2016, 13(2): 343-367. doi: 10.3934/mbe.2015006

Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay

1. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049

Received  April 2015 Revised  October 2015 Published  November 2015

A within-host viral infection model with both virus-to-cell and cell-to-cell transmissions and time delay in immune response is investigated. Mathematical analysis shows that delay may destabilize the infected steady state and lead to Hopf bifurcation. Moreover, the direction of the Hopf bifurcation and the stability of the periodic solutions are investigated by normal form and center manifold theory. Numerical simulations are done to explore the rich dynamics, including stability switches, Hopf bifurcations, and chaotic oscillations.
Citation: Jinhu Xu, Yicang Zhou. Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay. Mathematical Biosciences & Engineering, 2016, 13 (2) : 343-367. doi: 10.3934/mbe.2015006
References:
[1]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo,, SIAM Rev., 41 (1999), 3.  doi: 10.1137/S0036144598335107.  Google Scholar

[2]

P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis,, SIAM J. Appl. Math., 63 (2003), 1313.  doi: 10.1137/S0036139902406905.  Google Scholar

[3]

M. S. Ciupe, B. L. Bivort, D. M. Bortz and P. W. Nelson, Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models,, Math. Biosci., 200 (2006), 1.  doi: 10.1016/j.mbs.2005.12.006.  Google Scholar

[4]

V. Herz, S. Bonhoeffer and R. Anderson, et al., Viral dynamics in vivo: Limitations on estimations on intracellular delay and virus delay,, Proc. Natl. Acad. Sci. USA, 93 (1996), 7247.  doi: 10.1073/pnas.93.14.7247.  Google Scholar

[5]

K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data,, Math. Biosci., 235 (2012), 98.  doi: 10.1016/j.mbs.2011.11.002.  Google Scholar

[6]

R. V. Culshaw and S. G. Ruan, A delay-differential equation model of HIV infection of $CD4^+$ T-cells,, Math. Biosci., 165 (2000), 27.  doi: 10.1016/S0025-5564(00)00006-7.  Google Scholar

[7]

J. Mittler, B. Sulzer, A. Neumann and A. S. Perelson, Influence of delayed virus production on viral dynamics in HIV-1 infected patients,, Math. Biosci., 152 (1998), 143.  doi: 10.1016/S0025-5564(98)10027-5.  Google Scholar

[8]

P. W. Nelson, J. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay,, Math. Biosci., 163 (2000), 201.  doi: 10.1016/S0025-5564(99)00055-3.  Google Scholar

[9]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bull. Math. Biol., 72 (2010), 1492.  doi: 10.1007/s11538-010-9503-x.  Google Scholar

[10]

S. Liu and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy,, Math. Biosci. Eng., 7 (2010), 675.  doi: 10.3934/mbe.2010.7.675.  Google Scholar

[11]

R. V. Culshaw, S. Ruan and G. Webb, A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay,, J. Math. Biol., 46 (2003), 425.  doi: 10.1007/s00285-002-0191-5.  Google Scholar

[12]

H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immume response and intracellular delay,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 513.  doi: 10.3934/dcdsb.2009.12.511.  Google Scholar

[13]

H. Y. Zhu, Y. Luo and M. L. Chen, Stability and Hopf bifurcation of a HIV infection model with CTL-response delay,, Comput. Math. Appl., 62 (2011), 3091.  doi: 10.1016/j.camwa.2011.08.022.  Google Scholar

[14]

Y. Wang, Y. Zhou, J. Wu and J. Heffernan, Oscillatory viral dynamics in a delayed HIV pathogenesis model,, Math. Biosci., 219 (2009), 104.  doi: 10.1016/j.mbs.2009.03.003.  Google Scholar

[15]

M. Y. Li and H. Shu, Multiple stable periodic oscillations in a mathematical model of CTL response to HTLV-I infection,, Bull. Math. Biol., 73 (2011), 1774.  doi: 10.1007/s11538-010-9591-7.  Google Scholar

[16]

H. Song, W. Jiang and S. Liu, Virus dynamics model with intracellular delays and immune response,, Math. Biosci. Eng., 12 (2015), 185.  doi: 10.3934/mbe.2015.12.185.  Google Scholar

[17]

