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December  2013, 18(10): 2597-2625. doi: 10.3934/dcdsb.2013.18.2597

Pattern formation of the attraction-repulsion Keller-Segel system

Ping Liu 1, , Junping Shi 2, and  Zhi-An Wang 3,

1. 

Y.Y. Tseng Functional Analysis Research Center and School of Mathematics Science, Harbin Normal University, Harbin, Heilongjiang, 150025
2. 

Department of Mathematics, College of William and Mary, Williamsburg, Virginia, 23187-8795, United States
3. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received  December 2012 Revised  June 2013 Published  October 2013

In this paper, the pattern formation of the attraction-repulsion Keller-Segel (ARKS) system is studied analytically and numerically. By the Hopf bifurcation theorem as well as the local and global bifurcation theorem, we rigorously establish the existence of time-periodic patterns and steady state patterns for the ARKS model in the full parameter regimes, which are identified by a linear stability analysis. We also show that when the chemotactic attraction is strong, a spiky steady state pattern can develop. Explicit time-periodic rippling wave patterns and spiky steady state patterns are obtained numerically by carefully selecting parameter values based on our theoretical results. The study in the paper asserts that chemotactic competitive interaction between attraction and repulsion can produce periodic patterns which are impossible for the chemotaxis model with a single chemical (either chemo-attractant or chemo-repellent).
Keywords: Attraction-repulsion chemotaxis, time-periodic patterns., steady state bifurcation, Hopf bifurcation.
Mathematics Subject Classification: 35K55, 35K57, 35K45, 35K50, 92C15, 92C17, 92B9.
Citation: Ping Liu, Junping Shi, Zhi-An Wang. Pattern formation of the attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2597-2625. doi: 10.3934/dcdsb.2013.18.2597
References:
[1]

J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. Google Scholar

[2]

H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75.  Google Scholar

[3]

H. Amann, Hopf bifurcation in quasilinear reaction-diffusion systems, in Delay differential equations and dynamical systems (Claremont, CA, 1990), vol. 1475 of Lecture Notes in Math., Springer, Berlin, 1991, 53-63. doi: 10.1007/BFb0083479.  Google Scholar

[4]

E. Budrene and H. Berg, Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633. doi: 10.1038/349630a0.  Google Scholar

[5]

M. Chaplain and A. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumor angiogenesis factor, IMA J. Math. Appl. Med., 10 (1993), 149-168. Google Scholar

[6]

M. Chuai, W. Zeng, X. Yang, V. Boychenko, J. Glazier and C. Weijer, Cell movement during chick primitive streak formation, Dev. Biol., 296 (2006), 137-149. doi: 10.1016/j.ydbio.2006.04.451.  Google Scholar

[7]

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

[8]

M. Crandall and P. Rabinowitz, The Hopf bifurcation theorem in infinite dimensions, Arch. Rational Mech. Anal., 67 (1977), 53-72. doi: 10.1007/BF00280827.  Google Scholar

[9]

G. Da Prato and A. Lunardi, Hopf bifurcation for fully nonlinear equations in Banach space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 315-329.  Google Scholar

[10]

A.-K. Drangeid, The principle of linearized stability for quasilinear parabolic evolution equations, Nonlinear Anal., 13 (1989), 1091-1113. doi: 10.1016/0362-546X(89)90097-7.  Google Scholar

[11]

R. Firtel, Dictyostelium cinema,, http://people.biology.ucsd.edu/firtel/video.htm., ().   Google Scholar

[12]

A. Gamba, D. Ambrosi, A. Coniglio, A. de Candia, S. Di Talia, E. Giraudo, G. Serini, L. Preziosi and F. Bussolino, Percolation, Morphogenesis, and Burgers dynamics in blood vessels Formation, Phys. Rev. Lett., 90 (2003), 118101. doi: 10.1103/PhysRevLett.90.118101.  Google Scholar

[13]

