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Local sensitivity analysis for the Cucker-Smale model with random inputs
Traveling wave and aggregation in a flux-limited Keller-Segel model
1. | Institut Camille Jordan, UMR 5208 CNRS/Université Claude Bernard Lyon 1, and Project-team Inria NUMED, Lyon, France |
2. | Sorbonne Universités, UPMC Univ Paris 06, Laboratoire Jacques-Louis Lions UMR CNRS 7598, Université Paris Diderot, Inria de Paris, F75005 Paris, France |
3. | Graduate School of Simulation Studies, University of Hyogo, Kobe 650-0047, Japan |
Flux-limited Keller-Segel (FLKS) model has been recently derived from kinetic transport models for bacterial chemotaxis and shown to represent better the collective movement observed experimentally. Recently, associated to the kinetic model, a new instability formalism has been discovered related to stiff chemotactic response. This motivates our study of traveling wave and aggregation in population dynamics of chemotactic cells based on the FLKS model with a population growth term.
Our study includes both numerical and theoretical contributions. In the numerical part, we uncover a variety of solution types in the one-dimensional FLKS model additionally to standard Fisher/KPP type traveling wave. The remarkable result is a counter-intuitive backward traveling wave, where the population density initially saturated in a stable state transits toward an unstable state in the local population dynamics. Unexpectedly, we also find that the backward traveling wave solution transits to a localized spiky solution as increasing the stiffness of chemotactic response.
In the theoretical part, we obtain a novel analytic formula for the minimum traveling speed which includes the counter-balancing effect of chemotactic drift vs. reproduction/diffusion in the propagating front. The front propagation speeds of numerical results only slightly deviate from the minimum traveling speeds, except for the localized spiky solutions, even for the backward traveling waves. We also discover an analytic solution of unimodal traveling wave in the large-stiffness limit, which is certainly unstable but exists in a certain range of parameters.
References:
[1] |
J. Adler, Chemotaxis in Bacteria, Science, 153 (1966), 708-716. Google Scholar |
[2] |
D. G. Aronson and H. F. Weinberger, Lecture Notes in Mathematics, Springer, 1975. Google Scholar |
[3] |
E. O. Budrene and H. C. Berg,
Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633.
doi: 10.1038/349630a0. |
[4] |
E. O. Budrene and H. C. Berg,
Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature, 376 (1995), 49-53.
doi: 10.1038/376049a0. |
[5] |
N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler,
Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Model Methods Appl. Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[6] |
N. Bellomo and M. Winkler,
A degenerate chemotaxis system with flux limitation: Maximally extended solutions and abasence of gradient blow-up, Commun. Part. Diff. Eq., 42 (2017), 436-473.
doi: 10.1080/03605302.2016.1277237. |
[7] |
S. M. Block, J. E. Segall and H. C. Berg, Adaptation kinetics in bacterial chemotaxis, J. Bacteriol., 154 (1983), 312-323. Google Scholar |
[8] |
V. Calvez, Chemotactic waves of bacteria at the mesoscale, preprint, arXiv: 1607.00429. Google Scholar |
[9] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu,
On a chemotaxis model with saturated chemotactic flux, Kinetic and related models, 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[10] |
Y. Dolak and C. Schmeiser,
Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms, J. Math. Biol., 51 (2005), 595-615.
doi: 10.1007/s00285-005-0334-6. |
[11] |
S. Ei, H. Izuhara and M. Mimura,
Spatio-temporal oscillations in the Keller-Segel system with logistic growth, Physica D, 277 (2014), 1-21.
doi: 10.1016/j.physd.2014.03.002. |
[12] |
C. Emako, C. Gayrard, A. Buguin, L. Almeida and N. Vauchelet, Traveling pulses for a two-species chemotaxis model,
PLoS Comput. Biol., 12 (2016), e1004843.
doi: 10. 1371/journal. pcbi. 1004843. |
[13] |
R. A. Fisher, The advance of advantageous genes, Ann. Eugenics, 65 (1937), 335-369. Google Scholar |
[14] |
T. Hillen and K. J. Painter,
A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[15] |
F. James and N. Vauchelet,
Chemotaxis: from kinetic equations to aggregate dynamics, Nonlinear Differ. Equ. Appl., 20 (2013), 101-127.
doi: 10.1007/s00030-012-0155-4. |
[16] |
Y. V. Kalinin, L. Jiang, Y. Tu and M. Wu, Logarithmic sensing in Escherichia coli bacterial chemotaxis, Biophys. J., 96 (2009), 2439-2448. Google Scholar |
[17] |
E. F. Keller and L. A. Segel,
Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[18] |
E. F. Keller and L. A. Segel,
Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[19] |
A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l’équation de la diffusion avec croissance de la quantité de |
[20] |
C. Liu, X. Fu, L. Liu, X. Ren, C. K. L. Chau, S. Li, L. Xiang, H. Zeng, G. Chen, L. Tang, P. Lenz, X. Cui, W. Huang, T. Hwa and J.-.D. Huang,
Sequential Establishment of stripe patterns in an expanding cell population, Science, 334 (2011), 238-241.
