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November  2013, 18(9): 2457-2485. doi: 10.3934/dcdsb.2013.18.2457

A study on the positive nonconstant steady states of nonlocal chemotaxis systems

Tian Xiang 1,

1. 

Department of Mathematics, Tulane University, New Orleans, LA 70118, United States

Received  January 2013 Revised  August 2013 Published  September 2013

In one spatial dimension, we perform global bifurcation analysis on a general nonlocal Keller-Segel chemotaxis model, showing that positive nonconstant steady states exist, if the chemotactic coefficient $\chi$ is larger than a bifurcation value $\overline{\chi}_1$, which is expressible in terms of the parameters and the nonlocal sampling radius in the model. We then show that the positive solutions of the nonlocal model converge at least in $C^2([0,l])\times C^2([0,l])$ to that of the corresponding ``local" model as the nonlocal sampling radius $\rho\rightarrow 0+$. Finally, we use Helly's compactness theorem to establish the profiles of these steady states, when the ratio of the chemotactic coefficient and the cell diffusion rate is large and the nonlocal sampling radius is small, exhibiting whether they are either spiky, of transition layer structure or just flat everywhere. Our results supply understandings on how the biological parameters affect pattern formation for the nonlocal model. In the limit of $\rho\rightarrow 0+$, our results agree with those of local models studied in Wang and Xu [29].
Keywords: Chemotaxis, steady states, transition layer., nonlocal gradient, bifurcation, spike.
Mathematics Subject Classification: Primary: 92C17, 35B32; Secondary: 35J67, 35B41, 35B3.
Citation: Tian Xiang. A study on the positive nonconstant steady states of nonlocal chemotaxis systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2457-2485. doi: 10.3934/dcdsb.2013.18.2457
References:
[1]

M. Burger, Y. Dolak-Struss and C. Schmeiser, Asymptotic analysis of an advection-dominated chemotaxis model in multiple spatial dimensions, Communications in Mathematical Sciences, 6 (2008), 1-28. doi: 10.4310/CMS.2008.v6.n1.a1.  Google Scholar

[2]

X. Chen, J. Hao, X. Wang, Y. Wu and Y. Zhang, On Keller-Segel's minimal chemotaxis model,, in process., ().   Google Scholar

[3]

A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinetic and Related Models (KRM), 5 (2012), 51-95. doi: 10.3934/krm.2012.5.51.  Google Scholar

[4]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[5]

Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity, SIAM Journal on Applied Mathematics, 66 (2005), 286-308. doi: 10.1137/040612841.  Google Scholar

[6]

A. Fasano, A. Mancini and M. Primicerio, Equilibrium of two populations subject to chemotaxis, Mathematical Models and Methods in Applied Sciences, 14 (2004), 503-533. doi: 10.1142/S0218202504003337.  Google Scholar

[7]

P. Grindrod, J. Murray and S. Sinha, Steady-state spatial patterns in a cell-chemotaxis model, IMA J. Math. Appl. Med. Biol., 6 (1989), 69-79. doi: 10.1093/imammb/6.2.69.  Google Scholar

[8]

T. Hillen, K. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius, Discrete and Continuous Dynamical Systems - Series B (DCDS-B), 7 (2007), 125-144. doi: 10.3934/dcdsb.2007.7.125.  Google Scholar

[9]

T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

E. Keller and L. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[14]

C. Lin, W. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, Journal of Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7.  Google Scholar

[15]

M. Myerscough, P. Maini and K. Painter, Pattern formation in a generalized chemo-tactic model, Bull. Math. Biol., 60 (1998), 1-26. Google Scholar

[16]

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

[17]

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

[18]

H. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations, SIAM Journal on Applied Mathematics, 62 (2002), 1222-1250. doi: 10.1137/S0036139900382772.  Google Scholar

[19]

K. Painter, H. Othmer and P. Maini, Stripe formation in juvenile pomacanthus via chemotactic response to a reaction-diffusion mechanism, Proc. Natl. Acad. Sci. USA, 96 (1999), 5549-5554. Google Scholar

[20]

C. Patlak, Random walk with persistence and external bias, Bulletin Mathematical Biophysics, 15 (1953), 311-338. doi: 10.1007/BF02476407.  Google Scholar

[21]

J. Pejsachowicz and P. Rabier, Degree theory for C1 Fredholm mappings of index 0, Journal d'Analyse Mathematique, 76 (1998), 289-319. doi: 10.1007/BF02786939.  Google Scholar

[22]

A. Potapov, T. Hillen, Metastability in chemotaxis models, Journal of Dynamics and Differential Equations, 17 (2005), 293-330. doi: 10.1007/s10884-005-2938-3.  Google Scholar

[23]

P. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math., 3 (1973), 161-202. doi: 10.1216/RMJ-1973-3-2-161.  Google Scholar

[24]

P. Rabinowitz, Some global results for nonlinear eigenvalue problems, Journal of Functional Analysis, 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[25]

R. Schaaf, Stationary solutions of chemotaxis systems, Transactions of the American Mathematical Society, 292 (1985), 531-556. doi: 10.1090/S0002-9947-1985-0808736-1.  Google Scholar

[26]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, Journal of Differential Equations, 246 (2009), 2788-2812. doi: 10.1016/j.jde.2008.09.009.  Google Scholar

[27]