M. Gates, V. Coupe, E. Torres, R. Fricker-Gates and S. Dunnnett, Spatially and temporally restricted chemoattractant and repulsive cues direct the formation of the nigro-sriatal circuit, Euro. J. Neuroscience, 19 (2004), 831-844. Google Scholar

[14]

R. E. Goldstein, Traveling-wave chemotaxis, Phys. Rev. Lett., 77 (1996), 775-778. doi: 10.1103/PhysRevLett.77.775.  Google Scholar

[15]

P. Grindrod, J. D. 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

[16]

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

[17]

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

[18]

D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x.  Google Scholar

[19]

A. Huttenlocher and M. Poznansky, Reverse leukocyte migration can be attractive or repulsive, Trends in Cell Biology, 18 (2008), 298-306. doi: 10.1016/j.tcb.2008.04.001.  Google Scholar

[20]

O. Igoshin and D. Kaiser, Rippling of myxobacteria, Topics in biomathematics and related computational problems. Math. Biosci., 188 (2004), 221-233. doi: 10.1016/j.mbs.2003.04.001.  Google Scholar

[21]

O. Igoshin, R. Welch, D. Kaiser and G. Oster, Waves and aggregation patterns in myxobacteria, Proceedings of the National Academy of Sciences, 101 (2004), 4256-4261. doi: 10.1073/pnas.0400704101.  Google Scholar

[22]

Y. Kabeya and W.-M. Ni, Stationary Keller-Segel model with the linear sensitivity, Variational problems and related topics (Japanese) (Kyoto, 1997). RIMS Kokyuroku, 1025 (1998), 44-65.  Google Scholar

[23]

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

[24]

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

[25]

J. Liu and Z.-A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 31-41. doi: 10.1080/17513758.2011.571722.  Google Scholar

[26]

J. Liu, F. Yi and J. Wei, Multiple bifurcation analysis and spatiotemporal patterns in a 1-D Gierer-Meinhardt model of morphogenesis, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 1007-1025. doi: 10.1142/S0218127410026289.  Google Scholar

[27]

P. Liu, J. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal., 251 (2007), 573-600. doi: 10.1016/j.jfa.2007.06.015.  Google Scholar

[28]

M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: is there a connection? Bull. Math. Biol., 65 (2003), 693-730. doi: 10.1016/S0092-8240(03)00030-2.  Google Scholar

[29]

P. Maini, M. Myerscough, K. Winters and J. Murray, Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation, Bull. Math. Biol., 53 (1991), 701-719. Google Scholar

[30]

S. Martínez and W.-M. Ni, Periodic solutions of a $3 \times 3$ competitive system with cross-diffusion, Discrete Contin. Dyn. Syst., 15 (2006), 725-746. doi: 10.3934/dcds.2006.15.725.  Google Scholar

[31]

J. Murray, Mathematical Biology I: An Introduction, 3rd edition, Springer, Berlin, 2002.  Google Scholar

[32]

M. Myerscough, P. Maini and K. Painter, Pattern formation in a generalized chemotactic model, Bull. Math. Biol., 60 (1998), 1-26. doi: 10.1006/bulm.1997.0010.  Google Scholar

[33]

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

[34]

W.-M. Ni, Qualitative properties of solutions to elliptic problems, in Stationary partial differential equations. Vol. I, Handb. Differ. Equ., North-Holland, Amsterdam, 2004, 157-233. doi: 10.1016/S1874-5733(04)80005-6.  Google Scholar

[35]

K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.  Google Scholar

[36]

K. Painter, P. Maini and H. Othmer, Stripe formation in juvenile pomacanthus explained by a generalized turing mechanism with chemotaxis, Proc. Natl. Acad. Sci., 96 (1999), 5549-5554. doi: 10.1073/pnas.96.10.5549.  Google Scholar

[37]

