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September  2020, 19(9): 4327-4348. doi: 10.3934/cpaa.2020195

Existence of monotone positive solutions of a neighbour based chemotaxis model and aggregation phenomenon

Xin Xu

Faculty of Science and Technology, University of Macau, Macau, China, co. Department of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, China

Received  September 2019 Revised  February 2020 Published  June 2020

Fund Project: The author is supported by NSFC-11671190, NNSFC11571390, Macao S.A.R. (FDCT 038/ 2017/A1)

Using bifurcation theory, we prove the existence of spiky steady states and investigate the stability of bifurcating solutions of the one-dimensional continuous neighbour based chemotaxis model, in which the one-step jumping probability rate of cells is determined only by the chemoattractant concentration at the destination. These spiky steady states are crucial when we model cell aggregation, the most important phenomenon in chemotaxis.

Keywords: Chemotaxis, neighbour based, spiky steady states, bifurcation, existence.
Mathematics Subject Classification: Primary: 35A01, 35B40, 35K57, 35Q92; Secondary: 92C15, 92C17.
Citation: Xin Xu. Existence of monotone positive solutions of a neighbour based chemotaxis model and aggregation phenomenon. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4327-4348. doi: 10.3934/cpaa.2020195
References:
[1]

J. Ahn and C. Yoon, Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing, Nonlinearity, 32(4) (2019), 1327-1351.  doi: 10.1088/1361-6544/aaf513.

[2]

H. B. Chen, T. Bo and Q. Wang, Existence and stability of nonconstant positive steady states of morphogenesis models, Math. Meth. Appl. Sci., 38 (2015), 3833-3850.  doi: 10.1002/mma.3321.

[3]

X. F. Chen, J. H. Hao, X. F. Wang, Y. P. Wu and Y. J. Zhang, Stability of spiky solution of Keller-Segel's minimal chemotaxis model, J. Differ. Equ., 257 (2014), 3102-3134.  doi: 10.1016/j.jde.2014.06.008.

[4]

A. Chertock, A. Kurganov, X. F. Wang and Y. P. Wu, On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51-95.  doi: 10.3934/krm.2012.5.51.

[5]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 311-338.  doi: 10.1016/0025-5564(81)90055-9.

[6]

M. Crandall and P. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch.Rational Mech.Anal, 52 (1973), 161-180.  doi: 10.1007/bf00282325.

[7]

P. M. Fitzpatrick and J. Pejsachowicz, Parity and generilized multiplicity, Trans. Amer. Math. Soc., 326 (1991), 281-305.  doi: 10.2307/2001865.

[8]

C. F. Gui and J. C. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differ. Equ, 158 (1999), 1-27.  doi: 10.1016/s0022-0396(99)80016-3.

[9]

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.

[10]

D. Horstmann, From 1970 untill present: the Keller-Segel model in chemotaxis, Jahresber DMV, 105 (2003), 103-165. 

[11]

D. Horstmann, From 1970 untill present: the Keller-Segel model in chemotaxis, Jahresber DMV, 106 (2004), 51-69. 

[12]

H. Y. Jin, Y. J. Kim and Z. A. Wang, Boundedness, stabilization, and pattern forma-tion driven by density-suppressed motility, SIAM J. Appl. Math., 78(3) (2018), 1632-1657.  doi: 10.1137/17M1144647.

[13]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[14]

E. Keller and L. Segel, Models for chemotaxis, Kinet. J. Theor. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.

[15]

H. C. Li, Spiky steady states of a chemotaxis system with singular sensitivity, J. Dyn. Differ. Equ., 30 (2018), 1775-1795.  doi: 10.1007/s10884-017-9621-3.

[16]

C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differ. Equ., 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.

[17]

M. Ma, R. Wang and Z. A. Wang, Stationary and non-stationary patterns of the density-suppressed motility model, Physica D, (2019). doi: 10.1016/j.physd.2019.132259.

[18]

W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices. Amer. Math. Soc., 45 (1998), 9-18. 

[19]

W. M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke. Math. J., 70 (1993), 247-281.  doi: 10.1215/S0012-7094-93-07004-4.

[20]

H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: the ABCs of taxis in reinforced random walks, SIAM. J. Math. Anal. Vol., 57 (1997), 9-18.  doi: 10.1137/s0036139995288976.

[21]

K. J. Painter and J. A. Sherratt, Modelling the movement of interacting cell populations, J. Theor. Biol., 225 (2003), 327-339.  doi: 10.1016/s0022-5193(03)00258-3.

[22]

J. Pejsachowicz and P. J. Rabier, Degree theory for $C^1$ Fredholm mappings of index $0$, J. Anal. Math., 76 (1998), 289-319.  doi: 10.1007/BF02786939.