R. Tyson, S. R. Lubkin and J. D. Murray, A minimal mechanism for bacterial pattern formation, Proceedings of the Royal Society, 266 (1999), 299-304. doi: 10.1098/rspb.1999.0637.  Google Scholar

[28]

X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility and chemotaxis and dynamics, SIAM Journal on Mathematical Analysis, 31 (2000), 535-560. doi: 10.1137/S0036141098339897.  Google Scholar

[29]

X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation method and Helly's compactness theorem, Journal of Mathematical Biology, 66 (2013), 1241-1266. doi: 10.1007/s00285-012-0533-x.  Google Scholar

[30]

J. Wei, Existence and stability of spikes for the Gierer-Meinhardt system, Handbook of Differential Equations: Stationary Partial Differential Equations, 5 (2008), 487-585. doi: 10.1016/S1874-5733(08)80013-7.  Google Scholar

show all references

References:
[1]

M. Burger, Y. Dolak-Struss and C. Schmeiser, Asymptotic analysis of an advection-dominated chemotaxis model in multiple spatial dimensions, Communications in Mathematical Sciences, 6 (2008), 1-28. doi: 10.4310/CMS.2008.v6.n1.a1.  Google Scholar

[2]

X. Chen, J. Hao, X. Wang, Y. Wu and Y. Zhang, On Keller-Segel's minimal chemotaxis model,, in process., ().   Google Scholar

[3]

A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinetic and Related Models (KRM), 5 (2012), 51-95. doi: 10.3934/krm.2012.5.51.  Google Scholar

[4]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[5]

Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity, SIAM Journal on Applied Mathematics, 66 (2005), 286-308. doi: 10.1137/040612841.  Google Scholar

[6]

A. Fasano, A. Mancini and M. Primicerio, Equilibrium of two populations subject to chemotaxis, Mathematical Models and Methods in Applied Sciences, 14 (2004), 503-533. doi: 10.1142/S0218202504003337.  Google Scholar

[7]

P. Grindrod, J. Murray and S. Sinha, Steady-state spatial patterns in a cell-chemotaxis model, IMA J. Math. Appl. Med. Biol., 6 (1989), 69-79. doi: 10.1093/imammb/6.2.69.  Google Scholar

[8]

T. Hillen, K. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius, Discrete and Continuous Dynamical Systems - Series B (DCDS-B), 7 (2007), 125-144. doi: 10.3934/dcdsb.2007.7.125.  Google Scholar

[9]

T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

E. Keller and L. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[14]

C. Lin, W. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, Journal of Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7.  Google Scholar

[15]

M. Myerscough, P. Maini and K. Painter, Pattern formation in a generalized chemo-tactic model, Bull. Math. Biol., 60 (1998), 1-26. Google Scholar

[16]

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

[17]

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

[18]

H. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations, SIAM Journal on Applied Mathematics, 62 (2002), 1222-1250. doi: 10.1137/S0036139900382772.  Google Scholar

[19]

K. Painter, H. Othmer and P. Maini, Stripe formation in juvenile pomacanthus via chemotactic response to a reaction-diffusion mechanism, Proc. Natl. Acad. Sci. USA, 96 (1999), 5549-5554. Google Scholar

[20]

C. Patlak, Random walk with persistence and external bias, Bulletin Mathematical Biophysics, 15 (1953), 311-338. doi: 10.1007/BF02476407.  Google Scholar

[21]

J. Pejsachowicz and P. Rabier, Degree theory for C1 Fredholm mappings of index 0, Journal d'Analyse Mathematique, 76 (1998), 289-319. doi: 10.1007/BF02786939.  Google Scholar

[22]

A. Potapov, T. Hillen, Metastability in chemotaxis models, Journal of Dynamics and Differential Equations, 17 (2005), 293-330. doi: 10.1007/s10884-005-2938-3.  Google Scholar

[23]

P. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math., 3 (1973), 161-202. doi: 10.1216/RMJ-1973-3-2-161.  Google Scholar

[24]

P. Rabinowitz, Some global results for nonlinear eigenvalue problems, Journal of Functional Analysis, 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[25]

R. Schaaf, Stationary solutions of chemotaxis systems, Transactions of the American Mathematical Society, 292 (1985), 531-556. doi: 10.1090/S0002-9947-1985-0808736-1.  Google Scholar

[26]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, Journal of Differential Equations, 246 (2009), 2788-2812. doi: 10.1016/j.jde.2008.09.009.  Google Scholar

[27]

R. Tyson, S. R. Lubkin and J. D. Murray, A minimal mechanism for bacterial pattern formation, Proceedings of the Royal Society, 266 (1999), 299-304. doi: 10.1098/rspb.1999.0637.  Google Scholar

[28]

X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility and chemotaxis and dynamics, SIAM Journal on Mathematical Analysis, 31 (2000), 535-560. doi: 10.1137/S0036141098339897.  Google Scholar

[29]

X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation method and Helly's compactness theorem, Journal of Mathematical Biology, 66 (2013), 1241-1266. doi: 10.1007/s00285-012-0533-x.  Google Scholar

[30]

J. Wei, Existence and stability of spikes for the Gierer-Meinhardt system, Handbook of Differential Equations: Stationary Partial Differential Equations, 5 (2008), 487-585. doi: 10.1016/S1874-5733(08)80013-7.  Google Scholar

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