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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

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).
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.

[2]

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

[3]

H. Amann, Hopf bifurcation in quasilinear reaction-diffusion systems,, in Delay differential equations and dynamical systems (Claremont, (1990), 53. doi: 10.1007/BFb0083479.

[4]

E. Budrene and H. Berg, Complex patterns formed by motile cells of Escherichia coli,, Nature, 349 (1991), 630. 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.

[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. doi: 10.1016/j.ydbio.2006.04.451.

[7]

M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321. 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. 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.

[10]

A.-K. Drangeid, The principle of linearized stability for quasilinear parabolic evolution equations,, Nonlinear Anal., 13 (1989), 1091. 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). 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.

[14]

R. E. Goldstein, Traveling-wave chemotaxis,, Phys. Rev. Lett., 77 (1996), 775. 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. 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. 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.

[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. 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. 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. 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. 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, 1025 (1998), 44.

[23]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability,, Journal of Theoretical Biology, 26 (1970), 399. 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. 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. 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. doi: 10.1142/S0218127410026289.

[27]

P. Liu, J. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations,, J. Funct. Anal., 251 (2007), 573. 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. 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.

[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. doi: 10.3934/dcds.2006.15.725.

[31]

J. Murray, Mathematical Biology I: An Introduction,, 3rd edition, (2002).

[32]

M. Myerscough, P. Maini and K. Painter, Pattern formation in a generalized chemotactic model,, Bull. Math. Biol., 60 (1998), 1. 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.

[34]

W.-M. Ni, Qualitative properties of solutions to elliptic problems,, in Stationary partial differential equations. Vol. I, (2004), 157. 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.

[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. 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.

[38]

B. Perthame, Transport Equations in Biology,, Birkhäuser Verlag, (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. 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.

[41]

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

[42]

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

[43]

J. Shi, Persistence and bifurcation of degenerate solutions,, J. Funct. Anal., 169 (1999), 494. 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. 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.

[46]

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

[47]

T. Suzuki, Free energy and self-interaction particles,, Birkhäuser, (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. doi: 10.1142/S0218202512500443.

[49]

A. Turing, The chemical basis of morphogenesis,, Philosophical Transactions of the Royal Society of London. Series B, 237 (1952), 37.

[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. 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. 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.

[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. 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). 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. 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. 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. 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. doi: 10.1016/j.jde.2008.10.024.

show all references

References:
[1]

J. Adler, Chemotaxis in bacteria,, Science, 153 (1966), 708.

[2]

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

[3]

H. Amann, Hopf bifurcation in quasilinear reaction-diffusion systems,, in Delay differential equations and dynamical systems (Claremont, (1990), 53. doi: 10.1007/BFb0083479.

[4]

E. Budrene and H. Berg, Complex patterns formed by motile cells of Escherichia coli,, Nature, 349 (1991), 630. 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.

[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. doi: 10.1016/j.ydbio.2006.04.451.

[7]

M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321. 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. 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.

[10]

A.-K. Drangeid, The principle of linearized stability for quasilinear parabolic evolution equations,, Nonlinear Anal., 13 (1989), 1091. 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). 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.

[14]

R. E. Goldstein, Traveling-wave chemotaxis,, Phys. Rev. Lett., 77 (1996), 775. 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. 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. 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.

[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. 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. 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. 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. 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, 1025 (1998), 44.

[23]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability,, Journal of Theoretical Biology, 26 (1970), 399. 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. 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. 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. doi: 10.1142/S0218127410026289.

[27]

P. Liu, J. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations,, J. Funct. Anal., 251 (2007), 573. 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. 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.

[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. doi: 10.3934/dcds.2006.15.725.

[31]

J. Murray, Mathematical Biology I: An Introduction,, 3rd edition, (2002).

[32]

M. Myerscough, P. Maini and K. Painter, Pattern formation in a generalized chemotactic model,, Bull. Math. Biol., 60 (1998), 1. 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.

[34]

W.-M. Ni, Qualitative properties of solutions to elliptic problems,, in Stationary partial differential equations. Vol. I, (2004), 157. 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.

[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. 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.

[38]

B. Perthame, Transport Equations in Biology,, Birkhäuser Verlag, (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. 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.

[41]

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

[42]

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

[43]

J. Shi, Persistence and bifurcation of degenerate solutions,, J. Funct. Anal., 169 (1999), 494. 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. 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.

[46]

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

[47]

T. Suzuki, Free energy and self-interaction particles,, Birkhäuser, (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. doi: 10.1142/S0218202512500443.

[49]

A. Turing, The chemical basis of morphogenesis,, Philosophical Transactions of the Royal Society of London. Series B, 237 (1952), 37.

[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. 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. 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.

[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. 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). 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. 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. 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. 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. doi: 10.1016/j.jde.2008.10.024.

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