doi: 10.1126/science.1209042. |
[21] |
P. K. Maini, M. R. Myerscough, K. H. Winters and J. D. Murray, Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation, Bull. Math. Biol., 53 (1991), 701-719. Google Scholar |
[22] |
M. Mimura and T. Tsujikawa,
Aggregating pattern dynamics in a chemotaxis model including growth, Physica A, 230 (1996), 499-543.
doi: 10.1016/0378-4371(96)00051-9. |
[23] |
G. Nadin, B. Perthame and L. Ryzhik,
Traveling waves for the Keller-Segel system with Fisher birth terms, Interface free bound., 10 (2008), 517-538.
doi: 10.4171/IFB/200. |
[24] |
G. Nadin, B. Perthame and M. Tang,
Can a traveling wave connect two unstable state? The case of the nonlocal Fisher equation, C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 553-557.
doi: 10.1016/j.crma.2011.03.008. |
[25] |
K. J. Painter and T. Hillen,
Spatio-temporal chaos in a chemotaxis model, Physica D, 240 (2011), 363-375.
doi: 10.1016/j.physd.2010.09.011. |
[26] |
B. Perthame,
Parabolic Equations in Biology, Springer, 2015.
doi: 10.1007/978-3-319-19500-1. |
[27] |
B. Perthame, N. Vauchelet and Z. Wang, Modulation of stiff response in E. coli bacterial populations, (in preparation). Google Scholar |
[28] |
B. Perthame and S. Yasuda, Stiff-response-induced instability for chemotactic bacteria and flux-limited Keller-Segel equation, preprint, arXiv: 1703.08386. Google Scholar |
[29] |
J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses,
PLoS Comput. Biol., 6 (2010), e1000890, 12pp.
doi: 10.1371/journal.pcbi.1000890. |
[30] |
J. Saragosti, V. Calvez, N. Bournaveas, B. Perthame, A. Buguin and P. Silberzan,
Directional persistence of chemotactic bacteria in a traveling concentration wave, PNAS, 108 (2011), 16235-16240.
doi: 10.1073/pnas.1101996108. |
[31] |
M. J. Tindall, P. K. Maini, S. L. Porter and J. P. Armitage,
Overview of mathematical approaches used to model bacterial chemotaxis Ⅱ: Bacterial populations, Bull. Math. Biol., 70 (2008), 1570-1607.
doi: 10.1007/s11538-008-9322-5. |
show all references
References:
[1] |
J. Adler, Chemotaxis in Bacteria, Science, 153 (1966), 708-716. Google Scholar |
[2] |
D. G. Aronson and H. F. Weinberger, Lecture Notes in Mathematics, Springer, 1975. Google Scholar |
[3] |
E. O. Budrene and H. C. Berg,
Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633.
doi: 10.1038/349630a0. |
[4] |
E. O. Budrene and H. C. Berg,
Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature, 376 (1995), 49-53.
doi: 10.1038/376049a0. |
[5] |
N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler,
Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Model Methods Appl. Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[6] |
N. Bellomo and M. Winkler,
A degenerate chemotaxis system with flux limitation: Maximally extended solutions and abasence of gradient blow-up, Commun. Part. Diff. Eq., 42 (2017), 436-473.
doi: 10.1080/03605302.2016.1277237. |
[7] |
S. M. Block, J. E. Segall and H. C. Berg, Adaptation kinetics in bacterial chemotaxis, J. Bacteriol., 154 (1983), 312-323. Google Scholar |
[8] |
V. Calvez, Chemotactic waves of bacteria at the mesoscale, preprint, arXiv: 1607.00429. Google Scholar |
[9] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu,
On a chemotaxis model with saturated chemotactic flux, Kinetic and related models, 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[10] |
Y. Dolak and C. Schmeiser,
Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms, J. Math. Biol., 51 (2005), 595-615.
doi: 10.1007/s00285-005-0334-6. |
[11] |
S. Ei, H. Izuhara and M. Mimura,
Spatio-temporal oscillations in the Keller-Segel system with logistic growth, Physica D, 277 (2014), 1-21.
doi: 10.1016/j.physd.2014.03.002. |
[12] |
C. Emako, C. Gayrard, A. Buguin, L. Almeida and N. Vauchelet, Traveling pulses for a two-species chemotaxis model,
PLoS Comput. Biol., 12 (2016), e1004843.
