Advertisement

Invasion Waves in a Higher-Dimensional Lattice Competitive System with Stage Structure

  • 14 Accesses

Abstract

In this paper, we use Schauder’s fixed point theorem to establish the existence of invasion waves in a stage-structured competitive system on higher-dimensional lattices. To illustrate our results, we construct a pair of upper and lower solutions.

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

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.

References

  1. 1.

    Al-Omari, J., Gourly, S.A.: Monotone travelling fronts in an age-structured reaction–diffusion model of a single species. J. Math. Biol. 45, 294–312 (2002)

  2. 2.

    Al-Omari, J., Gourly, S.A.: Stability and traveling fronts in Lotka–Volterra competition models with stage structure. SIAM J. Appl. Math. 63, 2063–2086 (2003)

  3. 3.

    Al-Omari, J., Gourly, S.A.: A nonlocal reaction–diffusion model for a single species with stage structure and distributed maturation delay. Euro. J. Appl. Math. 16, 37–51 (2005)

  4. 4.

    Al-Omari, J., Gourly, S.A.: Monotone wave-fronts in a structured population model with distributed maturation delay. IMA J. Appl. Math. 70, 858–879 (2005)

  5. 5.

    Cahn, J.W., Mallet-Paret, J., Van Vleck, E.S.: Traveling wave solutions for systems of ODE’s on a two-dimensional spatial lattice. SIAM J. Appl. Math. 59, 455–493 (1998)

  6. 6.

    Chen, X., Guo, J.: Uniqueness and existence of travelling waves for discrete quasilinear monostable dynamics. Math. Ann. 326, 123–146 (2003)

  7. 7.

    Chen, X., Guo, J.: Existence and asymptotic stability of travelling waves of discrete quasilinear monostable equations. J. Differ. Equ. 184, 549–569 (2002)

  8. 8.

    Cheng, C.P., Li, W.T., Wang, Z.C., Zheng, S.Z.: Traveling waves connecting equilibrium and periodic orbit for a delayed population model on a two-dimensional spatial lattice. Int. J. Bifurcat. Chaos Appl. Sci. Engrg. 26 1650049: 1–13 (2016)

  9. 9.

    Chow, S.-N., Mallet-Paret, J., Shen, W.: Traveling waves in lattice dynamical systems. J. Differ. Equ. 149, 248–291 (1998)

  10. 10.

    Gourley, S.A., Kuang, Y.: Wavefronts and global stability in a time-delayed population model with stage structure. Proc. R. Soc. Lond. Ser. A 459, 1563–1579 (2003)

  11. 11.

    Gourly, S.A., Kuang, Y.: A stage structured predator-prey model and its dependence on maturation delay and death rate. J. Math. Biol. 49, 188–200 (2004)

  12. 12.

    Guo, J.S., Wu, C.H.: Existence and uniqueness of traveling waves for a monostable 2-D lattice dynamical system. Osaka J. Math. 45, 327–346 (2008)

  13. 13.

    Guo, J., Wu, C.: Traveling wave front for a two-component lattice dynamical system arising in competition models. J. Differ. Equ. 252, 4357–4391 (2012)

  14. 14.

    Huang, J., Lu, G., Ruan, S.: Existence of traveling wave fronts of delayed lattice differential equations. J. Math. Anal. Appl. 298, 538–558 (2004)

  15. 15.

    Huang, J., Lu, G.: Travelling wave solutions in delayed lattice dynamical system. Chin. Ann. Math. Ser. A 25, 153–164 (2004)

  16. 16.

    Huang, J., Huang, L.: Traveling wavefronts in systems of delayed reaction diffusion equations on higher dimensional lattices. Acta Math. Appl. Sin. Ser. A 28, 100–113 (2005)

  17. 17.

    Hou, X.J., Leung, A.W.: Traveling wave solutions for a competitive reaction–diffusion system and their asymptotics. Nonlinear Anal. RWA 9, 2196–2213 (2008)

  18. 18.

    Kan-on, Y.: Parameter dependence of propagation speed of traveling waves for competition–diffusion equations. SIAM J. Math. Anal. 26, 340–363 (1995)

  19. 19.

    Kuang, Y., So, J.: Analysis of a delayed two-stage population with space-limited recruitment. SIAM J. Appl. Math. J 55, 1675–1695 (1995)

  20. 20.

