, around Eq. (C6)] The finite Fourier truncation |l| <= m is postulated rather than derived. Since the source term S(l) contains p_{m-1} with support |l| <= m-1 and the shift operator T can in principle generate |l| = m+1 from a term f_m(l +/- 1), it is not evident that the truncation is exact. The active stress in Eq. (33) and the flow prediction Eq. (41) depend on p_2, so the expansion's exactness is load-bearing. Please provide an inductive proof of the support bound or verify explicitly that the displayed p_1 and p_2 satisfy Eq. (C3), including the boundary and normalization conditions.
[Sec. III C 1 and III D] The model sets C_B=0 and is then compared quantitatively to M. gryphiswaldense, which is elongated; this is acknowledged as a limitation, but the paper does not estimate the magnitude of the omitted Jeffery flux at the inferred shear rates. Because the fluid velocity is O(b^2) and the leading orientation distribution is isotropic, the Jeffery term is not a priori negligible. Please estimate C_B v'(x) relative to the retained terms, or restrict the quantitative claims accordingly and make the lower-bound argument in footnote [119] quantitative.

minor comments (5)

[Eq. (15)] The division by k is not clear in the typeset formula; make l identical to v_o lambda_o^{-1} bar_alpha_o / k explicit, and ensure consistency with the statement that the band width decreases when the tumbling modulation is stronger.
[Fig. 1 caption] The caption cites 'Codutti et al., PLoS Comput. Biol. 15(12) e1007548 (2009)', but the reference list gives the year as 2019; please correct the caption.
[Sec. III D 2] The phrase 'without any free free parameter' contains a duplicated word; also, while the profile shape is parameter-free, the magnitude involves sigma_o inferred from the same data set, so the wording should be qualified.
[Appendix C, Eq. (C8b)] The notations D[1,4]_r and D[4]_r are hard to follow; define them explicitly in the text before first use, for example as (1+D_r)(1+4D_r) and (1+4D_r).
[Eq. (28a) and following text] The orientation modulation is written for x>0; the text says the negative side is obtained by substitutions, but it would help to state explicitly that p(x,theta) is continuous at x=0 under those substitutions.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Laplace band and flow profile are derived from the model's own Fokker–Planck and Stokes equations, with the a posteriori dipole-moment inference explicitly labeled as an estimate.

full rationale

The central band result, Eq. (15), is obtained by substituting the inner ansatz Eqs. (13a)-(13b) into the model's own dimensionless band equation Eq. (9a) with the binary kernel Eq. (11); no fitted parameter enters the exponential form, and the comparison to the Codutti et al. density profile uses the independently measured band width l = 73 µm. The flow result, Eq. (41), is likewise derived within the paper: the Fokker–Planck expansion at small magnetic field gives the active stress, Eq. (33), and the velocity follows by convolving that stress with the Green function of the Stokes problem, Eqs. (39)-(40). This is a parameter-free shape prediction, and the force dipole sigma_o is inferred only afterwards in Sec. III D 2(v) from the measured maximal velocity, with the text explicitly saying it 'allows to estimate' sigma_o, not that sigma_o was predicted. The main external imports are Schnitzer's continuum random-walk solution [62], standard active-stress results [56,58], and the hydrodynamics of Jeffery/Bretherton; none are self-citations, and the paper's own self-citations ([16] and [45]) are contextual or technical rather than load-bearing. The acknowledged limitations (C_B = 0 quasi-spherical swimmers, transient experimental band, two-dimensional orientation) are scoped assumptions, not means of importing the target results. A reviewer concern that the claimed inner solution Eq. (14) may balance Eq. (9a) only for k = 1 is an internal algebraic accuracy question, not a circularity, and therefore does not affect this circularity score, though it should be checked separately.

