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Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type
Pattern formation of the attraction-repulsion Keller-Segel system
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 |
References:
[1] |
J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. |
[2] |
H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. |
[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. |
[4] |
E. Budrene and H. Berg, Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633.
doi: 10.1038/349630a0. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
R. Firtel, Dictyostelium cinema,, http://people.biology.ucsd.edu/firtel/video.htm., ().
|
[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. |
[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. |
[14] |
R. E. Goldstein, Traveling-wave chemotaxis, Phys. Rev. Lett., 77 (1996), 775-778.
doi: 10.1103/PhysRevLett.77.775. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[31] |
J. Murray, Mathematical Biology I: An Introduction, 3rd edition, Springer, Berlin, 2002. |
[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. |
[33] |
W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. |
[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. |
[35] |
K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. |
[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. |
[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. |
[38] |
B. Perthame, Transport Equations in Biology, Birkhäuser Verlag, Basel, 2007. |
[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. |
[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. |
[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. |
[42] |
R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556.
doi: 10.1090/S0002-9947-1985-0808736-1. |
[43] |
J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531.
doi: 10.1006/jfan.1999.3483. |
[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. |
[45] |
W. Shi and D. Zusman, Sensory adaptation during negative chemotaxis in myxococcus xanthus, J. Bacteriol, 176 (1994), 1517-1520. |
[46] |
G. Simonett, Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations, 8 (1995), 753-796. |
[47] |
T. Suzuki, Free energy and self-interaction particles, Birkhäuser, Boston, 2005.
doi: 10.1007/0-8176-4436-9. |
[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. |
[49] |
A. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237 (1952), 37-72. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
show all references
References:
[1] |
J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. |
[2] |
H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. |
[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. |
[4] |
E. Budrene and H. Berg, Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633.
doi: 10.1038/349630a0. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
R. Firtel, Dictyostelium cinema,, http://people.biology.ucsd.edu/firtel/video.htm., ().
|
[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. |
[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. |
[14] |
R. E. Goldstein, Traveling-wave chemotaxis, Phys. Rev. Lett., 77 (1996), 775-778.
doi: 10.1103/PhysRevLett.77.775. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[31] |
J. Murray, Mathematical Biology I: An Introduction, 3rd edition, Springer, Berlin, 2002. |
[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. |
[33] |
W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. |
[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. |
[35] |
K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. |
[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. |
[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. |
[38] |
B. Perthame, Transport Equations in Biology, Birkhäuser Verlag, Basel, 2007. |
[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. |
[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. |
[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. |
[42] |
R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556.
doi: 10.1090/S0002-9947-1985-0808736-1. |
[43] |
J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531.
doi: 10.1006/jfan.1999.3483. |
[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. |
[45] |
W. Shi and D. Zusman, Sensory adaptation during negative chemotaxis in myxococcus xanthus, J. Bacteriol, 176 (1994), 1517-1520. |
[46] |
G. Simonett, Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations, 8 (1995), 753-796. |
[47] |
T. Suzuki, Free energy and self-interaction particles, Birkhäuser, Boston, 2005.
doi: 10.1007/0-8176-4436-9. |
[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. |
[49] |
A. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237 (1952), 37-72. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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