doi: 10. 1371/journal. pcbi. 1004843. |
[13] |
R. A. Fisher, The advance of advantageous genes, Ann. Eugenics, 65 (1937), 335-369. Google Scholar |
[14] |
T. Hillen and K. J. Painter,
A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[15] |
F. James and N. Vauchelet,
Chemotaxis: from kinetic equations to aggregate dynamics, Nonlinear Differ. Equ. Appl., 20 (2013), 101-127.
doi: 10.1007/s00030-012-0155-4. |
[16] |
Y. V. Kalinin, L. Jiang, Y. Tu and M. Wu, Logarithmic sensing in Escherichia coli bacterial chemotaxis, Biophys. J., 96 (2009), 2439-2448. Google Scholar |
[17] |
E. F. Keller and L. A. Segel,
Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[18] |
E. F. Keller and L. A. Segel,
Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[19] |
A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l’équation de la diffusion avec croissance de la quantité de |
[20] |
C. Liu, X. Fu, L. Liu, X. Ren, C. K. L. Chau, S. Li, L. Xiang, H. Zeng, G. Chen, L. Tang, P. Lenz, X. Cui, W. Huang, T. Hwa and J.-.D. Huang,
Sequential Establishment of stripe patterns in an expanding cell population, Science, 334 (2011), 238-241.
doi: 10.1126/science.1209042. |
[21] |
P. K. Maini, M. R. Myerscough, K. H. Winters and J. D. Murray, Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation, Bull. Math. Biol., 53 (1991), 701-719. Google Scholar |
[22] |
M. Mimura and T. Tsujikawa,
Aggregating pattern dynamics in a chemotaxis model including growth, Physica A, 230 (1996), 499-543.
doi: 10.1016/0378-4371(96)00051-9. |
[23] |
G. Nadin, B. Perthame and L. Ryzhik,
Traveling waves for the Keller-Segel system with Fisher birth terms, Interface free bound., 10 (2008), 517-538.
doi: 10.4171/IFB/200. |
[24] |
G. Nadin, B. Perthame and M. Tang,
Can a traveling wave connect two unstable state? The case of the nonlocal Fisher equation, C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 553-557.
doi: 10.1016/j.crma.2011.03.008. |
[25] |
K. J. Painter and T. Hillen,
Spatio-temporal chaos in a chemotaxis model, Physica D, 240 (2011), 363-375.
doi: 10.1016/j.physd.2010.09.011. |
[26] |
B. Perthame,
Parabolic Equations in Biology, Springer, 2015.
doi: 10.1007/978-3-319-19500-1. |
[27] |
B. Perthame, N. Vauchelet and Z. Wang, Modulation of stiff response in E. coli bacterial populations, (in preparation). Google Scholar |
[28] |
B. Perthame and S. Yasuda, Stiff-response-induced instability for chemotactic bacteria and flux-limited Keller-Segel equation, preprint, arXiv: 1703.08386. Google Scholar |
[29] |
J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses,
PLoS Comput. Biol., 6 (2010), e1000890, 12pp.
doi: 10.1371/journal.pcbi.1000890. |
[30] |
J. Saragosti, V. Calvez, N. Bournaveas, B. Perthame, A. Buguin and P. Silberzan,
Directional persistence of chemotactic bacteria in a traveling concentration wave, PNAS, 108 (2011), 16235-16240.
doi: 10.1073/pnas.1101996108. |
[31] |
M. J. Tindall, P. K. Maini, S. L. Porter and J. P. Armitage,
Overview of mathematical approaches used to model bacterial chemotaxis Ⅱ: Bacterial populations, Bull. Math. Biol., 70 (2008), 1570-1607.
doi: 10.1007/s11538-008-9322-5. |










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10000 -5000 | | ||
20000 -10000 | | |
| | | |
10000 -5000 | | ||
20000 -10000 | | |
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7.0 | 1.30 | 3.08 | 21.0 | 1.065 | 5.57 | |
8.0 | 1.26 | 2.96 | 23.0 | 1.062 | 5.44 | |
9.0 | 1.23 | 2.79 | 25.0 | 1.059 | 5.28 | |
10.0 | 1.20 | 2.61 | 20.0 | 1.058 | 5.16 | |
| 1.00 | 3.09 | | 1.00 | 6.95 |
| ||||||
7.0 | 1.30 | 3.08 | 21.0 | 1.065 | 5.57 | |
8.0 | 1.26 | 2.96 | 23.0 | 1.062 | 5.44 | |
9.0 | 1.23 | 2.79 | 25.0 | 1.059 | 5.28 | |
10.0 | 1.20 | 2.61 | 20.0 | 1.058 | 5.16 | |
| 1.00 | 3.09 | | 1.00 | 6.95 |
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