M. Sourisseau, N. Sol-Foulon, F. Porrot, F. Blanchet and O. Schwartz, Inefficient human immunodeficiency virus replication in mobile lymphocytes,, J. Virol., 81 (2007), 1000.  doi: 10.1128/JVI.01629-06.  Google Scholar

[18]

D. S. Dimitrov, R. L. Willey and H. Sato, et al., Quantitation of human immunodeficiency virus type 1 infection kinetics,, J. Virol., 67 (1993), 2182.   Google Scholar

[19]

H. Sato, J. Orenstein, D. S. Dimitrov and M. A. Martin, Cell-to-cell spread of HIV-1 occurs with minutes and may not involve the participation of virus particles,, Virology, 186 (1992), 712.  doi: 10.1016/0042-6822(92)90038-Q.  Google Scholar

[20]

S. Gummuluru, C. M. Kinsey and M. Emerman, An in vitro rapid-turnover assay for human immunodeficiency virus type 1 replication selects for cell-to-cell spread of virus,, J. Virol., 74 (2000), 10882.  doi: 10.1128/JVI.74.23.10882-10891.2000.  Google Scholar

[21]

J. Witteveldt, M. J. Evans and J. Bitzegeio, et al., CD81 is dispensable for hepatitis C virus cell-to-cell transmission in hepatoma cells,, J. Gen. Virol., 90 (2009), 48.  doi: 10.1099/vir.0.006700-0.  Google Scholar

[22]

J. Lupberger, J. M. B. Zeisel and F. Xiao et al., EGFR and EphA2 are hepatitis C virus host entry factors and targets for antiviral therapy,, Nat. Med., 17 (2011), 589.  doi: 10.1038/nm.2341.  Google Scholar

[23]

I. Fofana, F. Xiao and C. Thumann, et al., A novel monoclonal anti-CD81 antibody produced by genetic immunization efficiently inhibits Hepatitis C virus cell-cell transmission,, PloS one., 8 (2013).  doi: 10.1371/journal.pone.0064221.  Google Scholar

[24]

S. Imai, J. Nishikawa ang K. Takada, Cell-to-cell contact as an efficient mode of Epstein-Barr virus infection of diverse human epithelial cells,, J. Virol., 72 (1998), 4371.   Google Scholar

[25]

Q. J. Sattentau, The direct passage of animal viruses between cells,, Current opinion in virology, 1 (2011), 396.  doi: 10.1016/j.coviro.2011.09.004.  Google Scholar

[26]

S. Benovic, T. Kok, A. Stephenson and J. McInnes, et al., De novo reverse transcription of HTLV-1 following cell-to-cell transmission of infection,, Virology., 244 (1998), 294.  doi: 10.1006/viro.1998.9111.  Google Scholar

[27]

M. L. Dustin and J. A. Cooper, The immunological synapse and the actin cytoskeleton: Molecular hardware for T cell signaling,, Nat Immunol., 1 (2000), 23.  doi: 10.1038/76877.  Google Scholar

[28]

N. Martin and Q. Sattentau, Cell-to-cell HIV-1 spread and its implications for immune evasion,, Curr Opin HIV AIDS., 4 (2009), 143.  doi: 10.1097/COH.0b013e328322f94a.  Google Scholar

[29]

Q. Sattentau, Avoiding the void: Cell-to-cell spread of human viruses,, Nat Rev Microbiol, 6 (2008), 815.  doi: 10.1038/nrmicro1972.  Google Scholar

[30]

G. Carloni, A. Crema and M. B Valli, et al., HCV infection by cell-to-cell transmission: Choice or necessity?,, Current molecular medicine., 12 (2012), 83.  doi: 10.2174/156652412798376152.  Google Scholar

[31]

X. L. Lai and X. F. Zou, Modeling HIV-1 virus dynamics with both virus-to-cell infection and cell-to-cell transmission,, SIAM J. Appl. Math., 74 (2014), 898.  doi: 10.1137/130930145.  Google Scholar

[32]

X. L. Lai and X. F. Zou, Modeling cell-to-cell spread of HIV-1 with logistic target cell growth,, J. Math. Anal. Appl., 426 (2015), 563.  doi: 10.1016/j.jmaa.2014.10.086.  Google Scholar

[33]

M. Nowak and C. Bangham, Population dynamics of immune responses to persistent viruses,, Science., 272 (1996), 74.  doi: 10.1126/science.272.5258.74.  Google Scholar

