# American Institute of Mathematical Sciences

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

 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].
Citation: Tian Xiang. A study on the positive nonconstant steady states of nonlocal chemotaxis systems. Discrete and 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. [2] X. Chen, J. Hao, X. Wang, Y. Wu and Y. Zhang, On Keller-Segel's minimal chemotaxis model,, in process., (). [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. [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. [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. [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. [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. [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. [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. [10] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I, Jahresber DMV, 105 (2003), 103-165. [11] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences II, Jahresber DMV, 106 (2004), 51-69. [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. [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. [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. [15] M. Myerscough, P. Maini and K. Painter, Pattern formation in a generalized chemo-tactic model, Bull. Math. Biol., 60 (1998), 1-26. [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. [17] W. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. [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. [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. [20] C. Patlak, Random walk with persistence and external bias, Bulletin Mathematical Biophysics, 15 (1953), 311-338. doi: 10.1007/BF02476407. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [2] X. Chen, J. Hao, X. Wang, Y. Wu and Y. Zhang, On Keller-Segel's minimal chemotaxis model,, in process., (). [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. [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. [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. [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. [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. [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. [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. [10] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I, Jahresber DMV, 105 (2003), 103-165. [11] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences II, Jahresber DMV, 106 (2004), 51-69. [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. [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. [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. [15] M. Myerscough, P. Maini and K. Painter, Pattern formation in a generalized chemo-tactic model, Bull. Math. Biol., 60 (1998), 1-26. [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. [17] W. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. [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. [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. [20] C. Patlak, Random walk with persistence and external bias, Bulletin Mathematical Biophysics, 15 (1953), 311-338. doi: 10.1007/BF02476407. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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