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

• Tang, Xiaosong ^[1] ; Ouyang Peichang ^[1]
  1. [1] Jinggangshan University

     Jinggangshan University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 3, 2020
• Idioma: inglés
• DOI: 10.1007/s12346-020-00422-0
• Enlaces
  • Texto completo
• Resumen

  • In this article, we study the effect of chemotaxis on the dynamics of a diffusive bacterial and viral diseases propagation model. From three aspects: χ>0, χ=0 and χ<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.

• Referencias bibliográficas
  • 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....
  • 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...
  • 3. Muqbel, K., Vas, G., Röst, G.: Periodic orbits and global stability for a discontinuous SIR model with delayed control. Qual. Theory Dyn....
  • 4. El Fatini, M., Pettersson, R., Sekkak, I., et al.: A stochastic analysis for a triple delayed SIQR epidemic model with vaccination and...
  • 5. Tang, X., Yu, T., Deng, Z., Liu, D.: NSFD scheme and dynamic consistency of a delayed diffusive humoral immunity viral infection model....
  • 6. Wang, X., Tang, X., Wang, Z., Li, X.: Global dynamics of a diffusive viral infection model with general incidence function and distributed...
  • 7. Tang, X., Wang, Z., Yang, J.: Threshold dynamics and competitive exclusion in a virus infection model with general incidence function and...
  • 8. Capasso, V., Maddalena, L.: Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of...
  • 9. Thieme, H., Zhao, X.: Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction diffusion models. J....
  • 10. Wu, S., Liu, S.: Asymptotic speed of spread and traveling fronts for a nonlocal reaction-diffusion model with distributed delay. Appl....
  • 11. Wu, S., Hsu, H., Xiao, Y.: Global attractivity, spreading speeds and traveling waves of delayed nonlocal reaction-diffusion systems. J....
  • 12. Hu, H., Tan, Y., Huang, J.: Hopf bifurcation analysis on a delayed reaction-diffusion system modelling the spatial spread of bacterial...
  • 13. Yan, S., Lian, X., Wang, W., Upadhyay, R.K.: Spatiotemporal dynamics in a delayed diffusive predator model. Appl. Math. Comput. 224, 524–534...
  • 14. Li, J., Sun, G., Jin, Z.: Pattern formation of an epidemic model with time delay. Physica A 403, 100–109 (2014)
  • 15. Sun, G., Wang, C., Chang, L., Wu, Y., Li, L., Jin, Z.: Effects of feedback regulation on vegetation patterns in semi-arid environments....
  • 16. Song, Y., Zou, X.: Bifurcation analysis of a diffusive ratio-dependent predator-prey model. Nonlinear Dyn. 78, 49–70 (2014)
  • 17. Yang, R., Wei, J.: Bifurcation analysis of a diffusive predator-prey system with nonconstant death rate and Holling III functional response....
  • 18. Wang, J.: The global stability and pattern formations of a predator-prey system with consuming resource. Appl. Math. Lett. 58, 49–55 (2016)
  • 19. Yang, W.: Analysis on existence of bifurcation solutions for a predator-prey model with herd behavior. Appl. Math. Model. 53, 433–446...
  • 20. Jiang, H.: Turing bifurcation in a diffusive predator-prey model with schooling behavior. Appl. Math. Lett. 96, 230–235 (2019)
  • 21. Yuan, S., Xu, C., Zhang, T.: Spatial dynamics in a predator-prey model with herd behavior. Chaos 23, 0331023 (2013)
  • 22. Tang, X., Song, Y.: Bifurcation analysis and turing instability in a diffusive predator-prey model with herd behavior and hyperbolic mortality....
  • 23. Wu, D., Zhao, M.: Qualitative analysis for a diffusive predator-prey model with hunting cooperative. Physica A 515, 299–309 (2019)
  • 24. Capone, F., Carfora, M.F., De Luca, R., Torcicollo, I.: Turing patterns in a reaction-diffusion system modeling hunting cooperation. Math....
  • 25. Tang, X., Song, Y.: Cross-diffusion induced spatiotemporal patterns in a predator-prey model with herd behavior. Nonlinear Anal. RWA 24,...
  • 26. Liu, B., Wu, R., Chen, L.: Patterns induced by super cross-diffusion in a predator-prey system with Michaelis–Menten type harvesting....
  • 27. Wu, S., Song, Y.: Stability and spatiotemporal dynamics in a diffusive predator-prey model with nonlocal prey competition. Nonlinear Anal....
  • 28. Song, Y., Wu, S., Wang, H.: Spatiotemporal dynamics in the single population model with memorybased diffusion and nonlocal effect. J....
  • 29. Pal, S., Ghorai, S., Banerjee, M.: Effect of kernels on spatio-temporal patterns of a non-local preypredator model. Math. Biosci. 310,...
  • 30. Song, Y., Tang, X.: Stability, steady-state bifurcations, and Turing patterns in a predator-prey model with herd behavior and prey-taxis....
  • 31. Zhang, T., Liu, X., Meng, Z., Zhang, Q.: Spatio-temporal dynamics near the steady state of a planktonic system. Comput. Math. Appl. 75,...
  • 32. Ma, M., Gao, M., Carretero-González, R.: Pattern formation for a two-dimensional reaction-diffusion model with chemotaxis. J. Math. Anal....
  • 33. Tang, X., Li, J.: Chemotaxis induced Turing bifurcation in a partly diffusive bacterial and viral diseases propagation model. Appl. Math....
  • 34. Patlak, C.S.: Random walk with persistence and external bias. Bull. Math. Biophys. 15, 311–338 (1953)
  • 35. Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)
  • 36. Keller, E.F., Segel, L.A.: Model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971)
  • 37. Li, C.: Global existence of classical solutions to the cross-diffusion three-species model with prey-taxis. Comput. Math. Appl. 72, 1394–1401...
  • 38. Zhao, X., Zheng, S.: Global existence and boundedness of solutions to a chemotaxis system with singular sensitivity and logistic-type...
  • 39. Li, Y.: Finite-time blow-up in quasilinear parabolic-elliptic chemotaxis system with nonlinear signal production. J. Math. Anal. Appl....
  • 40. Fuest, M.: Finite-time blow-up in a two-dimensional Keller–Segel system with an environmental dependent logistic source. Nonlinear Anal....
  • 41. Barresi, R., Bilotta, E., Gargano, F., Lombardo, M.C., Pantano, P., Sammartino, M.: Wavefront invasion for a chemotaxis model of multiple...
  • 42. Li, D., Guo, S.: Periodic traveling waves in a reaction-diffusion model with chemotaxis and nonlocal delay effect. J. Math. Anal. Appl....
  • 43. Dubey, B., Das, B., Hussain, J.: A predator-prey interaction model with self and cross-diffusion. Ecol. Model. 141, 67–76 (2001)
  • 44. Jorn, J.: Negative ionic cross diffusion coefficients in electrolytic solutions. J. Theor. Biol. 55, 529–532 (1975)
  • 45. Faria, T.: Normal forms and Hopf bifurcation for partial differential equations with delay. Trans. Am. Math. Soc. 352, 2217–2238 (2000)
  • 46. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (2003)
  • 47. Tan, Y., Huang, C., Sun, B., Wang, T.: Dynamics of a class of delayed reaction-diffusion systems with Neumann boundary condition. J. Math....