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Uniqueness and Stability of Coexistence States in Two Species Models With/Without Chemotaxis on Bounded Heterogeneous Environments

  • Tahir Bachar Issa
  • Wenxian Shen
Article
  • 28 Downloads

Abstract

The current paper is concerned with the asymptotic dynamics of two species competition systems with/without chemotaxis in heterogeneous media. In the previous work (Issa and Shen in Persistence, coexistence and extinction in two species chemotaxis models on bounded heterogeneous environments, 2017. https://arxiv.org/pdf/1709.10040.pdf), we find conditions on the parameters in such systems for the persistence of the two species and the existence of positive coexistence states. In this paper, we find conditions on the parameters for the uniqueness and stability of positive coexistence states of such systems. The established results are new even for the two species competition systems without chemotaxis but with space dependent coefficients.

Keywords

Ultimates bounds of solutions Asymptotic stability and uniqueness of coexistence states 

Mathematics Subject Classification

35B40 35B41 37J25 92C17 

Notes

Acknowledgements

The authors also would like to thank the referee for valuable comments and suggestions which improved the presentation of this paper considerably.

References

  1. 1.
    Ahmad, S.: Convergence and ultimate bounds of solutions of the nonautonomous Volterra–Lotka competition equations. J. Math. Anal. Appl. 127(2), 377–387 (1987)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alvarez, C., Lazek, A.C.: An application of topological degree to the periodic competing species problem. J. Austral. Morh. SIC. Ser. B 28, 202–219 (1986)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bellomo, N., Bellouquid, A., Tao, Y., Winkler, M.: Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues. Math. Models Methods Appl. Sci. 25(9), 1663–1763 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Black, T., Lankeit, J., Mizukami, M.: On the weakly competitive case in a two-species chemotaxis model. IMA J. Appl. Math. 81(5), 860–876 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fu, S., Ma, R.: Existence of a global coexistence state for periodic competition diffusion systems. Nonlinear Anal. 28, 1265–1271 (1977)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer, Berlin (1977)Google Scholar
  7. 7.
    Herrero, M.A., Velzquez, J.J.L.: Finite-time aggregation into a single point in a reaction–diffusion system. Nonlinearity 10, 1739–1754 (1997)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hetzer, G., Shen, W.: Convergence in almost periodic competition diffusion systems. J. Math. Anal. Appl. 262, 307–338 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hetzer, G., Shen, W.: Uniform persistence, coexistence, and extinction in almost periodic/nonautonomous competition diffusion systems. SIAM J. Math Anal. 34(1), 204–227 (2002)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hillen, T., Painter, K.J.: A users guide to PDE models for chemotaxis. Math. Biol. 58, 183–217 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Horstmann, D.: From 1970 until present: The Keller–Segel model in chemotaxis and its consequences. I. Jber. DMW 105, 103–165 (2003)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Isenbach, M.: Chemotaxis. Imperial College Press, London (2004)CrossRefGoogle Scholar
  13. 13.
    Issa, T.B., Shen, W.: Dynamics in chemotaxis models of parabolic–elliptic type on bounded domain with time and space dependent logistic sources. SIAM J. Appl. Dyn. Syst. 16(2), 926–973 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Issa, T.B., Salako, R.: Asymptotic dynamics in a two-species chemotaxis model with non-local terms. Discret. Cont. Dyn. Syst. Ser. B 22(10), 3839–3874 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Issa, T.B., Shen, W.: Persistence, coexistence and extinction in two species chemotaxis models on bounded heterogeneous environments, preprint (2017). https://arxiv.org/pdf/1709.10040.pdf
  16. 16.
    Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)CrossRefGoogle Scholar
  17. 17.
    Keller, E.F., Segel, L.A.: A model for chemotaxis. J. Theoret. Biol. 30, 225–234 (1971)CrossRefGoogle Scholar
  18. 18.
    Lauffenburger, D.A.: Quantitative studies of bacterial chemotaxis and microbial population dynamics. Microbial. Ecol. 22(1991), 175–85 (1991)CrossRefGoogle Scholar
  19. 19.
    Negreanu, M., Tello, J.I.: On a competitive system under chemotaxis effects with non-local terms. Nonlinearity 26, 1083–1103 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Salako, R.B., Shen, W.: Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on \({\mathbb{R}}^N\). J. Differ. Equ. 262, 5635–5690 (2017)CrossRefGoogle Scholar
  21. 21.
    Salako, R.B., Shen, W.: Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on \({\mathbb{R}}^N\). I. Persistence and asymptotic spreading. https://arxiv.org/pdf/1709.05785.pdf
  22. 22.
    Salako, R.B., Shen, W.: Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on \({\mathbb{R}}^N\). II. Existence, uniqueness and stability of positive entire solutions. https://arxiv.org/abs/1801.05310
  23. 23.
    Stinner, C., Tello, J.I., Winkler, M.: Competitive exclusion in a two-species chemotaxis model. J. Math. Biol. 68, 1607–1626 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Tello, J.I., Winkler, M.: Stabilization in two-species chemotaxis with a logistic source. Nonlinearity 25, 1413–1425 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Wang, Q., Yang, J., Zhang, L.: Time-periodic and stable patterns of a two-competing-species Keller–Segel chemotaxis model: EECT of cellular growth. Discret. Cont. Dyn. Syst. Ser. B 22(9), 3547–3574 (2017)CrossRefGoogle Scholar
  26. 26.
    Winkler, M.: Finite time blow-up in th higher-dimensional parabolic-parabolic Keller–Segel system. J. Math. Pures Appl. 100, 748–767 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsAuburn UniversityAuburnUSA

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