Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Coexistence and extinction for two competing species in patchy environments

1 Department of Applied Mathematics, National Pingtung University, Pingtung, Taiwan 900
2 Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan 300

A system of two competing species $u$ and $v$ that diffuse over a two-patch environment is investigated. When $u$-species has smaller birth rate in the first patch and larger birth rate in the second patch than $v$-species, and the average birth rate for $u$-species is larger than or equal to $v$-species, it was shown in a previous publication that two species coexist in a slow diffusion environment, whereas $u$-species drives $v$-species into extinction in a fast diffusion environment. In this paper, we analyze global dynamics and bifurcations for the same model with identical order of birth rates, but with opposite order of average birth rates, i.e., the average birth rate of $u$-species is less than that of $v$-species. We observe richer dynamics with two scenarios, depending on the relative difference between the variation in the birth rates of $v$-species on two patches and the variation in the average birth rates of two species. When the variation in average birth rates is relatively large, there is no stability switch for the semitrivial equilibria. On the other hand, such a stability switch takes place when the variation in average birth rates is relatively mild. In both cases, $v$-species, with larger average birth rate, prevails in a fast diffusion environment, whereas in a slow diffusion environment, the two species can coexist or $u$-species that has the greatest birth rate among both species and patches will persist and drive $v$-species to extinction.
  Article Metrics

Keywords competing species; dispersal rate; patchy environment; spatial heterogeneity; global dynamics; monotone dynamics

Citation: Chang-Yuan Cheng, Kuang-Hui Lin, Chih-Wen Shih. Coexistence and extinction for two competing species in patchy environments. Mathematical Biosciences and Engineering, 2019, 16(2): 909-946. doi: 10.3934/mbe.2019043


  • 1. E. Braveman and Md. Kamrujjamam, Lotka systems with directed dispersal dynamics: Competition and influence of diffusion strategies, Math. Biosci., 279 (2016), 1–12.
  • 2. R. S. Cantrell and C. Cosner, Spatial ecology via reaction-diffusion equations, Series in Mathematical and Computational Biology, John Wiley and Sons, Chichester, UK, 2003.
  • 3. R. S. Cantrell, C. Cosner and Y. Lou, Multiple reversals of competitive dominance in ecological reserves via external habitat degradation, J. Dyn. Diff. Eqs., 16 (2004), 973–1010.
  • 4. R. S. Cantrell, C. Cosner, D. L. DeAngelis and V. Padron, The ideal free distribution as an evolutionarily stable strategy, J. Biol. Dyn., 1 (2007), 249–271.
  • 5. R. S. Cantrell, C. Cosner and Y. Lou, Evolutionary stability of ideal free dispersal strategies in patchy environments, J. Math. Biol., 65 (2012), 943–965.
  • 6. X. Chen, K. Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discret. Contin. Dyn. Syst., 32 (2012), 3841–3859.
  • 7. J. Clobert, E. Danchin, A. A. Dhondt and J. D. Nichols, Dispersal, Oxford University Press, 2001.
  • 8. R. Cressman and V. Krivan, Two-patch population models with adaptive dispersal the effects of varying dispersal speeds, J. Math. Biol., 67 (2013), 329–358.
  • 9. J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: a reaction diffusion model, J. Math. Biol., 37 (1998), 61–83.
  • 10. G. F. Gause, The Struggle for Existence, Williams Wilkins, Baltimore, MD, 1934.
  • 11. S. A. Gourley and Y. Kuang, Two-species competition with high dispersal: the winning strategy, Math. Biosci. Eng., 2 (2005), 345–362.
  • 12. A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Popul. Biol., 24 (1983), 244–251.
  • 13. X. Q. He and W. M. Ni, The effects of diffusion and spatial variation in LotkaVolterra competitiondi ffusion system I: Heterogeneity vs. homogeneity, J. Differ. Eqs., 254 (2013), 528–546.
  • 14. X. Q. He and W. M. Ni, The effects of diffusion and spatial variation in LotkaVolterra competitiondi ffusion system II: The general case, J. Differ. Eqs., 254 (2013), 4088–4108.
  • 15. X. Q. He and W. M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: diffusion and spatial heterogeneity I, Commun. Pure Appl. Math., LXIX (2016), 981–1014.
  • 16. R. D. Holt, Population dynamics in two-patch environments some anomalous consequences of an optimal habitat distribution, Theor. Popul. Biol., 28 (1985), 181–208.
  • 17. S. B. Hsu, H. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083–4094.
  • 18. V. Hutson, J. Lopez-Gomez, K. Mischaikow and G. Vickers, Limit behaviour for a competing species problem with diffusion, in Dynamical Systems and Applications, in World Sci. Ser. Appl. Anal. 4, World Scientific, River Edge, NJ, (1995), 343–358.
  • 19. V. Hutson, Y. Lou, K. Mischaikow and P. Poláčik, Competing species near a degenerate limit, SIAM. J. Math. Anal., 35 (2003), 453–491.
  • 20. V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483–517.
  • 21. J. Jiang and X. Liang, Competitive systems with migration and the Poincaré-Bendixson theorem for a 4-dimensional case, Quar. Appl. Math., LXIV (2006), 483–498.
  • 22. Y. Kuang and Y. Takeuchi, Predator-prey dynamics in models of prey dispersal in two-patch environments, Math. Biosci., 120 (1994), 77–98.
  • 23. K. Y. Lam and W. M. Ni, Uniqueness and complete dynamics in heterogeneous competitiondi ffusion systems, SIAM J. Appl. Math., 72 (2012), 1695–1712.
  • 24. K. H. Lin, Y. Lou, C. W. Shih and T. H. Tsai, Global dynamics for two-species competition in patchy environment, Math. Biosci. Eng., 11 (2014), 947–970.
  • 25. Y. Lou, S. Martinez and P. Polacik, Loops and branches of coexistence states in a Lotka-Volterra competition model, J. Differ. Eqs., 230 (2006), 720–742.
  • 26. A. Okubo and S. A. Levin, Diffusion and ecological problems: modern perspectives, second ed., Interdisciplinary Applied Mathematics, 14, Springer, New York, 2001.
  • 27. H. L. Smith, Competing subcommunities of mutualists and a generalized Kamke theorem, SIAM J. Appl. Math., 46 (1986), 856–874.
  • 28. H. L. Smith, Monotone Dynamical Systems: an introduction to the theory of competitive and cooperative systems, Math. Surveys and Monographs, Amer. Math. Soc., 41, 1995.
  • 29. Y. Takeuchi, Diffusion-mediated persistence in two-species competition Lotka-Volterra model, Math. Biosci., 95 (1989), 65–83.
  • 30. Y. Takeuchi and Z. Lu, Permanence and global stability for competitive Lotka-Volterra diffusion systems, Nonlinear Anal. TMA, 24 (1995), 91–104.
  • 31. Y. Takeuchi, Global dynamical properties of Lotka-Volterra systems, River Edge, NJ 07661, World Scientic, 1996.


Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved