Skip to main content
SpringerLink
Log in

Spatiotemporal Dynamics in a Diffusive Bacterial and Viral Diseases Propagation Model with Chemotaxis

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

In this article, we study the effect of chemotaxis on the dynamics of a diffusive bacterial and viral diseases propagation model. From three aspects: \(\chi >0\), \(\chi =0\) and \(\chi <0\), we investigate the existence of Turing bifurcations and stability of positive equilibrium under Neumann boundary conditions. We find that Turing bifurcations can induced by chemotaxis, which does not occur in the original model. Moreover, for the model with diffusion and chemotaxis, we need explore the new expression of the normal form on Turing bifurcation. By the newly obtained normal form, we can determine the properties of Turing bifurcation. Finally, we perform some numerical simulations to verify the theoretical analysis and obtain stable steady state solutions, spots pattern and spots-strip pattern, which also expand the main results in this article.

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

Similar content being viewed by others

References

  1. Nowak, M.A., Bonhoeffer, S., Hill, A.M., Boehme, R., Thomas, H.C., McDade, H.: Viral dynamics in hepatitis B virus infection. Proc. Natl. Acad. Sci. 93, 4398–4402 (1996)

    Google Scholar 

  2. Wang, W., Cai, Y., Wu, M., Wang, K., Li, Z.: Complex dynamics of a reaction-diffusion epidemic model. Nonlinear Anal. RWA 13, 2240–2258 (2012)

    MathSciNet  MATH  Google Scholar 

  3. Muqbel, K., Vas, G., Röst, G.: Periodic orbits and global stability for a discontinuous SIR model with delayed control. Qual. Theory Dyn. Syst. 19, 59 (2020)

    MathSciNet  MATH  Google Scholar 

  4. El Fatini, M., Pettersson, R., Sekkak, I., et al.: A stochastic analysis for a triple delayed SIQR epidemic model with vaccination and elimination strategies. J. Appl. Math. Comput. (2020). https://doi.org/10.1007/s12190-020-01380-1

    Article  MathSciNet  Google Scholar 

  5. Tang, X., Yu, T., Deng, Z., Liu, D.: NSFD scheme and dynamic consistency of a delayed diffusive humoral immunity viral infection model. J. Appl. Math. Comput. (2020). https://doi.org/10.1007/s12190-020-01362-3

    Article  MathSciNet  Google Scholar 

  6. Wang, X., Tang, X., Wang, Z., Li, X.: Global dynamics of a diffusive viral infection model with general incidence function and distributed delays. Ricerche Mat. (2020). https://doi.org/10.1007/s11587-020-00481-0

    Article  Google Scholar 

  7. Tang, X., Wang, Z., Yang, J.: Threshold dynamics and competitive exclusion in a virus infection model with general incidence function and density-dependent diffusion. Complexity 2020, 4923856 (2020). https://doi.org/10.1155/2020/4923856

    Article  MATH  Google Scholar 

  8. Capasso, V., Maddalena, L.: Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases. J. Math. Biol. 13, 173–184 (1981)

    MathSciNet  MATH  Google Scholar 

  9. Thieme, H., Zhao, X.: Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction diffusion models. J. Differ. Equ. 195, 430–470 (2003)

    MathSciNet  MATH  Google Scholar 

  10. Wu, S., Liu, S.: Asymptotic speed of spread and traveling fronts for a nonlocal reaction-diffusion model with distributed delay. Appl. Math. Model. 33, 2757–2765 (2009)

    MathSciNet  MATH  Google Scholar 

  11. Wu, S., Hsu, H., Xiao, Y.: Global attractivity, spreading speeds and traveling waves of delayed nonlocal reaction-diffusion systems. J. Differ. Equ. 258, 1058–1105 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Hu, H., Tan, Y., Huang, J.: Hopf bifurcation analysis on a delayed reaction-diffusion system modelling the spatial spread of bacterial and viral diseases. Chaos Solitons Fract. 125, 152–162 (2019)

    MathSciNet  MATH  Google Scholar 

  13. Yan, S., Lian, X., Wang, W., Upadhyay, R.K.: Spatiotemporal dynamics in a delayed diffusive predator model. Appl. Math. Comput. 224, 524–534 (2013)