    Leung, A.W., Hou, X.J., Li, Y.: Exclusive traveling waves for competitive reaction-diffusion systems and their stabilities. J. Math. Anal. Appl. 338, 902–924 (2008)

  21. 21.

    Li, W.T., Zhang, L., Zhang, G.B.: Invasion entire solutions in a competition system with nonlocal dispersal. Discrete Contin. Dyn. Syst. Ser. A 35, 1531–1560 (2014)

  22. 22.

    Lin, G., Li, W.: Traveling waves in delayed lattice dynamical systems with competition interactions. Nonlinear Anal. RWA 11, 3666–3679 (2010)

  23. 23.

    Liu, S., Chen, L., Liu, Z.: Extinction and permanence in nonautonomous competitive system with stage structure. J. Math. Anal. Appl. 274, 667–684 (2002)

  24. 24.

    Pan, S., Lin, G.: Invasion traveling wave solutions of a competitive system with dispersal. Bound. Value Probl. 2012, 1–11 (2012)

  25. 25.

    So, J.W.H., Wu, J., Zou, X.: A reaction–diffusion model for a single species with age structure. I, Travelling wavefronts on unbounded domains. Proc. R. Soc. Lond. Ser. A 457, 1841–1853 (2001)

  26. 26.

    Weng, P.X.: Spreading speed and traveling wavefront of an age-structured population diffusing in a 2D lattice strip. Discrete Contin. Dyn. Syst. Ser. B 12, 883–904 (2009)

  27. 27.

    Wang, Y., Li, X.: Some entire solutions to the competitive reaction diffusion system. J. Math. Anal. Appl. 430, 993–1008 (2015)

  28. 28.

    Wu, S.L., Liu, S.Y.: Travelling waves in delayed reaction–diffusion equations on higher dimensional lattices. J. Differ. Equ. Appl. 19, 384–401 (2013)

  29. 29.

    Wu, J., Zou, X.: Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations. J. Differ. Equ. 135, 315–357 (1997)

  30. 30.

    Xu, R., Chaplain, M., Davidson, F.: Travelling wave and convergence in stage-structured reaction–diffusion competitive models with nonlocal delays. Chaos Solitons Fractals 30, 974–992 (2006)

  31. 31.

    Yu, Z.X., Yuan, R.: Nonlinear stability of wavefronts for a delayed stage-structured population model on a 2-D lattice. Osaka J. Math. 50, 963–976 (2013)

  32. 32.

    Yu, Z.X., Zhang, W.G., Wang, X.M.: Spreading speeds and travelling waves for non-monotone time-delayed 2D lattice systems. Math. Comput. Model. 58, 1510–1521 (2013)

  33. 33.

    Zhao, H.Q., Wu, S.L.: Wave propagation for a reaction–diffusion model with a quiescent stage on a 2D spatial lattice. Nonlinear Anal. RWA 12, 1178–1191 (2011)

  34. 34.

    Zhao, H.Q.: Asymptotic stability of traveling fronts in delayed reaction-diffusion monostable equations on higher-dimensional lattices. Electron. J. Differ. Equ. 2013(119), 1–15 (2013)

  35. 35.

    Zou, X.: Traveling wave fronts in spatially discrete reaction-diffusion equations on higher-dimensional lattices. In: Proceedings of the Third Mississippi State Conference on Difference Equations and Computational Simulations (Mississippi State, MS, 1997), pp. 211-221, Electron. J. Differ. Equ. Conf. 1, Southwest Texas State Univ., San Marcos, TX (1998)

Download references

Author information

Correspondence to Kun Li.

Additional information

Publisher's Note

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

Supported by the National Natural Science Foundation of China (Grant No. 11971160) and the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 18B472).

Communicated by See Keong Lee.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Li, K. Invasion Waves in a Higher-Dimensional Lattice Competitive System with Stage Structure. Bull. Malays. Math. Sci. Soc. (2020) doi:10.1007/s40840-020-00890-2

Download citation

Keywords

  • Higher-dimensional lattice
  • Stage structure
  • Traveling wave solution
  • Schauder’s fixed point theorem
  • Upper and lower solutions

Mathematics Subject Classification

  • 37L60
  • 34K10
  • 39A10