[34]

K. Wang, W. Wang and X. Liu, Global stability in a viral infection model with lytic and nonlytic immune response,, Comput. Math. Appl., 51 (2006), 1593.  doi: 10.1016/j.camwa.2005.07.020.  Google Scholar

[35]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Springer-Verlag, (1993).  doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[36]

W. M. Hirsch, H. Hanisch and J. P. Gabril, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior,, Commun. Pure. Appl. Math., 38 (1985), 733.  doi: 10.1002/cpa.3160380607.  Google Scholar

[37]

J. Hale and W. Z. Huang, Global geometry of the stable regions for two delay differential equations,, J. Math. Anal. Appl., 178 (1993), 344.  doi: 10.1006/jmaa.1993.1312.  Google Scholar

[38]

S. Busenberg and K. L. Cooke, Vertically Transmitted Diseases: Models and Dynamics,, vol. 23. Biomathematics. New York: Springer, (1993).  doi: 10.1007/978-3-642-75301-5.  Google Scholar

[39]

X. Li and J. Wei, On the zeros of a fourth degree exponential polynomial with applications to an eural network model with delays,, Chaos Solitions Fractals., 26 (2005), 519.  doi: 10.1016/j.chaos.2005.01.019.  Google Scholar

[40]

J. Hale, Theory of Function Differential Equations,, Springer, (1977).   Google Scholar

[41]

B. D. Hassard, N. D. Kazariniff and Y. H. Wan, Theory and Application of Hopf Bifurcation,, London math society lecture note series, (1981).   Google Scholar

show all references

References:
[1]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo,, SIAM Rev., 41 (1999), 3.  doi: 10.1137/S0036144598335107.  Google Scholar

[2]

P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis,, SIAM J. Appl. Math., 63 (2003), 1313.  doi: 10.1137/S0036139902406905.  Google Scholar

[3]

M. S. Ciupe, B. L. Bivort, D. M. Bortz and P. W. Nelson, Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models,, Math. Biosci., 200 (2006), 1.  doi: 10.1016/j.mbs.2005.12.006.  Google Scholar

[4]

V. Herz, S. Bonhoeffer and R. Anderson, et al., Viral dynamics in vivo: Limitations on estimations on intracellular delay and virus delay,, Proc. Natl. Acad. Sci. USA, 93 (1996), 7247.  doi: 10.1073/pnas.93.14.7247.  Google Scholar

[5]

K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data,, Math. Biosci., 235 (2012), 98.  doi: 10.1016/j.mbs.2011.11.002.  Google Scholar

[6]

R. V. Culshaw and S. G. Ruan, A delay-differential equation model of HIV infection of $CD4^+$ T-cells,, Math. Biosci., 165 (2000), 27.  doi: 10.1016/S0025-5564(00)00006-7.  Google Scholar

[7]

J. Mittler, B. Sulzer, A. Neumann and A. S. Perelson, Influence of delayed virus production on viral dynamics in HIV-1 infected patients,, Math. Biosci., 152 (1998), 143.  doi: 10.1016/S0025-5564(98)10027-5.  Google Scholar

[8]

P. W. Nelson, J. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay,, Math. Biosci., 163 (2000), 201.  doi: 10.1016/S0025-5564(99)00055-3.  Google Scholar

[9]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bull. Math. Biol., 72 (2010), 1492.  doi: 10.1007/s11538-010-9503-x.  Google Scholar

[10]

S. Liu and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy,, Math. Biosci. Eng., 7 (2010), 675.  doi: 10.3934/mbe.2010.7.675.  Google Scholar

[11]

R. V. Culshaw, S. Ruan and G. Webb, A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay,, J. Math. Biol., 46 (2003), 425.  doi: 10.1007/s00285-002-0191-5.  Google Scholar

[12]

H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immume response and intracellular delay,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 513.  doi: 10.3934/dcdsb.2009.12.511.  Google Scholar

[13]

H. Y. Zhu, Y. Luo and M. L. Chen, Stability and Hopf bifurcation of a HIV infection model with CTL-response delay,, Comput. Math. Appl., 62 (2011), 3091.  doi: 10.1016/j.camwa.2011.08.022.  Google Scholar

[14]

