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On the generic complete synchronization of the discrete Kuramoto model
Stability of a non-local kinetic model for cell migration with density dependent orientation bias
Department of Mathematical Sciences "G. L. Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy |
The aim of the article is to study the stability of a non-local kinetic model proposed in [
References:
[1] |
N. J. Armstrong, K. J. Painter and J. A. Sherratt,
A continuum approach to modelling cell-cell adhesion, J. Theoret. Biol., 243 (2006), 98-113.
doi: 10.1016/j.jtbi.2006.05.030. |
[2] |
V. Bitsouni and R. Eftimie,
Non-local parabolic and hyperbolic models for cell polarisation in heterogeneous cancer cell populations, Bull. Math. Biol., 80 (2018), 2600-2632.
doi: 10.1007/s11538-018-0477-4. |
[3] |
A. Buttenschön, Integro-partial Differential Equation Models for Cell-cell Adhesion and its Application, Ph.D thesis, University of Alberta, 2018. |
[4] |
A. Buttenschön and T. Hillen, Non-local adhesion models for microorganisms on bounded domains, SIAM J. Appl. Math., 80(1), (2020) 382–401.
doi: 10.1137/19M1250315. |
[5] |
A. Buttenschön and T. Hillen, Non-local cell adhesion models: Steady states and bifurcations, 2020, arXiv: 2001.00286. |
[6] |
A. Buttenschön, T. Hillen, A. Gerisch and K. J. Painter,
A space-jump derivation for non-local models of cell-cell adhesion and non-local chemotaxis, J. Math. Biol., 76 (2018), 429-456.
doi: 10.1007/s00285-017-1144-3. |
[7] |
J. Carrillo, F. Hoffmann and R. Eftimie,
Non-local kinetic and macroscopic models for self-organised animal aggregations, Kin. Rel. Models, 8 (2015), 413-441.
doi: 10.3934/krm.2015.8.413. |
[8] |
A. Colombi, M. Scianna and L. Preziosi,
Coherent modelling switch between pointwise and distributed representations of cell aggregates, J. Math. Biol., 74 (2017), 783-808.
doi: 10.1007/s00285-016-1042-0. |
[9] |
A. Colombi, M. Scianna and A. Tosin,
Differentiated cell behavior: A multiscale approach using measure theory, J. Math. Biol., 71 (2015), 1049-1079.
doi: 10.1007/s00285-014-0846-z. |
[10] |
R. Eftimie,
Hyperbolic and kinetic models for self-organized biological aggregations and movement: A brief review, J. Math. Biol., 65 (2012), 35-75.
doi: 10.1007/s00285-011-0452-2. |
[11] |
R. Eftimie, G. de Vries, M. A Lewis and F. Lutscher,
Modeling group formation and activity patterns in self-organizing collectives of individuals, Bull. Math. Biol., 69 (2007), 1537-1565.
doi: 10.1007/s11538-006-9175-8. |
[12] |
R. Eftimie, M. Perez and P.-L. Buono,
Pattern formation in a nonlocal mathematical model for the multiple roles of the TGF-$\beta$ pathway in tumour dynamics, Math. Biosci., 289 (2017), 96-115.
doi: 10.1016/j.mbs.2017.05.003. |
[13] |
R. Eftimie, G. Vries and M. Lewis,
Complex spatial group patterns result from different animal communication mechanisms, PNAS USA, 104 (2007), 6974-6979.
doi: 10.1073/pnas.0611483104. |
[14] |
F. Filbet and K. Yang, Numerical simulation of kinetic models for chemotaxis, SIAM J. Sci. Comput., 36 (2014), B348–B366.
doi: 10.1137/130910208. |
[15] |
T. Hillen, K. J. Painter and C. Schmeiser,
Global existence for chemotaxis with finite sampling radius, Discr. Cont. Dyn. Sys. - B, 7 (2007), 125-144.
doi: 10.3934/dcdsb.2007.7.125. |
[16] |
N. Loy and L. Preziosi,
Modelling physical limits of migration by a kinetic model with non-local sensing, J. Math. Biol., 80 (2020), 1759-1801.
doi: 10.1007/s00285-020-01479-w. |
[17] |
N. Loy and L. Preziosi,
Kinetic models with non-local sensing determining cell polarization and speed according to independent cues, J. Math. Biol., 80 (2019), 373-421.
