Skip to main content
SpringerLink
Log in

Invasion Traveling Waves for a Discrete Diffusive Ratio-Dependent Predator-Prey Model

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

This article is concerned with the existence of traveling wave solutions for a discrete diffusive ratio-dependent predator-prey model. By applying Schauder’s fixed point theorem with the help of suitable upper and lower solutions, we prove that there exists a positive constant c* such that when c > c*, the discrete diffusive predator-prey system admits an invasion traveling wave. The existence of an invasion traveling wave with c = c* is also established by a limiting argument and a delicate analysis of the boundary conditions. Finally, by the asymptotic spreading theory and the comparison principle, the non-existence of invasion traveling waves with speed c < c* is also proved.

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

Similar content being viewed by others

References

  1. Anderson A R A, Sleeman B D. Wave front propagation and its failure in coupled systems of discrete bistable cells modeled by Fitzhugh-Nagumo dynamics. Int J Bifurcat Chaos, 1995, 5: 63–74

    Article  Google Scholar 

  2. Chen X, Fu S C, Guo J S. Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices. SIAM J Math Anal, 2006, 38: 233–258

    Article  MathSciNet  Google Scholar 

  3. Chen X, Guo J S. Existence and asymptotic stability of travelling waves of discrete quasilinear monostable equations. J Diff Eq, 2002, 184: 549–569

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Chen Y Y, Guo J S, Yao C H. Traveling wave solutions for a continuous and discrete diffusive predator-prey model. J Math Anal Appl, 2017, 445: 212–239

    Article  MathSciNet  Google Scholar 

  6. Cheng C, Li W T, Wang Z C. Spreading speeds and travelling waves in a delayed population model with stage structure on a 2D spatial lattice. IMA J Appl Math, 2008, 73: 592–618

    Article  MathSciNet  Google Scholar 

  7. Cheng H M, Yuan R. Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion. Discrete Contin Dyn Syst, 2017, 37: 5433–5454

    Article  MathSciNet  Google Scholar 

  8. Cheng H M, Yuan R. Traveling waves of some Holling-Tanner predator-prey system with nonlocal diffusion. Appl Math Comput, 2018, 338: 12–24

    MathSciNet  MATH  Google Scholar 

  9. Chow S N, Mallet-Paret J, Shen W. Travelling waves in lattice dynamical systems. J Diff Eq, 1998, 149: 248–291

    Article  Google Scholar 

  10. Guo J S, Liang X. The minimal speed of traveling fronts for the Lotka-Volterra competition system. J Dyn Diff Eq, 2011, 23: 353–363

    Article  MathSciNet  Google Scholar 

  11. Guo J S, Wu C H. Wave propagation for a two-component lattice dynamical system arising in strong competition models. J Diff Eq, 2011, 250: 3504–3533

    Article  MathSciNet  Google Scholar 

  12. Guo J S, Wu C H. Traveling wave front for a two-component lattice dynamical system arising in competition models. J Diff Eq, 2012, 252: 4357–4391

    Article  MathSciNet  Google Scholar 

  13. Huang J, Zou X. Existence of traveling wave fronts of delayed reaction-diffusion systems without monotonicity. Discrete Contin Dyn Syst, 2003, 9: 925–936

    Article  MathSciNet  Google Scholar 

  14. Huang J, Lu G, Ruan S. Traveling wave solutions in delayed lattice differential equations with partial monotonicity. Nonlinear Anal, 2005, 60: 1331–1350

    Article  MathSciNet  Google Scholar 

  15. Huang Y L, Lin G. Traveling wave solutions in a diffusive system with two preys and one predator. J Math Anal Appl, 2014, 418: 163–184

    Article  MathSciNet  Google Scholar 

  16. Hsu C H, Lin S S. Existence and multiplicity of travelling waves in a lattice dynamical system. J Diff Eq, 2000, 164: 431–450

