doi: 10.3934/dcds.2021071

Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent

Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA

* Corresponding author: Harumi Hattori

Received  December 2020 Published  April 2021

We study global existence and asymptotic behavior of the solutions for a chemotaxis system with chemoattractant and repellent in three dimensions. To accomplish this, we use the Fourier transform and energy method. We consider the case when the mass is conserved and we use the Lotka-Volterra type model for chemoattractant and repellent. Also, we establish $ L^q $ time-decay for the linear homogeneous system by using a Fourier transform and finding Green's matrix. Then, we find $ L^q $ time-decay for the nonlinear system using solution representation by Duhamel's principle and time-weighted estimates.

Citation: Harumi Hattori, Aesha Lagha. Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021071
References:
[1]

D. AmbrosiF. Bussolino and L. Preziosi, A review of vasculogenesis models, J. Theor. Med., 6 (2005), 1-19.  doi: 10.1080/1027366042000327098.  Google Scholar

[2]

D. AmbrosiA. Gamba and G. Serini, Cell directional and chemotaxis in vascular morphogenesis, Bulletin of Mathematical Biology, 66 (2004), 1851-1873.  doi: 10.1016/j.bulm.2004.04.004.  Google Scholar

[3]

I. M. Bomze, Lotka-Volterra equation and replicator dynamics: A two-dimensional classification, Biological Cybernetics, 48 (1983), 201-211.  doi: 10.1007/BF00318088.  Google Scholar

[4]

I. M. Bomze, Lotka-Volterra equation and replicator dynamics: New issues in classification, Biological Cybernetics, 72 (1995), 447-453.  doi: 10.1007/BF00201420.  Google Scholar

[5]

M. ChaeK. Kang and J. Lee, Existence of smooth solutions to chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297.  doi: 10.3934/dcds.2013.33.2271.  Google Scholar

[6]

R. J. Duan, Global smooth flows for the compressible Euler-Maxwell system: Relaxation case, J. Hyperbolic Differential Equations, 8 (2011), 375-413.  doi: 10.1142/S0219891611002421.  Google Scholar

[7]

R. DuanQ. Liu and C. Zhu, The Cauchy problem on the compressible two-fluids Euler-Maxwell Equations, SIAM J. Math. Anal., 44 (2012), 102-133.  doi: 10.1137/110838406.  Google Scholar

[8]

R. DuanA. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Diff. Equations, 35 (2010), 1635-1673.  doi: 10.1080/03605302.2010.497199.  Google Scholar

[9]

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, Physical Review Letters, 90 (2003), 118101. doi: 10.1103/PhysRevLett.90.118101.  Google Scholar

[10]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 81-205.  doi: 10.1007/BF00280740.  Google Scholar

[11]

E. F. Keller and L. A. Segel, A model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[12]

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

[13]

E. Lankeit and J. Lankeit, Classical solutions to a logistic chemotaxis model with singular sensitivity and signal absorption, Nonlinear Anal. Real World Appl., 46 (2019), 421-445.  doi: 10.1016/j.nonrwa.2018.09.012.  Google Scholar

[14]

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

[15]

M. LucaA. Chavez-RossL. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and alzheimer's disease senile plaques: Is there a connection?, Bulletin of Mathematical Biology, 65 (2003), 693-730.  doi: 10.1016/S0092-8240(03)00030-2.  Google Scholar

[16]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[17]

A. D. RodriguezL. C. F. Ferreira and É. J. Villamizar-Roa, Global existence for an attraction-repulsion chemotaxis fluid model with logistic source, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 423-447.  doi: 10.3934/dcdsb.2018180.  Google Scholar

[18]

G. SeriniD. AmbrosiE. GiraudoA. GambaL. Preziosi and F. Bussolino, Modeling the early stages of vascular network assembly, EMBO Journal, 22 (2003), 1771-1779.  doi: 10.1093/emboj/cdg176.  Google Scholar

[19]

