Skip to main content
Log in

Spatial versus non-spatial dynamics for diffusive Lotka–Volterra competing species models

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

This paper studies the dynamics of the diffusive Lotka–Volterra competition model under small dispersion rates and Dirichlet boundary conditions. Its main goal is ascertaining the connections between the qualitative behavior of the positive solutions of the parabolic model for small diffusions and the dynamics of its associated non-spatial model given by switching them off. After sharpening very substantially some previous results of Furter and López-Gómez (Proc R Soc Edinb 127A:281–336, 1997), we characterizes the singular limit, as diffusions go to zero, of any sequence of coexistence steady-state solutions. It turns out that they must approximate, point-wise in the inhabiting territory, the global attractor of the non-spatial model, uniformly on compact subsets of the habitat zones where a global hyperbolic attractor exist. As a very special consequence of our general theorem, the Dirichlet counterpart of the singular perturbation theorem of Hutson et al. (World Sci Ser Appl Anal 4:501–533, 1995) holds. Further, a multiplicity result is given when the underlying non-spatial model exhibits a founder control competition somewhere in the territory. This is the first multiplicity theorem available in the literature for small diffusivities.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Amann, H.: On the existence of positive solutions of nonlinear elliptic boundary value problems. Indiana Univ. Math. J. 21, 125–146 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cano-Casanova, S., López-Gómez, J.: Permanence under strong aggressions is possible. Ann. I. H. Poincaré AN 20, 999–1041 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cantrell, R.S., Cosner, C.: On the effects of spatial heterogeneity on the persistence of interacting species. J. Math. Biol. 37, 103–145 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cantrell, R.S., Cosner, C.: Spatial Ecology via Reaction–Diffusion Equations. Mathematical and Computational Biology. Wiley, Chippenham (2003)

    MATH  Google Scholar 

  5. Cosner, C., Lazer, A.C.: Stable coexistence states in the Lotka–Volterra competition model with diffusion. SIAM J. Appl. Math. 44, 1112–1132 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dancer, E.N.: On the existence and uniqueness of positive solutions for competing species models with diffusion. Trans. AMS 326, 829–859 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dockery, J., Hutson, V., Mischaikow, K., Pernarowski, M.: The evolution of slow dispersal rates: a reaction diffusion model. J. Math. Biol. 37, 61–83 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fernández-Rincón, S., López-Gómez, J.: A singular perturbation result in competition theory. J. Math. Anal. Appl. 445, 280–296 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fisher, R.A.: The wave of advances of advantageous genes. Ann. Eugen. 7, 355–369 (1937)

    Article  Google Scholar 

  10. Furter, J.E., López-Gómez, J.: On the existence and uniqueness of coexistence states for the Lotka–Volterra competition model with diffusion and spatially dependent coefficients. Nonlinear Anal. Theory Methods Appl. 25, 363–398 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Furter, J.E., López-Gómez, J.: Diffusion-mediated permanence problem for a heterogeneous Lotka–Volterra competition model. Proc. R. Soc. Edinb. 127A, 281–336 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hale, J.K., Waltman, P.: Persistence in infinite-dimensional systems. SIAM J. Math. Anal. 20, 388–395 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. He, X., Ni, W.M.: The effects of diffusion and spatial variation in Lotka–Volterra competition diffusion system I: heterogeneity vs. homogeneity. J. Differ. Equ. 254, 528–546 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. He, X., Ni, W.M.: The effects of diffusion and spatial variation in Lotka–Volterra competition diffusion system II: the general case. J. Differ. Equ. 254, 4088–4108 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hess, P.: Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics Series, 247. Longman Scientific & Technical, Harlow, Essex (1991)

  16. Hess, P., Lazer, A.C.: On an abstract competition model and applications. Nonlinear Anal. Theory Methods Appl. 16, 917–940 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hutson, V., López-Gómez, J., Mischaikow, K., Vickers, G.: Limit behavior for a competing species problem with diffusion. World Sci. Ser. Appl. Anal. 4, 501–533 (1995)

    MATH  Google Scholar 

  18. Hutson, V., Lou, Y., Mischaikow, K.: Spatial heterogeneity of resources versus Lotka–Volterra dynamics. J. Differ. Equ. 185, 97–136 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hutson, V., Lou, Y., Mischaikow, K.: Convergence in competition models with small diffusion coefficients. J. Differ. Equ. 211, 135–161 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hutson, V., Lou, Y., Mischaikow, K., Poláčik, P.: Competing species near the degenerate limit. SIAM J. Math. Anal. 35, 453–491 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995)

    Book  Google Scholar 

  22. Kolmogorov, A.N., Petrovsky, I.G., Piskunov, N.S.: Etude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bull. de l’Univ. d’Etat à Mosc. (Sèr. Int.) A I, 1–26 (1937)

    Google Scholar 

  23. López-Gómez, J.: Positive periodic solutions of Lotka–Volterra reaction–diffusion systems. Differ. Integral Equ. 5, 55–72 (1992)

    MathSciNet  MATH  Google Scholar 

  24. López-Gómez, J.: Coexistence and metacoexistence states in competing species models. Houst. J. Math. 29, 485–538 (2003)

    Google Scholar 

  25. López-Gómez, J.: Linear Second Order Elliptic Operators. WSP, Singapore (2013)

    Book  MATH  Google Scholar 

  26. López-Gómez, J.: Metasolutions of Parabolic Equations in Population Dynamics. CRC Press, Boca Raton (2015)

    Book  MATH  Google Scholar 

  27. López-Gómez, J., Sabina de Lis, J.C.: Coexistence states and global attractivity for the convective diffusive competition model. Trans. AMS 347, 3797–3833 (1995)

    Article  MATH  Google Scholar 

  28. Schrödinger, E.: The Physical Aspect of the Living Cell. Cambridge University Press, Cambridge (1944)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Julián López-Gómez.

Additional information

Communicated by P. Rabinowitz.

Partially supported by the Ministry of Economy and Competitiveness of Spain under Research Grants MTM2012-30669 and MT2015-65899-P, and by the Institute of Interdisciplinary Mathematics (IMI) of Complutense University. The first author has been also supported by the Ministry of Education and Culture of Spain under Fellowship Grant FPU15/04755.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fernández-Rincón, S., López-Gómez, J. Spatial versus non-spatial dynamics for diffusive Lotka–Volterra competing species models. Calc. Var. 56, 71 (2017). https://doi.org/10.1007/s00526-017-1161-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-017-1161-5

Mathematics Subject Classification

Navigation