Skip to main content

Advertisement

SpringerLink
Log in

Pattern formations in nonlinear dynamics of hepatitis B virus

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

We investigate theoretically and numerically a spatiotemporal dynamics of hepatitis B virus (HBV) infection model through pattern formation of a prey–predation, competition and commensalism species. The system is modeled by a reaction diffusion equations with two competing and the third species presents itself both as a predator and a host respectively for the first and second species. We determined the equilibrium points of the model and by making use of the Routh–Hurwitz criteria the analysis of the bifurcation are made which permits us to derive the conditions of existence of the Turing patterns. Numerical results are given in order to illustrate how biological processes affect spatiotemporal pattern formation and rise out the influences of diffusion on the density the concentrations zones through Turing and non-Turing patterns.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. A.M. Turing, On the chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B Biol. Sci. 237, 37–72 (1952)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Q. Ouyang, H.L. Swinney, Transition from a uniform state to hexagonal and striped Turing patterns. Nature 352, 610–612 (1991)

    Article  ADS  Google Scholar 

  3. M.C. Cross, P.C. Hohenberg, Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993)

    Article  ADS  MATH  Google Scholar 

  4. A.B. Medvinsky, S.V. Petrovskii, I.A. Tikhonova, H. Malchow, B.L. Li, Spatiotemporal complexity of plankton and fish dynamics. SIAM Rev. 44, 311–370 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. L.A. Segel, J.L. Jackson, Dissipative structure: an explanation and an ecological example. J. Theor. Biol. 37, 545–559 (1972)

    Article  Google Scholar 

  6. S. Ghorai, S. Poria, Turing patterns induced by cross-diffusion in a predator–prey system in presence of habitat complexity. Chaos Solitons Fractals 91, 421–429 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. G.Q. Sun, J. Zhang, L.P. Song, Z. Jin, B.L. Li, Pattern formation of a spatial predator–prey system. Appl. Math. Comput. 218, 11162 (2012)

    MathSciNet  MATH  Google Scholar 

  8. G. Nicolis, I. Prigogine, Self-Organization in Nonequilibrium Systems (Wiley, New York, 1977)

    MATH  Google Scholar 

  9. V. Castets, E. Dulos, J. Boissonade, P.D. Kepper, Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. Phys. Rev. Lett. 64(24), 2953 (1990)

    Article  ADS  Google Scholar 

  10. L.A. Segel, J.L. Jackson, Dissipative structure: an explanation and an ecological example. J. Theor. Biol. 37(3), 545–559 (1972)

    Article  Google Scholar 

  11. C.B. Huffaker, Experimental studies on predation: dispersion factors and predator–prey oscillations. Hilgardia 27, 343–383 (1958)

    Article  Google Scholar 

  12. J. Ma, Y. Xu, G. Ren, C. Wang, Prediction for breakup of spiral wave in a regular neuronal network. Nonlinear Dyn. 84, 497–509 (2016)

    Article  MathSciNet  Google Scholar 

  13. J.X. Chen, M.M. Guo, J. MA, Termination of pinned spirals by local stimuli. Europhys. Lett. 113(3), 38004 (2016)

    Article  ADS  Google Scholar 

  14. Y. Xu, W. Jin, J. Ma, Emergence and robustness of target wave in a neuronal network. In. J. Mod. Phys. B 29(23), 1550164 (2015)

    Article  ADS  MATH  Google Scholar 

  15. H. Qin, Y. Wu, C. Wang, J. Ma, Emitting waves from defects in network with autapses. Commun. Nonlinear Sci. Numer. Simul. 23(1), 164–174 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. T.B. Liu, J. Ma, Q. Zhao, J. Tang, Force exerted on the spiral tip by the heterogeneity in an excitable medium. Europhys. Lett. 104(5), 58005 (2014)

    Article  ADS  Google Scholar 

  17. X. Song, C. Wang, J. MA, G. Ren, Collapse of ordered spatial pattern in neuronal network. Phys. A Stat. Mech. Appl. 451, 95–112 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Ma, Y. Xu, C. Wang, W. Jin, Pattern selection and selforganization induced by random boundary initial values in a neuronal network. Phys. A Stat. Mech. Appl. 461, 585–594 (2016)

    Article  Google Scholar 

  19. G.Q. Sun, G. Zhang, Z. Jin, L. Li, Predator cannibalism can give rise to regular spatial pattern in a predator–prey system. Nonlinear Dyn. 58, 75–84 (2009)

    Article  MATH  Google Scholar 

  20. S. Ghorai, S. Poria, Pattern formation and control of spatiotemporal chaos in a reaction–diffusion prey–predator system supplying additional food. Chaos Solitons Fractals 85, 57–67 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. K. Chakraborty, V. Manthena, Modelling and analysis of spatio-temporal dynamics of a marine ecosystem. Nonlinear Dyn. 81(4), 1895–1906 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. L.N. Guin, P.K. Mandal, Spatiotemporal dynamics of reaction–diffusion models of interacting populations. Appl. Math. Model. 38(17), 4417–4427 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Y. Song, X. Zou, Bifurcation analysis of a diffusive ratio dependent predator–prey model. Nonlinear Dyn. 78(1), 49–70 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Hastings, T. Powell, Chaos in a three species food chain. Ecology 72(3), 896–903 (1991)