Assumptions & free parameters 5 free parameters · 8 assumptions · 0 invented entities

The model introduces no new physical entities. Its free parameters are the tumbling modulation strength, the preferred oxygen concentration, the gradient scale in the linear-sensing variant, and the band width; the force dipole strength is inferred a posteriori from experiment. The main axioms are the kinetic equation, Markovian cosine-modulated tumbling, constant consumption, the binary-sensitivity toy kernel, the low-field truncation, the approximate Green function, and the quasi-spherical swimmer approximation.

free parameters (5)

k (tumbling modulation magnitude) = 0.5
Dimensionless modulation strength in Eq (11); chosen so that the predicted band width l = vo/(lambda_o alpha_bar_o k) matches the experimental l = 15 um.
c* (preferred oxygen concentration)
Target oxygen concentration where the tumbling bias switches sign; a model input set by the bacterium's microaerophilic preference.
c'_o (characteristic gradient in linear sensing)
Scale introduced in Eq (16) to make the kernel dimensionless; not fixed by theory.
sigma_o (force dipole moment) = -1.5e-19 J
Inferred from the measured maximal flow velocity via Eq (48a); used only to check magnitude, not predicted.
l (band width) = 15 um
Taken from the experimental density profile (Gaussian standard deviation); used as input to the flow profile comparison.

assumptions (8)

domain assumption The Fokker-Planck equation (Eq 1) describes the bacterial position-orientation distribution.
The microscopic model is built on this kinetic equation.
domain assumption Tumbling rate is Markovian with lambda(x,theta) = lambda_o (1 + k(x) cos theta) (Eq 4).
History effects are discarded and only the cosine Fourier mode is retained; justified by linear response and experiments [10].
domain assumption Steady-state solution p(x,theta) = N exp(-K/kappa) (Eq 5) from Schnitzer (1993) [62].
The exact solution of Eq (1) under local tumbling is imported from prior literature.
domain assumption Per-bacterium oxygen consumption is constant, Phi = q_o.
Simplifies Eq (7) to Eq (8); a standard approximation.
ad hoc to paper Binary sensitivity kernel k(c,c') = k sign(c* - c) (Eq 11).
A toy model chosen for tractability; the paper notes it is an idealization.
ad hoc to paper Low-field expansion in b with finite Fourier truncation |l| <= m (Appendix C).
The expansion order is assumed to truncate; not proven.
ad hoc to paper Green function approximation G_app = Lambda/2 exp(-|x|/Lambda) (Eq 39).
Exact to first order at small x, within 4% maximum deviation; used to make the flow integral analytical.
ad hoc to paper Quasi-spherical swimmers with CB = 0 (Sec III C 1).
Neglects flow-induced rotation to avoid nonlinear coupling; not valid for elongated bacteria.

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Cite this review

Pith. "Pith review of Exact model of aerotactic band: From Fokker-Planck equation to band structure and fluid flow." pith.science (2026). https://pith.science/paper/CB6VYSZP

@misc{pith2026250716314,
  author       = {Pith},
  title        = {Pith review of: Exact model of aerotactic band: From Fokker-Planck equation to band structure and fluid flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CB6VYSZP}},
  note         = {Machine review of arXiv:2507.16314}
}

read the original abstract

A variety of bacterial species spontaneously assemble in aerotactic band, local accumulation at a fixed distance from the air-water interface. Although the phenomenon is long known, its modelling is so far limited to mesoscopic, one-dimensional or numerical descriptions. We investigate band properties at the microscopic scale using exact solutions to the Fokker-Planck equation. First, we show that the interplay between oxygen consumption and tumbling modulation is governed by a third-order nonlinear differential equation relating the oxygen concentration to the aerotactic response. For two model aerotactic behaviors, we present analytical solutions and discuss the resulting band structure. Second, we investigate how an aerotactic band of magnetotactic bacteria in a magnetic field induces a spontaneous fluid flow, as observed in experiments [Marmol et al, arXiv 2025]. In the low field limit, we determine the bacterial distribution and the active stress tensor. Using the Green function of the hydrodynamic problem, we obtain a prediction for the fluid flow that is both simple and consistent with observations. Altogether, our results provide a model system of aerotactic band and solid ground to analyze aerotaxis-driven self-organization.