    MathSciNet  MATH  Google Scholar 

  14. Li, J., Sun, G., Jin, Z.: Pattern formation of an epidemic model with time delay. Physica A 403, 100–109 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Sun, G., Wang, C., Chang, L., Wu, Y., Li, L., Jin, Z.: Effects of feedback regulation on vegetation patterns in semi-arid environments. Appl. Math. Model. 61, 200–215 (2018)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Yang, R., Wei, J.: Bifurcation analysis of a diffusive predator-prey system with nonconstant death rate and Holling III functional response. Chaos Solitons Fract. 70, 1–13 (2015)

    MathSciNet  MATH  Google Scholar 

  18. Wang, J.: The global stability and pattern formations of a predator-prey system with consuming resource. Appl. Math. Lett. 58, 49–55 (2016)

    MathSciNet  MATH  Google Scholar 

  19. Yang, W.: Analysis on existence of bifurcation solutions for a predator-prey model with herd behavior. Appl. Math. Model. 53, 433–446 (2018)

    MathSciNet  MATH  Google Scholar 

  20. Jiang, H.: Turing bifurcation in a diffusive predator-prey model with schooling behavior. Appl. Math. Lett. 96, 230–235 (2019)

    MathSciNet  MATH  Google Scholar 

  21. Yuan, S., Xu, C., Zhang, T.: Spatial dynamics in a predator-prey model with herd behavior. Chaos 23, 0331023 (2013)

    MathSciNet  MATH  Google Scholar 

  22. Tang, X., Song, Y.: Bifurcation analysis and turing instability in a diffusive predator-prey model with herd behavior and hyperbolic mortality. Chaos Solitons Fract 81(A), 303–314 (2015)

    MathSciNet  MATH  Google Scholar 

  23. Wu, D., Zhao, M.: Qualitative analysis for a diffusive predator-prey model with hunting cooperative. Physica A 515, 299–309 (2019)

    MathSciNet  Google Scholar 

  24. Capone, F., Carfora, M.F., De Luca, R., Torcicollo, I.: Turing patterns in a reaction-diffusion system modeling hunting cooperation. Math. Comput. Simul. 165, 172–180 (2019)

    MathSciNet  Google Scholar 

  25. Tang, X., Song, Y.: Cross-diffusion induced spatiotemporal patterns in a predator-prey model with herd behavior. Nonlinear Anal. RWA 24, 36–49 (2015)

    MathSciNet  MATH  Google Scholar 

  26. Liu, B., Wu, R., Chen, L.: Patterns induced by super cross-diffusion in a predator-prey system with Michaelis–Menten type harvesting. Math. Biosci. 298, 71–79 (2018)

    MathSciNet  MATH  Google Scholar 

  27. Wu, S., Song, Y.: Stability and spatiotemporal dynamics in a diffusive predator-prey model with nonlocal prey competition. Nonlinear Anal. RWA 48, 12–39 (2019)

    MathSciNet  MATH  Google Scholar 

  28. Song, Y., Wu, S., Wang, H.: Spatiotemporal dynamics in the single population model with memory-based diffusion and nonlocal effect. J. Differ. Equ. 267, 6316–6351 (2019)

    MathSciNet  MATH  Google Scholar 

  29. Pal, S., Ghorai, S., Banerjee, M.: Effect of kernels on spatio-temporal patterns of a non-local prey-predator model. Math. Biosci. 310, 96–107 (2019)

    MathSciNet  MATH  Google Scholar 

  30. Song, Y., Tang, X.: Stability, steady-state bifurcations, and Turing patterns in a predator-prey model with herd behavior and prey-taxis. Stud. Appl. Math. 139(3), 371–404 (2017)

    MathSciNet  MATH  Google Scholar 

  31. Zhang, T., Liu, X., Meng, Z., Zhang, Q.: Spatio-temporal dynamics near the steady state of a planktonic system. Comput. Math. Appl. 75, 4490–4504 (2018)

    MathSciNet  MATH  Google Scholar 

  32. Ma, M., Gao, M., Carretero-González, R.: Pattern formation for a two-dimensional reaction-diffusion model with chemotaxis. J. Math. Anal. Appl. 475, 1883–1909 (2019)