Y. Wang, Y. Zhou, J. Wu and J. Heffernan, Oscillatory viral dynamics in a delayed HIV pathogenesis model,, Math. Biosci., 219 (2009), 104.  doi: 10.1016/j.mbs.2009.03.003.  Google Scholar

[15]

M. Y. Li and H. Shu, Multiple stable periodic oscillations in a mathematical model of CTL response to HTLV-I infection,, Bull. Math. Biol., 73 (2011), 1774.  doi: 10.1007/s11538-010-9591-7.  Google Scholar

[16]

H. Song, W. Jiang and S. Liu, Virus dynamics model with intracellular delays and immune response,, Math. Biosci. Eng., 12 (2015), 185.  doi: 10.3934/mbe.2015.12.185.  Google Scholar

[17]

M. Sourisseau, N. Sol-Foulon, F. Porrot, F. Blanchet and O. Schwartz, Inefficient human immunodeficiency virus replication in mobile lymphocytes,, J. Virol., 81 (2007), 1000.  doi: 10.1128/JVI.01629-06.  Google Scholar

[18]

D. S. Dimitrov, R. L. Willey and H. Sato, et al., Quantitation of human immunodeficiency virus type 1 infection kinetics,, J. Virol., 67 (1993), 2182.   Google Scholar

[19]

H. Sato, J. Orenstein, D. S. Dimitrov and M. A. Martin, Cell-to-cell spread of HIV-1 occurs with minutes and may not involve the participation of virus particles,, Virology, 186 (1992), 712.  doi: 10.1016/0042-6822(92)90038-Q.  Google Scholar

[20]

S. Gummuluru, C. M. Kinsey and M. Emerman, An in vitro rapid-turnover assay for human immunodeficiency virus type 1 replication selects for cell-to-cell spread of virus,, J. Virol., 74 (2000), 10882.  doi: 10.1128/JVI.74.23.10882-10891.2000.  Google Scholar

[21]

J. Witteveldt, M. J. Evans and J. Bitzegeio, et al., CD81 is dispensable for hepatitis C virus cell-to-cell transmission in hepatoma cells,, J. Gen. Virol., 90 (2009), 48.  doi: 10.1099/vir.0.006700-0.  Google Scholar

[22]

J. Lupberger, J. M. B. Zeisel and F. Xiao et al., EGFR and EphA2 are hepatitis C virus host entry factors and targets for antiviral therapy,, Nat. Med., 17 (2011), 589.  doi: 10.1038/nm.2341.  Google Scholar

[23]

I. Fofana, F. Xiao and C. Thumann, et al., A novel monoclonal anti-CD81 antibody produced by genetic immunization efficiently inhibits Hepatitis C virus cell-cell transmission,, PloS one., 8 (2013).  doi: 10.1371/journal.pone.0064221.  Google Scholar

[24]

S. Imai, J. Nishikawa ang K. Takada, Cell-to-cell contact as an efficient mode of Epstein-Barr virus infection of diverse human epithelial cells,, J. Virol., 72 (1998), 4371.   Google Scholar

[25]

Q. J. Sattentau, The direct passage of animal viruses between cells,, Current opinion in virology, 1 (2011), 396.  doi: 10.1016/j.coviro.2011.09.004.  Google Scholar

[26]

S. Benovic, T. Kok, A. Stephenson and J. McInnes, et al., De novo reverse transcription of HTLV-1 following cell-to-cell transmission of infection,, Virology., 244 (1998), 294.  doi: 10.1006/viro.1998.9111.  Google Scholar

[27]

M. L. Dustin and J. A. Cooper, The immunological synapse and the actin cytoskeleton: Molecular hardware for T cell signaling,, Nat Immunol., 1 (2000), 23.  doi: 10.1038/76877.  Google Scholar

[28]

N. Martin and Q. Sattentau, Cell-to-cell HIV-1 spread and its implications for immune evasion,, Curr Opin HIV AIDS., 4 (2009), 143.  doi: 10.1097/COH.0b013e328322f94a.  Google Scholar

[29]

Q. Sattentau, Avoiding the void: Cell-to-cell spread of human viruses,, Nat Rev Microbiol, 6 (2008), 815.  doi: 10.1038/nrmicro1972.  Google Scholar

[30]