Figures

Figures reproduced from arXiv: 2507.16314 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

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Reference graph

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The probability distribution is thus proportional to δ(θ − β)

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In particular, microaerophile bacteria seek oxy- gen levels well below the saturation ( c∗ ≪ cs)

Toy sensing: the Laplace band A commonly observed aerotactic behavior is the drive toward a preferred oxygen concentration c∗ [24, 25, 31, 32, 36]. In particular, microaerophile bacteria seek oxy- gen levels well below the saturation ( c∗ ≪ cs). Probably the simplest way to encapsulate the preference toward c∗ in the tumble rate kernel is to write k(c, c′...
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Dimensionless form and interior layer It is useful to express the problem in dimensionless form. To do so, the capillary length L is taken as unit length and the oxygen saturation value cs as unit con- centration, giving the governing equations −ϵ c′′′(x) = c′′(x)k(c, c′), (9a) c(0) = 1, c ′(0) = −γ−1, c ′(1) = 0, (9b) p(x) = γc′′(x), (9c) where from now ...
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Linear sensing: an asymmetric band We now consider linear sensing, i.e. an aerotactic re- sponse proportional to the perceived oxygen gradient, as is the case in linear response. Keeping the assumption of binary sensitivity, the rate modulation is now k(c, c′) = − c′(x) c′o sign (c∗ − c) , (16) where c′ o > 0 is a characteristic concentration gradient in-...
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One particular feature that may appear unphysical is the dis- continuity in aerotactic response

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Can we infer from the data the under- lying aerotactic behavior? If the kernel depends only on oxygen concentration c, the answer is positive

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Such a framework has been extensively used to describe chemo- tactic band, aggregation and patterns [50, 51]

Relationship with Keller-Segel approach As a final point, we discuss the relation between our microscopic description and a macroscopic modelling based on Keller-Segel type equation [47, 48]. Such a framework has been extensively used to describe chemo- tactic band, aggregation and patterns [50, 51]. In this approach, the bacterial flux J(x) is written as...

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Here we do not carry out a perturbative analysis on the bacterial distribution

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(48b) The characteristic field Bc delineates the boundary be- tween the low and high field regimes

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Reviewed August 6, 2026 · model on record in the stance chip above.

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The probability distribution is thus proportional to δ(θ − β)

work page

[3] [1]

In particular, microaerophile bacteria seek oxy- gen levels well below the saturation ( c∗ ≪ cs)

Toy sensing: the Laplace band A commonly observed aerotactic behavior is the drive toward a preferred oxygen concentration c∗ [24, 25, 31, 32, 36]. In particular, microaerophile bacteria seek oxy- gen levels well below the saturation ( c∗ ≪ cs). Probably the simplest way to encapsulate the preference toward c∗ in the tumble rate kernel is to write k(c, c′...

[4] [2]

Dimensionless form and interior layer It is useful to express the problem in dimensionless form. To do so, the capillary length L is taken as unit length and the oxygen saturation value cs as unit con- centration, giving the governing equations −ϵ c′′′(x) = c′′(x)k(c, c′), (9a) c(0) = 1, c ′(0) = −γ−1, c ′(1) = 0, (9b) p(x) = γc′′(x), (9c) where from now ...

[5] [3]

an aerotactic re- sponse proportional to the perceived oxygen gradient, as is the case in linear response

Linear sensing: an asymmetric band We now consider linear sensing, i.e. an aerotactic re- sponse proportional to the perceived oxygen gradient, as is the case in linear response. Keeping the assumption of binary sensitivity, the rate modulation is now k(c, c′) = − c′(x) c′o sign (c∗ − c) , (16) where c′ o > 0 is a characteristic concentration gradient in-...

[6] [4]

One particular feature that may appear unphysical is the dis- continuity in aerotactic response

Continuous aerotactic kernel Binary sensitivity has the advantage of tractability but is likely, for most bacteria, to remain an idealization. One particular feature that may appear unphysical is the dis- continuity in aerotactic response. Instead, one could ex- pect a smooth crossover between the two saturation val- ues, for at least two reasons. At the ...

[7] [5]

Can we infer from the data the under- lying aerotactic behavior? If the kernel depends only on oxygen concentration c, the answer is positive

Inference of aerotactic kernel Band profiles that are neither exponential nor sym- metric have been observed for several bacteria species in experiments [36]. Can we infer from the data the under- lying aerotactic behavior? If the kernel depends only on oxygen concentration c, the answer is positive. Indeed, 7 exploiting Eq. (8) and Eq. (9c) gives in this...

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Such a framework has been extensively used to describe chemo- tactic band, aggregation and patterns [50, 51]

Relationship with Keller-Segel approach As a final point, we discuss the relation between our microscopic description and a macroscopic modelling based on Keller-Segel type equation [47, 48]. Such a framework has been extensively used to describe chemo- tactic band, aggregation and patterns [50, 51]. In this approach, the bacterial flux J(x) is written as...