Skip to main content
SpringerLink
Log in

Spreading speeds determinacy for a cooperative Lotka–Volterra system with stacked fronts

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

Cooperation in a population system can result in the occurrence of a coexistence (win–win) equilibrium. When diffusion is incorporated, individual species possibly invade into far ends with different spreading speeds (co-speed or fast–slow speed). Predicting or determining them becomes challenging. This paper is devoted to the complete understanding of propagation dynamics to a cooperative Lotka–Volterra system, in addition of the determinacy of these speeds. The first major contribution of this paper is to establish a necessary and sufficient condition for the existence of a single spreading speed. In the existence of a single spreading speed (i.e., the two species share a common invasion speed), nonnegative traveling wave profiles, connecting the extinction equilibrium, exist, if the wave speed is not less than the common speed. We find that predicting or determining the invasion speed can be linked to the linearized system at the extinction state, and linear/nonlinear selections are established. The existence of multiple spreading speeds indicates new connections of traveling wave profiles into certain intermediate states. Due to this, the determinacy of multiple spreading speeds focuses not only on the extinction state but also on the corresponding intermediate states. Based on our constructions of upper–lower solutions, we also establish analytic criteria that can determine the fast–slow invasive speeds. Our speed selection mechanism can also help scientists greatly understand the movement of stacked fronts in cooperative systems with multiple species.

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

Similar content being viewed by others

References

  1. Alhasanat, A., Ou, C.: Minimal-speed selection of traveling waves to the Lotka–Volterra competition model. J. Differ. Equ. 266, 7357–7378 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alhasanat, A., Ou, C.: On a conjecture raised by Yuzo Hosono. J. Dyn. Differ. Equ. 31, 287–304 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alhasanat, A., Ou, C.: On the conjecture for the pushed wavefront to the diffusive Lotka–Volterra competition model. J. Math. Biol. 80, 1413–1422 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aronson, D.G., Weinberger, H.F.: Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Partial Differential Equations and Related Topics (Program, Tulane Univ., New Orleans, La., 1974), vol. 446, pp. 5–49 (1975)

  5. Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusion arising in population dynamics. Adv. Math. 30, 33–76 (1978)

    Article  MATH  Google Scholar 

  6. Fang, J., Zhao, X.-Q.: Traveling waves for monotone semiflows with weak compactness. SIAM J. Math. Anal. 46(6), 3678–3704 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Girardin, L., Lam, K.-Y.: Invasion of open space by two competitors: spreading properties of monostable two-species competition-diffusion systems. Proc. Lond. Math. Soc. 3(119), 1279–1335 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Huang, Z., Ou, C.: Speed determinacy of traveling waves to a stream-population model with Allee effect. SIAM J. Appl. Math. 80, 1820–1840 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  9. Huang, Z., Ou, C.: Speed selection for traveling waves of a reaction–diffusion–advection equation in a cylinder. Phys. D 402, 132225 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  10. Huang, Z., Ou, C.: Determining spreading speeds for abstract time-periodic monotone semiflows. J. Differ. Equ. 353, 339–384 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  11. Iida, M., Lui, R., Ninomiya, H.: Stacked fronts for cooperative systems with equal diffusion coefficients. SIAM J. Math. Anal. 43, 1369–1389 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kan-on, Y.: Fisher wave fronts for the Lotka–Volterra competition model with diffusion. Nonlinear Anal. 28, 145–164 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Khaikin, B.I., Filonenko, A.K., Khudyaev, S.I.: Flame propagation in the presence of two successive gas-phase reactions. Combust. Explos. Shock Waves 4, 343–349 (1968)

    Article  Google Scholar 

  14. Li, B., Weinberger, H.F., Lewis, M.A.: Spreading speeds as slowest wave speeds for cooperative systems. Math. Biosci. 196, 82–98 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liang, X., Zhao, X.-Q.: Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Commun. Pure Appl. Math. 60, 1–40 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lin, G.: Asymptotic spreading fastened by inter-specific coupled nonlinearities: a cooperative system. Phys. D 241, 705–710 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lin, G., Li, W.-T., Ma, M.: Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models. Discrete Contin. Dyn. Syst. Ser. B 13(2), 393–414 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Liu, X., Ouyang, Z., Huang, Z., Ou, C.: Spreading speed of the periodic Lotka–Volterra competition model. J. Differ. Equ. 275, 533–553 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lui, R.: Biological growth and spread modeled by systems of recursions. I. Mathematical theory. Math. Biosci. 93, 269–295 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ma, M., Huang, Z., Ou, C.: Speed of the traveling wave for the bistable Lotka–Volterra competition model. Nonlinearity 32, 3143–3162 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ma, M., Ou, C.: Asymptotic analysis of the perturbed Poisson–Boltzmann equation on unbounded domains. Asymptot. Anal. 91, 125–146 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Ma, M., Ou, C.: Linear and nonlinear speed selection for mono-stable wave propagations. SIAM J. Math. Anal 51(1), 321–345 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ma, M., Ou, C.: The minimal wave speed of a general reaction-diffusion equation with nonlinear advection. Z. Angew. Math. Phys. 72, 1–14 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ma, M., Ou, C.: Bistable wave-speed for monotone semiflows with applications. J. Differ. Equ. 323, 253–279 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ma, M., Yue, J., Ou, C.: Propagation direction of the bistable travelling wavefront for delayed non-local reaction diffusion equations. Proc. A 475, 20180898 (2019)