K. Painter, P. Maini and H. Othmer, A chemotactic model for the advance and retreat of the primitive streak in avian development, Bull. Math. Biol., 62 (2000), 501-525. Google Scholar

[38]

B. Perthame, Transport Equations in Biology, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[39]

B. Perthame, C. Schmeiser, M. Tang and N. Vauchelet, Traveling plateaus for a hyperbolic keller-segel system with attraction and repulsion-existence and branching instabilitiesn, Nonlinearity, 24 (2011), 1253-1270. doi: 10.1088/0951-7715/24/4/012.  Google Scholar

[40]

G. Petter, H. Byrne, D. Mcelwain and J. Norbury, A model of wound healing and angiogenesis in soft tissue, Math. Biosci., 136 (2003), 35-63. Google Scholar

[41]

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

[42]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556. doi: 10.1090/S0002-9947-1985-0808736-1.  Google Scholar

[43]

J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531. doi: 10.1006/jfan.1999.3483.  Google Scholar

[44]

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

[45]

W. Shi and D. Zusman, Sensory adaptation during negative chemotaxis in myxococcus xanthus, J. Bacteriol, 176 (1994), 1517-1520. Google Scholar

[46]

G. Simonett, Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations, 8 (1995), 753-796.  Google Scholar

[47]

T. Suzuki, Free energy and self-interaction particles, Birkhäuser, Boston, 2005. doi: 10.1007/0-8176-4436-9.  Google Scholar

[48]

Y. Tao and Z.-A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36. doi: 10.1142/S0218202512500443.  Google Scholar

[49]

A. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237 (1952), 37-72. Google Scholar

[50]

J. Wang, J. Shi and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Differential Equations, 251 (2011), 1276-1304. doi: 10.1016/j.jde.2011.03.004.  Google Scholar

[51]

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

[52]

X. Wang and Y. Wu, Qualitative analysis on a chemotactic diffusion model for two species competing for a limited resource, Quart. Appl. Math., 60 (2002), 505-531.  Google Scholar

[53]

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

[54]

Z.-A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108, 13 pp. doi: 10.1063/1.2766864.  Google Scholar

[55]

Z.-A. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods Appl. Sci., 31 (2008), 45-70. doi: 10.1002/mma.898.  Google Scholar

[56]

M. J. Ward and J. Wei, Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model, J. Nonlinear Sci., 13 (2003), 209-264. doi: 10.1007/s00332-002-0531-z.  Google Scholar

[57]

R. Welch and D. Kaiser, Cell behavior in traveling wave patterns of myxobacteria, Proceedings of the National Academy of Sciences, 98 (2001), 14907-14912. doi: 10.1073/pnas.261574598.  Google Scholar

[58]

F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977. doi: 10.1016/j.jde.2008.10.024.  Google Scholar

show all references

References:
[1]

J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. Google Scholar

[2]

H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75.  Google Scholar

[3]

H. Amann, Hopf bifurcation in quasilinear reaction-diffusion systems, in Delay differential equations and dynamical systems (Claremont, CA, 1990), vol. 1475 of Lecture Notes in Math., Springer, Berlin, 1991, 53-63. doi: 10.1007/BFb0083479.  Google Scholar

[4]

E. Budrene and H. Berg, Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633. doi: 10.1038/349630a0.  Google Scholar

[5]

M. Chaplain and A. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumor angiogenesis factor, IMA J. Math. Appl. Med., 10 (1993), 149-168. Google Scholar

[6]

M. Chuai, W. Zeng, X. Yang, V. Boychenko, J. Glazier and C. Weijer, Cell movement during chick primitive streak formation, Dev. Biol., 296 (2006), 137-149. doi: 10.1016/j.ydbio.2006.04.451.  Google Scholar

[7]

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

[8]

M. Crandall and P. Rabinowitz, The Hopf bifurcation theorem in infinite dimensions, Arch. Rational Mech. Anal., 67 (1977), 53-72. doi: 10.1007/BF00280827.  Google Scholar