    Article  Google Scholar 

  17. Liang X, Zhao X Q. Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Comm Pure Appl Math, 2007, 60: 1–40

    Article  MathSciNet  Google Scholar 

  18. Lin G. Invasion traveling wave solutions of a predator-prey system. Nonlinear Anal, 2014, 96: 47–58

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Liu X X, Weng P X. Asympotic speed of wave propagation for a discrete reaction-diffusion equation. Acta Math Appl Sin (Eng Ser), 2006, 22: 369–386

    Article  Google Scholar 

  21. Ma S, Zhao X Q. Global asymptotic stability of minimal fronts in monostable lattice equations. Discrete Contin Dyn Syst, 2008, 21: 259–275

    Article  MathSciNet  Google Scholar 

  22. Ma S, Zou X. Existence, uniqueness and stability of traveling waves in a discrete reaction-diffusion monostable equation with delay. J Diff Eq, 2005 217: 54–87

    Article  Google Scholar 

  23. Ma S, Zou X. Propagation and its failure in a lattice delayed differential equation with global interaction. J Diff Eq, 2005, 212: 129–190

    Article  MathSciNet  Google Scholar 

  24. Mallet-Paret J. The global structure of travelling waves in spatially discrete dynamical systems. J Dyn Diff Eq, 1999, 11: 49–127

    Article  Google Scholar 

  25. Wang X S, Wang H Y, Wu J. Traveling waves of diffusive predator-prey systems: Disease outbreak propagation. Discrete Contin Dyn Syst, 2012, 32: 3303–3324

    Article  MathSciNet  Google Scholar 

  26. Wei D, Wu J Y, Mei M. Remark on critical speed of traveling wavefronts for Nicholson’s blowflies equation with diffusion. Acta Math Sci, 2010, 30B(5): 1561–1566

    MathSciNet  MATH  Google Scholar 

  27. Weng P, Huang H, Wu J. Asymptotic speed of propagation of wave front in a lattice delay differential equation with global interaction. IMA J Appl Math, 2003, 68: 409–439

    Article  MathSciNet  Google Scholar 

  28. Wu S L, Weng P X, Ruan S. Spatial dynamics of a lattice population model with two age classes and maturation delay. Euro J Appl Math, 2015, 26: 61–91

    Article  MathSciNet  Google Scholar 

  29. Yu Z X, Yuan R. Travelling wave solutions in non-local convolution diffusive competitive-cooperative systems. IMA J Appl Math, 2011, 76: 493–513

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  31. Zuo W J, Shi J P. Traveling wave solutions of a diffusive ratio-dependent Holling-Tanner system with distributed delay. Comm pure Appl Anal, 2018, 17: 1179–1200

    Article  MathSciNet  Google Scholar 

  32. Zhang G B, Li W T, Lin G. Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure. Math Comput Model, 2009, 49: 1021–1029

    Article  MathSciNet  Google Scholar 

  33. Zhang G B, Tian G. Stability of traveling wavefronts for a two-component lattice dynamical system arising in competition models. Can Math Bull, 2018, 61: 423–437

    Article  MathSciNet  Google Scholar 

  34. Zhang L, Li W T, Wang Z C, Sun Y J. Entire solutions for nonlocal dispersal equations with bistable nonlinearity: asymmetric case. Acta Math Sin, 2019, 35: 1771–1794

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Colloge of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, China

    Tao Su & Guobao Zhang

Authors
  1. Tao Su
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guobao Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guobao Zhang.

Additional information

This work was supported by NSF of China (11861056), Gansu Provincial Natural Science Foundation (18JR3RA093).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Su, T., Zhang, G. Invasion Traveling Waves for a Discrete Diffusive Ratio-Dependent Predator-Prey Model. Acta Math Sci 40, 1459–1476 (2020). https://doi.org/10.1007/s10473-020-0517-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-020-0517-7

Key words

2010 MR Subject Classification

Search

Navigation