    Article  Google Scholar 

  25. J. Chattopadhyay, O. Arino, A predator–prey model with disease in the prey. Nonlinear Anal. Theory Methods Appl. 36(6), 747–766 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. R.M. May, W.J. Leonard, Nonlinear aspects of competition between three species. SIAM J. Appl. Math. 29(2), 243–253 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  27. P. Panja, S.K. Mondal, Stability analysis of coexistence of three species prey–predator model. Nonlinear Dyn. 81, 373–382 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. B. Sahoo, S. Poria, The chaos and control of a food chain model supplying additional food to top-predator. Chaos Solitons Fractals 58, 52–64 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. H.I. Freedman, P. Waltman, Persistence in models of three interacting predator–prey populations. Math. Biosci. 68(2), 213–231 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  30. H.I. Freedman, P. Waltman, Mathematical analysis of some three-species food-chain models. Math. Biosci. 33(3), 257–276 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  31. U. Saif, A.K. Muhammad, F. Muhammad, A new fractional model for the dynamics of the hepatitis B virus using the Caputo–Fabrizio derivative. Eur. Phys. J. Plus 133, 237 (2018)

    Article  Google Scholar 

  32. N. Gul, R. Bilal, E.A. Algehyne, M.G. Alshehri, M.A. Khan, Y.M. Chu, S. Islam, The dynamics of fractional order hepatitis B virus model with asymptomatic carriers. Alex. Eng. J. 60, 3945–3955 (2021)

    Article  Google Scholar 

  33. I. Podlubny, Fractional Differential Equations (Academic Press, Cambridge, 1999)

    MATH  Google Scholar 

  34. O.A. Ebraheem, A. Wisal, A.K. Muhammad, J.M. Sharaf, Optimal control strategies of Zika virus model with mutant. Commun. Nonlinear Sci. Numer. Simul. 93, 105532 (2020)

    MathSciNet  MATH  Google Scholar 

  35. A.K. Muhammad, A. Abdon, Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative. Alex. Eng. J. 59(4), 2379–2389 (2020)

    Article  Google Scholar 

  36. Z.P. Ma, J.L. Yue, Competitive exclusion and coexistence of a delayed reaction–diffusion system modeling two predators competing for one prey. Comput. Math. Appl. 71(9), 1799–1817 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Morozov, S. Ruan, B.L. Li, Patterns of patchy spread in multi-species reaction–diffusion models. Ecol. Complex. 5(4), 313–328 (2008)

    Article  Google Scholar 

  38. S. Hata, H. Nakao, A.S. Mikhailov, Sufficient conditions for wave instability in three-component reaction–diffusion systems. Prog. Theor. Exp. Phys. 2014(1), 013A01 (2014)

    Article  MATH  Google Scholar 

  39. K.A.J. White, C.A. Gilligan, Spatial heterogeneity in three species, plant-parasite-hyperparasite, systems. Philos. Trans. R. Soc. B 353(1368), 543–557 (1998)

    Article  Google Scholar 

  40. S. Hews, S. Eikenberry, J.D. Nagy, Y. Kuang, Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth. J. Math. Biol. 60, 573–590 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. A. Mitchell, D. Griffiths, The Finite Difference Method in Partial Differential (Wiley, Chichester, 1980).

    MATH  Google Scholar 

  42. W.F. Ames, Numerical Methods for Partial Differential Equations (Academic Press, London, 1992).

    MATH  Google Scholar 

  43. J.D. Murray, Mathematical Biology II Spatial Models and Biomedical Applications, vol. 18, 3rd edn. (Springer, Berlin, 2003).

    MATH  Google Scholar 

  44. T.S. Shaikh, N. Fayyaz, N. Ahmed et al., Numerical study for epidemic model of hepatitis-B virus. Eur. Phys. J. Plus 136, 367 (2021)

    Article  Google Scholar 

  45. Maya Mincheva, Marc R. Roussel, Turing–Hopf instability in biochemical reaction networks arising from pairs of subnetworks. Math. Biosci. 240, 1–11 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the Editor and to the anonymous Referees for their valuables remarks and suggestions which help us to improve the quality of the present paper.

Author information

Authors and Affiliations

  1. Laboratory of Biophysics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon

    B. Tamko Mbopda, S. Issa & H. P. Ekobena Fouda

  2. Department Refining and Petrochemistry, Faculty of Mines and Petroleum Industries, University of Maroua, P.O. Box 46, Maroua, Cameroon

    S. Issa

  3. Département des Sciences Physiques, Ecole Normale Supérieure, Université de Maroua, P.O. Box 55, Maroua, Cameroon

    S. Abdoulkary

  4. Department of Basic Science, Faculty of Mines and Petroleum Industries, University of Maroua, P.O. Box 46, Maroua, Cameroon

    S. Abdoulkary

  5. National Advanced School of Engineering of Maroua, Université de Maroua, P.O. Box 46, Maroua, Cameroon

    R. Guiem

Authors
  1. B. Tamko Mbopda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Issa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Abdoulkary
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. R. Guiem
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. H. P. Ekobena Fouda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Issa.

Ethics declarations

Conflict of interest

Authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mbopda, B.T., Issa, S., Abdoulkary, S. et al. Pattern formations in nonlinear dynamics of hepatitis B virus. Eur. Phys. J. Plus 136, 586 (2021). https://doi.org/10.1140/epjp/s13360-021-01569-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-021-01569-8

Search

Navigation