Z. Tan and J. Zhou, Global existence and time decay estimate of solutions to the Keller- Segel system, Math. Meth. Appl. Sci., 42 (2019), 375-402.  doi: 10.1002/mma.5352.  Google Scholar

[20]

Y. Wang, Boundedness in a three-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion and logistic source, Journal of Differential Equations, 2016 (2016), 21 pp.  Google Scholar

show all references

References:
[1]

D. AmbrosiF. Bussolino and L. Preziosi, A review of vasculogenesis models, J. Theor. Med., 6 (2005), 1-19.  doi: 10.1080/1027366042000327098.  Google Scholar

[2]

D. AmbrosiA. Gamba and G. Serini, Cell directional and chemotaxis in vascular morphogenesis, Bulletin of Mathematical Biology, 66 (2004), 1851-1873.  doi: 10.1016/j.bulm.2004.04.004.  Google Scholar

[3]

I. M. Bomze, Lotka-Volterra equation and replicator dynamics: A two-dimensional classification, Biological Cybernetics, 48 (1983), 201-211.  doi: 10.1007/BF00318088.  Google Scholar

[4]

I. M. Bomze, Lotka-Volterra equation and replicator dynamics: New issues in classification, Biological Cybernetics, 72 (1995), 447-453.  doi: 10.1007/BF00201420.  Google Scholar

[5]

M. ChaeK. Kang and J. Lee, Existence of smooth solutions to chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297.  doi: 10.3934/dcds.2013.33.2271.  Google Scholar

[6]

R. J. Duan, Global smooth flows for the compressible Euler-Maxwell system: Relaxation case, J. Hyperbolic Differential Equations, 8 (2011), 375-413.  doi: 10.1142/S0219891611002421.  Google Scholar

[7]

R. DuanQ. Liu and C. Zhu, The Cauchy problem on the compressible two-fluids Euler-Maxwell Equations, SIAM J. Math. Anal., 44 (2012), 102-133.  doi: 10.1137/110838406.  Google Scholar

[8]

R. DuanA. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Diff. Equations, 35 (2010), 1635-1673.  doi: 10.1080/03605302.2010.497199.  Google Scholar

[9]

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, Physical Review Letters, 90 (2003), 118101. doi: 10.1103/PhysRevLett.90.118101.  Google Scholar

[10]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 81-205.  doi: 10.1007/BF00280740.  Google Scholar

[11]

E. F. Keller and L. A. Segel, A model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[12]

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

[13]

E. Lankeit and J. Lankeit, Classical solutions to a logistic chemotaxis model with singular sensitivity and signal absorption, Nonlinear Anal. Real World Appl., 46 (2019), 421-445.  doi: 10.1016/j.nonrwa.2018.09.012.  Google Scholar

[14]

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

[15]

M. LucaA. Chavez-RossL. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and alzheimer's disease senile plaques: Is there a connection?, Bulletin of Mathematical Biology, 65 (2003), 693-730.  doi: 10.1016/S0092-8240(03)00030-2.  Google Scholar

[16]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[17]

A. D. RodriguezL. C. F. Ferreira and É. J. Villamizar-Roa, Global existence for an attraction-repulsion chemotaxis fluid model with logistic source, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 423-447.  doi: 10.3934/dcdsb.2018180.  Google Scholar

[18]

G. SeriniD. AmbrosiE. GiraudoA. GambaL. Preziosi and F. Bussolino, Modeling the early stages of vascular network assembly, EMBO Journal, 22 (2003), 1771-1779.  doi: 10.1093/emboj/cdg176.  Google Scholar

[19]

Z. Tan and J. Zhou, Global existence and time decay estimate of solutions to the Keller- Segel system, Math. Meth. Appl. Sci., 42 (2019), 375-402.  doi: 10.1002/mma.5352.  Google Scholar

[20]

Y. Wang, Boundedness in a three-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion and logistic source, Journal of Differential Equations, 2016 (2016), 21 pp.  Google Scholar

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