    MathSciNet  MATH  Google Scholar 

  33. Tang, X., Li, J.: Chemotaxis induced Turing bifurcation in a partly diffusive bacterial and viral diseases propagation model. Appl. Math. Lett. 100, 106037 (2020)

    MathSciNet  MATH  Google Scholar 

  34. Patlak, C.S.: Random walk with persistence and external bias. Bull. Math. Biophys. 15, 311–338 (1953)

    MathSciNet  MATH  Google Scholar 

  35. Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)

    MathSciNet  MATH  Google Scholar 

  36. Keller, E.F., Segel, L.A.: Model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971)

    MATH  Google Scholar 

  37. Li, C.: Global existence of classical solutions to the cross-diffusion three-species model with prey-taxis. Comput. Math. Appl. 72, 1394–1401 (2016)

    MathSciNet  MATH  Google Scholar 

  38. Zhao, X., Zheng, S.: Global existence and boundedness of solutions to a chemotaxis system with singular sensitivity and logistic-type source. J. Differ. Equ. 267, 826–865 (2019)

    MathSciNet  MATH  Google Scholar 

  39. Li, Y.: Finite-time blow-up in quasilinear parabolic-elliptic chemotaxis system with nonlinear signal production. J. Math. Anal. Appl. 480, 123376 (2019)

    MathSciNet  MATH  Google Scholar 

  40. Fuest, M.: Finite-time blow-up in a two-dimensional Keller–Segel system with an environmental dependent logistic source. Nonlinear Anal. RWA 52, 103022 (2020)

    MathSciNet  MATH  Google Scholar 

  41. Barresi, R., Bilotta, E., Gargano, F., Lombardo, M.C., Pantano, P., Sammartino, M.: Wavefront invasion for a chemotaxis model of multiple sclerosis. Ricerche di Matematica 65, 423–434 (2016)

    MathSciNet  MATH  Google Scholar 

  42. Li, D., Guo, S.: Periodic traveling waves in a reaction-diffusion model with chemotaxis and nonlocal delay effect. J. Math. Anal. Appl. 467, 1080–1099 (2018)

    MathSciNet  MATH  Google Scholar 

  43. Dubey, B., Das, B., Hussain, J.: A predator-prey interaction model with self and cross-diffusion. Ecol. Model. 141, 67–76 (2001)

    Google Scholar 

  44. Jorn, J.: Negative ionic cross diffusion coefficients in electrolytic solutions. J. Theor. Biol. 55, 529–532 (1975)

    Google Scholar 

  45. Faria, T.: Normal forms and Hopf bifurcation for partial differential equations with delay. Trans. Am. Math. Soc. 352, 2217–2238 (2000)

    MathSciNet  MATH  Google Scholar 

  46. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (2003)

    MATH  Google Scholar 

  47. Tan, Y., Huang, C., Sun, B., Wang, T.: Dynamics of a class of delayed reaction-diffusion systems with Neumann boundary condition. J. Math. Anal. Appl. 458, 1115–1130 (2018)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank the anonymous reviewers for the helpful comments and suggestions, which led to an improvement of the original manuscript. This research is supported by the National Natural Science Foundation of China (Granted No. 11761038, 11761039, 62062042), Science and Technology Project of Department of Education of Jiangxi Province (Granted No. GJJ180583, GJJ190542) and Natural Science Foundation of Jiangxi Province of China (Granted No. 20202ACBL211001).

Author information

Authors and Affiliations

  1. School of Mathematics and Physics, Jinggangshan University, Ji’an, 343009, Jiangxi, China

    Xiaosong Tang & Peichang Ouyang

Authors
  1. Xiaosong Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peichang Ouyang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peichang Ouyang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tang, X., Ouyang, P. Spatiotemporal Dynamics in a Diffusive Bacterial and Viral Diseases Propagation Model with Chemotaxis. Qual. Theory Dyn. Syst. 19, 91 (2020). https://doi.org/10.1007/s12346-020-00422-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-020-00422-0

Keywords

Mathematics Subject Classification

Search

Navigation