[23]

A. B. Potapov and T. Hillen, Metastability in chemotaxis models, J. Dyn. Differ. Equ., 17 (2005), 293-330.  doi: 10.1007/s10884-005-2938-3.

[24]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.

[25]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc, 292 (1985), 531-556.  doi: 10.2307/2000228.

[26]

J. P. Shi and X. F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differ. Equ., 246 (2009), 2788-2812.  doi: 10.1016/j.jde.2008.09.009.

[27]

B. Sleeman, M. Ward and J. C. Wei, The existence and stability of spike patterns in a chemotaxis model, SIAM J. Appl. Math., 65 (2005), 790-817.  doi: 10.2307/4096197.

[28]

J. Smith-Roberge, D. Iron and T. Kolokolnikov, Pattern formation in bacterial colonies with density-dependent diffusion, Eur. J. Appl. Math., 30 (2019), 196-218.  doi: 10.1017/S0956792518000013.

[29]

X. F Wang, Qualitative behavior of solutions of chemotactic diffusion systems: effects of motility and chemotaxis and dynamics, SIAM. J. Math. Anal., 31 (2000), 535-560.  doi: 10.1137/S0036141098339897.

[30]

Q. Wang, J. D. Yan and C. Y. Gai, Qualitative analysis of stationary Keller-Segel chemotaxis models with logistic growth, Z. Angew. Math. Phys., 67 (2016). doi: 10.1007/s00033-016-0648-9.

[31]

Q. Wang and X. F. Wang, Steady states and their qualitative properties of several classes of Keller-Segel models (in Chinese), Sci. Sin. Math., 49 (2019), 1911-1946. 

[32]

X. F Wang and Q. Xu, Spiky and transition layer steady state of chemotaxis systems via global bifurcation and Helly's compactness theorem, J. Math. Biol., 66 (2013), 1241-1266.  doi: 10.1007/s00285-012-0533-x.

[33]

L. Xin, X. F. Chen, M. X. Wang, C. Qin and Y. J. Zhang, Existence, uniqueness, and stability of bubble solutions of a chemotaxis model, Discrete Contin. Dyn. Syst., 36 (2016), 805-832.  doi: 10.3934/dcds.2016.36.805.

[34]

Q. Xu, The Stability of Bifurcation Steady State of Serveral Classes of Chemotaxis Systems, Discrete Contin. Dyn. Syst. Ser. B., 20 (2015), 231-248.  doi: 10.3934/dcdsb.2015.20.231.

[35]

C. Yoon and Y. J. Kim, Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion, Acta Appl. Math., 149 (2017), 101-123.  doi: 10.1007/s10440-016-0089-7.

show all references

References:
[1]

J. Ahn and C. Yoon, Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing, Nonlinearity, 32(4) (2019), 1327-1351.  doi: 10.1088/1361-6544/aaf513.

[2]

H. B. Chen, T. Bo and Q. Wang, Existence and stability of nonconstant positive steady states of morphogenesis models, Math. Meth. Appl. Sci., 38 (2015), 3833-3850.  doi: 10.1002/mma.3321.

[3]

X. F. Chen, J. H. Hao, X. F. Wang, Y. P. Wu and Y. J. Zhang, Stability of spiky solution of Keller-Segel's minimal chemotaxis model, J. Differ. Equ., 257 (2014), 3102-3134.  doi: 10.1016/j.jde.2014.06.008.

[4]

A. Chertock, A. Kurganov, X. F. Wang and Y. P. Wu, On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51-95.  doi: 10.3934/krm.2012.5.51.

[5]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 311-338.  doi: 10.1016/0025-5564(81)90055-9.

[6]

M. Crandall and P. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch.Rational Mech.Anal, 52 (1973), 161-180.  doi: 10.1007/bf00282325.

[7]

P. M. Fitzpatrick and J. Pejsachowicz, Parity and generilized multiplicity, Trans. Amer. Math. Soc., 326 (1991), 281-305.  doi: 10.2307/2001865.

[8]

C. F. Gui and J. C. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differ. Equ, 158 (1999), 1-27.  doi: 10.1016/s0022-0396(99)80016-3.

[9]

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.

[10]

D. Horstmann, From 1970 untill present: the Keller-Segel model in chemotaxis, Jahresber DMV, 105 (2003), 103-165. 

[11]

D. Horstmann, From 1970 untill present: the Keller-Segel model in chemotaxis, Jahresber DMV, 106 (2004), 51-69. 

[12]

H. Y. Jin, Y. J. Kim and Z. A. Wang, Boundedness, stabilization, and pattern forma-tion driven by density-suppressed motility, SIAM J. Appl. Math., 78(3) (2018), 1632-1657.  doi: 10.1137/17M1144647.