    MathSciNet  MATH  Google Scholar 

  26. Ma, M., Zhang, Q., Yue, J., Ou, C.: Bistable wave speed of the Lotka–Volterra competition model. J. Biol. Dyn. 14, 608–620 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mei, M., Ou, C., Zhao, X.-Q.: Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations. SIAM J. Math. Anal. 42 (6), 2762–2790 (2010)

  28. Morita, Y., Tachibana, K.: An entire solution to the Lotka–Volterra competition–diffusion equations. SIAM J. Math. Anal. 40, 2217–2240 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pan, C., Wang, H., Ou, C.: Invasive speed for a competition–diffusion system with three species. Discrete Contin. Dyn. Syst. Ser. B 27, 3515–3532 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  30. Weinberger, H.F.: Long-time behavior of a class of biological model. SIAM J. Math. Anal. 13(3), 353–396 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  31. Weinberger, H.F., Lewis, M.A., Li, B.: Analysis of linear determinacy for spread in cooperative models. J. Math. Biol. 45, 183–218 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, H., Huang, Z., Ou, C.: Speed selection for the wavefronts of the lattice Lotka–Volterra competition system. J. Differ. Equ. 268(7), 3880–3902 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang, Y., Lin, G., Ou, C.: Speed sign of traveling waves to a competitive Lotka–Volterra recursive system with bistable nonlinearity. Commun. Pure Appl. Anal. 21(12), 4287–4310 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang, H., Ou, C.: Propagation direction of the traveling wave for the Lotka–Volterra competitive lattice system. J. Dyn. Differ. Equ. 33, 1153–1174 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wang, H., Ou, C.: Propagation speed of the bistable traveling wave to the Lotka–Volterra competition system in a periodic habitat. J. Nonlinear Sci. 30, 3129–3159 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, H., Pan, C., Ou, C.: Propagation dynamics of forced pulsating waves of a time periodic Lotka–Volterra competition system in a shifting habitat. J. Differ. Equ. 340, 359–385 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, H., Pan, C., Ou, C.: Existence, uniqueness and stability of forced waves to the Lotka–Volterra competition system in a shifting environment. Stud. Appl. Math. 148(1), 186–218 (2022)

    Article  MathSciNet  Google Scholar 

  38. Wang, H., Wang, H., Ou, C.: Spreading dynamics of a Lotka–Volterra competition model in periodic habitats. J. Differ. Equ. 270, 664–693 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  39. Mei, M., Ou, C., Zhao, X.-Q.: Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations. SIAM J. Math. Anal. 42 (6), 2762–2790(2010)

  40. Ouyang, Z., Ou, C.: Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete Contin. Dyn. Syst. Ser. B 17 (3), 993–1007(2012)

  41. Ou, C. H., Wong, R.: Shooting method for nonlinear singularly perturbed boundary-value problems. Stud. Appl. Math. 112 (2), 161–200(2004)

  42. Ou, C. H., Wong, R.: On a two-point boundary-value problem with spurious solutions. Stud. Appl. Math. 111 (4), 377–408 (2003)

Download references

Author information

Authors and Affiliations

  1. School of International Programs, Guangdong University of Finance, Guangzhou, 510521, Guangdong, China

    Zhe Huang

  2. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7, Canada

    Zhe Huang & Chunhua Ou

Authors
  1. Zhe Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chunhua Ou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chunhua Ou.

Additional information

Publisher's Note

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

This work was partly supported by the Canadian NSERC discovery grants (RGPIN04709-2016 and RGPIN03842-2022).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, Z., Ou, C. Spreading speeds determinacy for a cooperative Lotka–Volterra system with stacked fronts. Z. Angew. Math. Phys. 74, 73 (2023). https://doi.org/10.1007/s00033-023-01965-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-023-01965-3

Keywords

Mathematics Subject Classification

Search

Navigation