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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

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.

Citation: Xin Xu. Existence of monotone positive solutions of a neighbour based chemotaxis model and aggregation phenomenon. Communications on Pure & 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.  Google Scholar

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H. B. ChenT. 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.  Google Scholar

[3]

X. F. ChenJ. H. HaoX. F. WangY. 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.  Google Scholar

[4]

A. ChertockA. KurganovX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

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

[12]

H. Y. JinY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

C. S. LinW. 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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[27]

B. SleemanM. 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.  Google Scholar

[28]

J. Smith-RobergeD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[33]

L. XinX. F. ChenM. X. WangC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

H. B. ChenT. 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.  Google Scholar

[3]

X. F. ChenJ. H. HaoX. F. WangY. 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.  Google Scholar

[4]

A. ChertockA. KurganovX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

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

[12]

H. Y. JinY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

C. S. LinW. 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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[27]

B. SleemanM. 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.  Google Scholar

[28]

J. Smith-RobergeD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[33]

L. XinX. F. ChenM. X. WangC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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
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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