[13]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[14]

E. Keller and L. Segel, Models for chemotaxis, Kinet. J. Theor. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.

[15]

H. C. Li, Spiky steady states of a chemotaxis system with singular sensitivity, J. Dyn. Differ. Equ., 30 (2018), 1775-1795.  doi: 10.1007/s10884-017-9621-3.

[16]

C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differ. Equ., 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.

[17]

M. Ma, R. Wang and Z. A. Wang, Stationary and non-stationary patterns of the density-suppressed motility model, Physica D, (2019). doi: 10.1016/j.physd.2019.132259.

[18]

W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices. Amer. Math. Soc., 45 (1998), 9-18. 

[19]

W. M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke. Math. J., 70 (1993), 247-281.  doi: 10.1215/S0012-7094-93-07004-4.

[20]

H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: the ABCs of taxis in reinforced random walks, SIAM. J. Math. Anal. Vol., 57 (1997), 9-18.  doi: 10.1137/s0036139995288976.

[21]

K. J. Painter and J. A. Sherratt, Modelling the movement of interacting cell populations, J. Theor. Biol., 225 (2003), 327-339.  doi: 10.1016/s0022-5193(03)00258-3.

[22]

J. Pejsachowicz and P. J. Rabier, Degree theory for $C^1$ Fredholm mappings of index $0$, J. Anal. Math., 76 (1998), 289-319.  doi: 10.1007/BF02786939.

[23]

A. B. Potapov and T. Hillen, Metastability in chemotaxis models, J. Dyn. Differ. Equ., 17 (2005), 293-330.  doi: 10.1007/s10884-005-2938-3.

[24]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.

[25]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc, 292 (1985), 531-556.  doi: 10.2307/2000228.

[26]

J. P. Shi and X. F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differ. Equ., 246 (2009), 2788-2812.  doi: 10.1016/j.jde.2008.09.009.

[27]

B. Sleeman, M. Ward and J. C. Wei, The existence and stability of spike patterns in a chemotaxis model, SIAM J. Appl. Math., 65 (2005), 790-817.  doi: 10.2307/4096197.

[28]

J. Smith-Roberge, D. Iron and T. Kolokolnikov, Pattern formation in bacterial colonies with density-dependent diffusion, Eur. J. Appl. Math., 30 (2019), 196-218.  doi: 10.1017/S0956792518000013.

[29]

X. F Wang, Qualitative behavior of solutions of chemotactic diffusion systems: effects of motility and chemotaxis and dynamics, SIAM. J. Math. Anal., 31 (2000), 535-560.  doi: 10.1137/S0036141098339897.

[30]

Q. Wang, J. D. Yan and C. Y. Gai, Qualitative analysis of stationary Keller-Segel chemotaxis models with logistic growth, Z. Angew. Math. Phys., 67 (2016). doi: 10.1007/s00033-016-0648-9.

[31]

Q. Wang and X. F. Wang, Steady states and their qualitative properties of several classes of Keller-Segel models (in Chinese), Sci. Sin. Math., 49 (2019), 1911-1946. 

[32]

X. F Wang and Q. Xu, Spiky and transition layer steady state of chemotaxis systems via global bifurcation and Helly's compactness theorem, J. Math. Biol., 66 (2013), 1241-1266.  doi: 10.1007/s00285-012-0533-x.

[33]

L. Xin, X. F. Chen, M. X. Wang, C. Qin and Y. J. Zhang, Existence, uniqueness, and stability of bubble solutions of a chemotaxis model, Discrete Contin. Dyn. Syst., 36 (2016), 805-832.  doi: 10.3934/dcds.2016.36.805.

[34]

Q. Xu, The Stability of Bifurcation Steady State of Serveral Classes of Chemotaxis Systems, Discrete Contin. Dyn. Syst. Ser. B., 20 (2015), 231-248.  doi: 10.3934/dcdsb.2015.20.231.

[35]

C. Yoon and Y. J. Kim, Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion, Acta Appl. Math., 149 (2017), 101-123.  doi: 10.1007/s10440-016-0089-7.

Figure 1.  On the horizontal axis is the average mass of chemical substance, which is $ v* $. On the vertical axis is the space of $ (u, v) $. Every point on the cure is a solution of (1.7). On the "upper", branch, the solutions are decreasing in $ \Omega $ and vice versa. $ \bar{v}_1^* $ is a positive constant and will be given in Section 2
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Figure 2.  $ D = \alpha = \beta = 1 $, $ u^* = 20 $, $ l = 2 $. This figure shows the graph of $ u $ and $ v $ at time $ 200 $. The $ u $ curve is red and the $ v $ curve is blue
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