Convergence and Traveling Wave Solutions in a Delayed Diffusive Competitive Model

• Shuxia Pan ^[1] ; Shengnan Hao ^[1]
  1. [1] Lanzhou University of Technology

     Lanzhou University of Technology

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 4, 2022
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • This paper studies the convergence and minimal wave speed of traveling wave solutions in a delayed competitive model. Due to the effect of time delays, this system may not satisfy the classical comparison principle. When the domain is bounded and equipped with homogeneous Neumann boundary condition, we obtain the persistence and stability of any positive mild solutions by constructing proper contracting rectangles. When the domain is whole space R, we study the minimal wave speed of traveling wave solutions by presenting the existence or nonexistence of nontrivial traveling wave solutions with any wave speeds, which models the coinvasioncoexistence of these species. Our main recipe on minimal wave speed is the generalized upper and lower solutions, contracting rectangles and spreading speeds.

• Referencias bibliográficas
  • 1. Ducrot, A., Giletti, T., Matano, H.: Existence and convergence to a propagating terrace in onedimensional reaction–diffusion equations....
  • 2. Fang, J., Zhao, X.-Q.: Existence and uniqueness of traveling waves for non-monotone integral equations with applications. J. Differ. Equ....
  • 3. Fu, S.C.: Traveling waves for a diffusive SIR model with delay. J. Math. Anal. Appl. 435, 20–37 (2016)
  • 4. Girardin, L., Lam, K.Y.: Invasion of open space by two competitors: spreading properties of monostable two-species competition-diffusion...
  • 5. Huang, J., Zou, X.: Existence of traveling wavefronts of delayed reaction-diffusion systems without monotonicity. Discrete Cont. Dyn. Sys....
  • 6. Huang, M., Wu, S.-L., Zhao, X.-Q.: Propagation dynamics for time-periodic and partially degenerate reaction–diffusion systems. SIAM J....
  • 7. Huston, M.A., DeAngelis, D.L.: Competition and coexistence: the effects of resource transport and supply rates. Am. Nat. 144, 954–977 (1994)
  • 8. Lewis, M.A., Petrovskii, S.V., Potts, J.R.: The Mathematics Behind Biological Invasions. Springer, Cham (2016)
  • 9. Li,W.-T., Lin, G., Ruan, S.: Existence of traveling wave solutions in delayed reaction–diffusion systems with applications to diffusion–competition...
  • 10. Liang, X., Zhao, X.-Q.: Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Commun. Pure Appl. Math....
  • 11. Lin, G.: Minimal wave speed of competitive diffusive systems with time delays. Appl. Math. Lett. 76, 164–169 (2018)
  • 12. Lin, G., Li, W.-T.: Asymptotic spreading of competition diffusion systems: the role of interspecific competitions. Eur. J. Appl. Math....
  • 13. Lin, G., Ruan, S.: Traveling wave solutions for delayed reaction-diffusion systems and applications to Lotka–Volterra competition–diffusion...
  • 14. Ma, S.: Traveling wavefronts for delayed reaction–diffusion systems via a fixed point theorem. J. Differ. Equ. 171, 294–314 (2001)
  • 15. Ma, S.: Traveling waves for non-local delayed diffusion equations via auxiliary equation. J. Differ. Equ. 237, 259–277 (2007)
  • 16. Martin, R.H., Smith, H.L.: Abstract functional differential equations and reaction–diffusion systems. Trans. Am. Math. Soc. 321, 1–44...
  • 17. Martin, R.H., Smith, H.L.: Reaction–diffusion systems with the time delay: monotonicity, invariance, comparison and convergence. J. Reine...
  • 18. Mackey, M.C., Glass, L.: Oscillation and chaos in physiological control systems. Science 197, 287–289 (1977)
  • 19. Murray, J.D.: Mathematical Biology, II. Spatial Models and Biomedical Applications, Interdisciplinary Applied Mathematics 18, 3rd edn....
  • 20. Ruiz-Herrera, A.: Delay reaction–diffusion systems via discrete dynamics. SIAM J. Math. Anal. 52, 6297–6312 (2020)
  • 21. Schaaf, K.W.: Asymptotic behavior and traveling wave solutions for parabolic functional differential equations. Trans. Am. Math. Soc....
  • 22. Shigesada, N., Kawasaki, K.: Biological Invasions: Theory and Practice. Oxford, New York (1997)
  • 23. Smith, H.L.: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. AMS, Providence (1995)
  • 24. Tang, M.M., Fife, P.: Propagating fronts for competing species equations with diffusion. Arch. Ration. Mech. Anal. 73, 69–77 (1980)
  • 25. Thieme, H.R., Zhao, X.-Q.: Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction–diffusion models....
  • 26. Wang, H.: On the existence of traveling waves for delayed reaction–diffusion equations. J. Differ. Equ. 247, 887–905 (2009)
  • 27. Wang, Z.-C., Li, W.-T., Ruan, S.: Traveling wave fronts of reaction-diffusion systems with spatiotemporal delays. J. Differ. Equ. 222,...
  • 28. Weinberger, H.F., Lewis, M.A., Li, B.: Analysis of linear determinacy for spread in cooperative models. J. Math. Biol. 45, 183–218 (2002)
  • 29. Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer, New York (1996)
  • 30. Wu, J., Zou, X.: Traveling wave fronts of reaction–diffusion systems with delay. J. Dyn. Differ. Equ. 13, 651–687 (2001)
  • 31. Xing, Y., Lin, G.: Traveling wave solutions in a delayed competitive model. J. Math. Anal. Appl., 507 (2022), Paper No. 125766
  • 32. Yi, T., Chen, Y., Wu, J.: Unimodal dynamical systems: comparison principles, spreading speeds and travelling waves. J. Differ. Equ. 254,...
  • 33. Ye, Q., Li, Z., Wang, M., Wu, Y.: Introduction to Reaction–Diffusion Equations, 2nd edn. Science Press, Beijing (2011)
  • 34. Zhao, X.-Q.: Dynamincal Systems in Population Biology, 2nd edn. Springer, New York (2017)