G. Carloni, A. Crema and M. B Valli, et al., HCV infection by cell-to-cell transmission: Choice or necessity?,, Current molecular medicine., 12 (2012), 83.  doi: 10.2174/156652412798376152.  Google Scholar

[31]

X. L. Lai and X. F. Zou, Modeling HIV-1 virus dynamics with both virus-to-cell infection and cell-to-cell transmission,, SIAM J. Appl. Math., 74 (2014), 898.  doi: 10.1137/130930145.  Google Scholar

[32]

X. L. Lai and X. F. Zou, Modeling cell-to-cell spread of HIV-1 with logistic target cell growth,, J. Math. Anal. Appl., 426 (2015), 563.  doi: 10.1016/j.jmaa.2014.10.086.  Google Scholar

[33]

M. Nowak and C. Bangham, Population dynamics of immune responses to persistent viruses,, Science., 272 (1996), 74.  doi: 10.1126/science.272.5258.74.  Google Scholar

[34]

K. Wang, W. Wang and X. Liu, Global stability in a viral infection model with lytic and nonlytic immune response,, Comput. Math. Appl., 51 (2006), 1593.  doi: 10.1016/j.camwa.2005.07.020.  Google Scholar

[35]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Springer-Verlag, (1993).  doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[36]

W. M. Hirsch, H. Hanisch and J. P. Gabril, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior,, Commun. Pure. Appl. Math., 38 (1985), 733.  doi: 10.1002/cpa.3160380607.  Google Scholar

[37]

J. Hale and W. Z. Huang, Global geometry of the stable regions for two delay differential equations,, J. Math. Anal. Appl., 178 (1993), 344.  doi: 10.1006/jmaa.1993.1312.  Google Scholar

[38]

S. Busenberg and K. L. Cooke, Vertically Transmitted Diseases: Models and Dynamics,, vol. 23. Biomathematics. New York: Springer, (1993).  doi: 10.1007/978-3-642-75301-5.  Google Scholar

[39]

X. Li and J. Wei, On the zeros of a fourth degree exponential polynomial with applications to an eural network model with delays,, Chaos Solitions Fractals., 26 (2005), 519.  doi: 10.1016/j.chaos.2005.01.019.  Google Scholar

[40]

J. Hale, Theory of Function Differential Equations,, Springer, (1977).   Google Scholar

[41]

B. D. Hassard, N. D. Kazariniff and Y. H. Wan, Theory and Application of Hopf Bifurcation,, London math society lecture note series, (1981).   Google Scholar

[1]

Tin Phan, Bruce Pell, Amy E. Kendig, Elizabeth T. Borer, Yang Kuang. Rich dynamics of a simple delay host-pathogen model of cell-to-cell infection for plant virus. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 515-539. doi: 10.3934/dcdsb.2020261

[2]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084

[3]

Rajendra K C Khatri, Brendan J Caseria, Yifei Lou, Guanghua Xiao, Yan Cao. Automatic extraction of cell nuclei using dilated convolutional network. Inverse Problems & Imaging, 2021, 15 (1) : 27-40. doi: 10.3934/ipi.2020049

[4]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003

[5]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344

[6]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[7]

Guihong Fan, Gail S. K. Wolkowicz. Chaotic dynamics in a simple predator-prey model with discrete delay. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 191-216. doi: 10.3934/dcdsb.2020263

[8]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[9]

Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342

[10]

Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021013

[11]

Simone Göttlich, Elisa Iacomini, Thomas Jung. Properties of the LWR model with time delay. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020032

[12]

Ali Wehbe, Rayan Nasser, Nahla Noun. Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020050

[13]

Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial & Management Optimization, 2021, 17 (1) : 317-337. doi: 10.3934/jimo.2019113

[14]

Maoli Chen, Xiao Wang, Yicheng Liu. Collision-free flocking for a time-delay system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1223-1241. doi: 10.3934/dcdsb.2020251

[15]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[16]

Jesús A. Álvarez López, Ramón Barral Lijó, John Hunton, Hiraku Nozawa, John R. Parker. Chaotic Delone sets. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021016

[17]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341

[18]

Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173

[19]

Marcos C. Mota, Regilene D. S. Oliveira. Dynamic aspects of Sprott BC chaotic system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1653-1673. doi: 10.3934/dcdsb.2020177

[20]

Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $ p $-Laplacians. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]