[9]

G. Da Prato and A. Lunardi, Hopf bifurcation for fully nonlinear equations in Banach space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 315-329.  Google Scholar

[10]

A.-K. Drangeid, The principle of linearized stability for quasilinear parabolic evolution equations, Nonlinear Anal., 13 (1989), 1091-1113. doi: 10.1016/0362-546X(89)90097-7.  Google Scholar

[11]

R. Firtel, Dictyostelium cinema,, http://people.biology.ucsd.edu/firtel/video.htm., ().   Google Scholar

[12]

A. Gamba, D. Ambrosi, A. Coniglio, A. de Candia, S. Di Talia, E. Giraudo, G. Serini, L. Preziosi and F. Bussolino, Percolation, Morphogenesis, and Burgers dynamics in blood vessels Formation, Phys. Rev. Lett., 90 (2003), 118101. doi: 10.1103/PhysRevLett.90.118101.  Google Scholar

[13]

M. Gates, V. Coupe, E. Torres, R. Fricker-Gates and S. Dunnnett, Spatially and temporally restricted chemoattractant and repulsive cues direct the formation of the nigro-sriatal circuit, Euro. J. Neuroscience, 19 (2004), 831-844. Google Scholar

[14]

R. E. Goldstein, Traveling-wave chemotaxis, Phys. Rev. Lett., 77 (1996), 775-778. doi: 10.1103/PhysRevLett.77.775.  Google Scholar

[15]

P. Grindrod, J. D. 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

[16]

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

[17]

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

[18]

D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x.  Google Scholar

[19]

A. Huttenlocher and M. Poznansky, Reverse leukocyte migration can be attractive or repulsive, Trends in Cell Biology, 18 (2008), 298-306. doi: 10.1016/j.tcb.2008.04.001.  Google Scholar

[20]

O. Igoshin and D. Kaiser, Rippling of myxobacteria, Topics in biomathematics and related computational problems. Math. Biosci., 188 (2004), 221-233. doi: 10.1016/j.mbs.2003.04.001.  Google Scholar

[21]

O. Igoshin, R. Welch, D. Kaiser and G. Oster, Waves and aggregation patterns in myxobacteria, Proceedings of the National Academy of Sciences, 101 (2004), 4256-4261. doi: 10.1073/pnas.0400704101.  Google Scholar

[22]

Y. Kabeya and W.-M. Ni, Stationary Keller-Segel model with the linear sensitivity, Variational problems and related topics (Japanese) (Kyoto, 1997). RIMS Kokyuroku, 1025 (1998), 44-65.  Google Scholar

[23]

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

[24]

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

[25]

J. Liu and Z.-A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 31-41. doi: 10.1080/17513758.2011.571722.  Google Scholar

[26]

J. Liu, F. Yi and J. Wei, Multiple bifurcation analysis and spatiotemporal patterns in a 1-D Gierer-Meinhardt model of morphogenesis, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 1007-1025. doi: 10.1142/S0218127410026289.  Google Scholar

[27]

P. Liu, J. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal., 251 (2007), 573-600. doi: 10.1016/j.jfa.2007.06.015.  Google Scholar

[28]

M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: is there a connection? Bull. Math. Biol., 65 (2003), 693-730. doi: 10.1016/S0092-8240(03)00030-2.  Google Scholar

[29]

P. Maini, M. Myerscough, K. Winters and J. Murray, Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation, Bull. Math. Biol., 53 (1991), 701-719. Google Scholar

[30]

S. Martínez and W.-M. Ni, Periodic solutions of a $3 \times 3$ competitive system with cross-diffusion, Discrete Contin. Dyn. Syst., 15 (2006), 725-746. doi: 10.3934/dcds.2006.15.725.  Google Scholar

[31]

J. Murray, Mathematical Biology I: An Introduction, 3rd edition, Springer, Berlin, 2002.  Google Scholar

[32]