doi: 10.1007/s00285-019-01411-x. |
[18] |
H. G. Othmer, S. R. Dunbar and W. Alt,
Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.
doi: 10.1007/BF00277392. |
[19] |
H. Othmer and T. Hillen,
The diffusion limit of transport equations Ⅱ: Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250.
doi: 10.1137/S0036139900382772. |
[20] |
J. K. Painter and T. Hillen,
Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (4) (2002), 501-543.
|
[21] |
K. J. Painter, N. J. Armstrong and J. A. Sherratt,
The impact of adhesion on cellular invasion processes in cancer and development, J. Theoret. Biol., 264 (2010), 1057-1067.
doi: 10.1016/j.jtbi.2010.03.033. |
[22] |
K. J. Painter, M. J. Bloomfield, J. A. Sherratt and A. Gerisch,
A nonlocal model for contact attraction and repulsion in heterogeneous cell populations, Bull. Math. Biol., 77 (2015), 1132-1165.
doi: 10.1007/s11538-015-0080-x. |
[23] |
B. Perthame and S. Yasuda,
Stiff-response-induced instability for chemotactic bacteria and flux-limited keller-segel equation, Nonlinearity, 31 (2018), 4065-4089.
doi: 10.1088/1361-6544/aac760. |
[24] |
R. G. Plaza,
Derivation of a bacterial nutrient-taxis system with doubly degenerate cross-diffusion as the parabolic limit of a velocity-jump process, J. Math. Biol., 78 (2019), 1681-1711.
doi: 10.1007/s00285-018-1323-x. |
[25] |
C. Schmeiser and A. Nouri,
Aggregated steady states of a kinetic model for chemotaxis, Kin. Rel. Models, 10 (2017), 313-327.
doi: 10.3934/krm.2017013. |
[26] |
D. W. Stroock,
Some stochastic processes which arise from a model of the motion of a bacterium, Zeit. Wahr. Ver. Geb., 28 (1974), 305-315.
doi: 10.1007/BF00532948. |
[27] |
A. Tosin and P. Frasca,
Existence and approximation of probability measure solutions to models of collective behaviors, Netw. Heter. Media, 6 (2011), 561-596.
doi: 10.3934/nhm.2011.6.561. |
show all references
References:
[1] |
N. J. Armstrong, K. J. Painter and J. A. Sherratt,
A continuum approach to modelling cell-cell adhesion, J. Theoret. Biol., 243 (2006), 98-113.
doi: 10.1016/j.jtbi.2006.05.030. |
[2] |
V. Bitsouni and R. Eftimie,
Non-local parabolic and hyperbolic models for cell polarisation in heterogeneous cancer cell populations, Bull. Math. Biol., 80 (2018), 2600-2632.
doi: 10.1007/s11538-018-0477-4. |
[3] |
A. Buttenschön, Integro-partial Differential Equation Models for Cell-cell Adhesion and its Application, Ph.D thesis, University of Alberta, 2018. |
[4] |
A. Buttenschön and T. Hillen, Non-local adhesion models for microorganisms on bounded domains, SIAM J. Appl. Math., 80(1), (2020) 382–401.
doi: 10.1137/19M1250315. |
[5] |
A. Buttenschön and T. Hillen, Non-local cell adhesion models: Steady states and bifurcations, 2020, arXiv: 2001.00286. |
[6] |
A. Buttenschön, T. Hillen, A. Gerisch and K. J. Painter,
A space-jump derivation for non-local models of cell-cell adhesion and non-local chemotaxis, J. Math. Biol., 76 (2018), 429-456.
doi: 10.1007/s00285-017-1144-3. |
[7] |
J. Carrillo, F. Hoffmann and R. Eftimie,
Non-local kinetic and macroscopic models for self-organised animal aggregations, Kin. Rel. Models, 8 (2015), 413-441.
doi: 10.3934/krm.2015.8.413. |
[8] |
A. Colombi, M. Scianna and L. Preziosi,
Coherent modelling switch between pointwise and distributed representations of cell aggregates, J. Math. Biol., 74 (2017), 783-808.
doi: 10.1007/s00285-016-1042-0. |
[9] |
A. Colombi, M. Scianna and A. Tosin,
Differentiated cell behavior: A multiscale approach using measure theory, J. Math. Biol., 71 (2015), 1049-1079.