M. Myerscough, P. Maini and K. Painter, Pattern formation in a generalized chemotactic model, Bull. Math. Biol., 60 (1998), 1-26. doi: 10.1006/bulm.1997.0010.  Google Scholar

[33]

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

[34]

W.-M. Ni, Qualitative properties of solutions to elliptic problems, in Stationary partial differential equations. Vol. I, Handb. Differ. Equ., North-Holland, Amsterdam, 2004, 157-233. doi: 10.1016/S1874-5733(04)80005-6.  Google Scholar

[35]

K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.  Google Scholar

[36]

K. Painter, P. Maini and H. Othmer, Stripe formation in juvenile pomacanthus explained by a generalized turing mechanism with chemotaxis, Proc. Natl. Acad. Sci., 96 (1999), 5549-5554. doi: 10.1073/pnas.96.10.5549.  Google Scholar

[37]

K. Painter, P. Maini and H. Othmer, A chemotactic model for the advance and retreat of the primitive streak in avian development, Bull. Math. Biol., 62 (2000), 501-525. Google Scholar

[38]

B. Perthame, Transport Equations in Biology, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[39]

B. Perthame, C. Schmeiser, M. Tang and N. Vauchelet, Traveling plateaus for a hyperbolic keller-segel system with attraction and repulsion-existence and branching instabilitiesn, Nonlinearity, 24 (2011), 1253-1270. doi: 10.1088/0951-7715/24/4/012.  Google Scholar

[40]

G. Petter, H. Byrne, D. Mcelwain and J. Norbury, A model of wound healing and angiogenesis in soft tissue, Math. Biosci., 136 (2003), 35-63. Google Scholar

[41]

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

[42]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556. doi: 10.1090/S0002-9947-1985-0808736-1.  Google Scholar

[43]

J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531. doi: 10.1006/jfan.1999.3483.  Google Scholar

[44]

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

[45]

W. Shi and D. Zusman, Sensory adaptation during negative chemotaxis in myxococcus xanthus, J. Bacteriol, 176 (1994), 1517-1520. Google Scholar

[46]

G. Simonett, Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations, 8 (1995), 753-796.  Google Scholar

[47]

T. Suzuki, Free energy and self-interaction particles, Birkhäuser, Boston, 2005. doi: 10.1007/0-8176-4436-9.  Google Scholar

[48]

Y. Tao and Z.-A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36. doi: 10.1142/S0218202512500443.  Google Scholar

[49]

A. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237 (1952), 37-72. Google Scholar

[50]

J. Wang, J. Shi and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Differential Equations, 251 (2011), 1276-1304. doi: 10.1016/j.jde.2011.03.004.  Google Scholar

[51]

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

[52]

X. Wang and Y. Wu, Qualitative analysis on a chemotactic diffusion model for two species competing for a limited resource, Quart. Appl. Math., 60 (2002), 505-531.  Google Scholar

[53]

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

[54]

Z.-A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108, 13 pp. doi: 10.1063/1.2766864.  Google Scholar

[55]

Z.-A. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods Appl. Sci., 31 (2008), 45-70. doi: 10.1002/mma.898.  Google Scholar

[56]

M. J. Ward and J. Wei, Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model, J. Nonlinear Sci., 13 (2003), 209-264. doi: 10.1007/s00332-002-0531-z.  Google Scholar

[57]

R. Welch and D. Kaiser, Cell behavior in traveling wave patterns of myxobacteria, Proceedings of the National Academy of Sciences, 98 (2001), 14907-14912. doi: 10.1073/pnas.261574598.  Google Scholar

[58]

F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977. doi: 10.1016/j.jde.2008.10.024.  Google Scholar

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Shijie Shi, Zhengrong Liu, Hai-Yang Jin. Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source. Kinetic & Related Models, 2017, 10 (3) : 855-878. doi: 10.3934/krm.2017034
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