doi: 10.1007/s00285-014-0846-z. |
[10] |
R. Eftimie,
Hyperbolic and kinetic models for self-organized biological aggregations and movement: A brief review, J. Math. Biol., 65 (2012), 35-75.
doi: 10.1007/s00285-011-0452-2. |
[11] |
R. Eftimie, G. de Vries, M. A Lewis and F. Lutscher,
Modeling group formation and activity patterns in self-organizing collectives of individuals, Bull. Math. Biol., 69 (2007), 1537-1565.
doi: 10.1007/s11538-006-9175-8. |
[12] |
R. Eftimie, M. Perez and P.-L. Buono,
Pattern formation in a nonlocal mathematical model for the multiple roles of the TGF-$\beta$ pathway in tumour dynamics, Math. Biosci., 289 (2017), 96-115.
doi: 10.1016/j.mbs.2017.05.003. |
[13] |
R. Eftimie, G. Vries and M. Lewis,
Complex spatial group patterns result from different animal communication mechanisms, PNAS USA, 104 (2007), 6974-6979.
doi: 10.1073/pnas.0611483104. |
[14] |
F. Filbet and K. Yang, Numerical simulation of kinetic models for chemotaxis, SIAM J. Sci. Comput., 36 (2014), B348–B366.
doi: 10.1137/130910208. |
[15] |
T. Hillen, K. J. Painter and C. Schmeiser,
Global existence for chemotaxis with finite sampling radius, Discr. Cont. Dyn. Sys. - B, 7 (2007), 125-144.
doi: 10.3934/dcdsb.2007.7.125. |
[16] |
N. Loy and L. Preziosi,
Modelling physical limits of migration by a kinetic model with non-local sensing, J. Math. Biol., 80 (2020), 1759-1801.
doi: 10.1007/s00285-020-01479-w. |
[17] |
N. Loy and L. Preziosi,
Kinetic models with non-local sensing determining cell polarization and speed according to independent cues, J. Math. Biol., 80 (2019), 373-421.
doi: 10.1007/s00285-019-01411-x. |
[18] |
H. G. Othmer, S. R. Dunbar and W. Alt,
Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.
doi: 10.1007/BF00277392. |
[19] |
H. Othmer and T. Hillen,
The diffusion limit of transport equations Ⅱ: Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250.
doi: 10.1137/S0036139900382772. |
[20] |
J. K. Painter and T. Hillen,
Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (4) (2002), 501-543.
|
[21] |
K. J. Painter, N. J. Armstrong and J. A. Sherratt,
The impact of adhesion on cellular invasion processes in cancer and development, J. Theoret. Biol., 264 (2010), 1057-1067.
doi: 10.1016/j.jtbi.2010.03.033. |
[22] |
K. J. Painter, M. J. Bloomfield, J. A. Sherratt and A. Gerisch,
A nonlocal model for contact attraction and repulsion in heterogeneous cell populations, Bull. Math. Biol., 77 (2015), 1132-1165.
doi: 10.1007/s11538-015-0080-x. |
[23] |
B. Perthame and S. Yasuda,
Stiff-response-induced instability for chemotactic bacteria and flux-limited keller-segel equation, Nonlinearity, 31 (2018), 4065-4089.
doi: 10.1088/1361-6544/aac760. |
[24] |
R. G. Plaza,
Derivation of a bacterial nutrient-taxis system with doubly degenerate cross-diffusion as the parabolic limit of a velocity-jump process, J. Math. Biol., 78 (2019), 1681-1711.
doi: 10.1007/s00285-018-1323-x. |
[25] |
C. Schmeiser and A. Nouri,
Aggregated steady states of a kinetic model for chemotaxis, Kin. Rel. Models, 10 (2017), 313-327.
doi: 10.3934/krm.2017013. |
[26] |
D. W. Stroock,
Some stochastic processes which arise from a model of the motion of a bacterium, Zeit. Wahr. Ver. Geb., 28 (1974), 305-315.
doi: 10.1007/BF00532948. |
[27] |
A. Tosin and P. Frasca,
Existence and approximation of probability measure solutions to models of collective behaviors, Netw. Heter. Media, 6 (2011), 561-596.
doi: 10.3934/nhm.2011.6.561. |






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