Journal of Dynamics and Differential Equations

, Volume 31, Issue 4, pp 1839–1871 | Cite as

Persistence, Coexistence and Extinction in Two Species Chemotaxis Models on Bounded Heterogeneous Environments

  • Tahir Bachar IssaEmail author
  • Wenxian Shen


In this paper, we consider two species chemotaxis systems with Lotka–Volterra type competition terms in heterogeneous media. We first find various conditions on the parameters which guarantee the global existence and boundedness of classical solutions with nonnegative initial functions. Next, we find further conditions on the parameters which establish the persistence of the two species. Then, under the same set of conditions for the persistence of two species, we prove the existence of coexistence states. Finally we prove the extinction phenomena in the sense that one of the species dies out asymptotically and the other reaches its carrying capacity as time goes to infinity. The persistence in general two species chemotaxis systems is studied for the first time. Several important techniques are developed to study the persistence and coexistence of two species chemotaxis systems. Many existing results on the persistence, coexistence, and extinction on two species competition systems without chemotaxis are recovered.


Global existence Classical solutions Persistence Coexistence states Entire solutions Periodic solutions Almost periodic solutions Steady state solutions Extinction Comparison principle 

Mathematics Subject Classification

35A01 35A02 35B08 35B40 35K57 35Q92 92C17 



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


  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.
    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
  3. 3.
    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). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fu, S., Ma, R.: Existence of a global coexistence state for periodic competition diffusion systems. Nonlin. Anal. 28, 1265–1271 (1977)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer, Berlin (1977)Google Scholar
  6. 6.
    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
  7. 7.
    Hetzer, G., Shen, W.: Convergence in almost periodic competition diffusion systems. J. Math. Anal. Appl. 262, 307–338 (2001)MathSciNetCrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    Hillen, T., Painter, K.J.: A users guide to PDE models for chemotaxis. Math. Biol. 58, 183–217 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Horstmann, D.: From 1970 until present: The Keller–Segel model in chemotaxis and its consequences I. Jber DMW 105, 103–165 (2003)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Isenbach, M.: Chemotaxis. Imperial College Press, London (2004)CrossRefGoogle Scholar
  12. 12.
    Issa, T.B., Salako, R.: Asymptotic dynamics in a two-species chemotaxis model with non-local terms. Discret. Contin. Dyn. Syst. Ser. B 22(10), 3839–3874 (2017)MathSciNetzbMATHGoogle 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, Shen, W.: Uniqueness and stability of coexistence states in two species models with/without chemotaxis on bounded heterogeneous environments, preprint (2017)
  15. 15.
    Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Keller, E.F., Segel, L.A.: A model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971)CrossRefGoogle Scholar
  17. 17.
    Lauffenburger, D.A.: Quantitative studies of bacterial chemotaxis and microbial population dynamics. Microb. Ecol. 22(1991), 175–85 (1991)CrossRefGoogle Scholar
  18. 18.
    Negreanu, M., Tello, J.I.: On a competitive system under chemotaxis effects with non-local terms. Nonlinearity 26, 1083–1103 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Stinner, C., Tello, J.I., Winkler, W.: Competive exclusion in a two-species chemotaxis. J. Math. Biol. 68, 1607–1626 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Tao, Y., Winkler, M.: Persistence of mass in a chemotaxis system with logistic source. J. Differ. Equ. 259(11), 6142–6161 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Tello, J.I., Winkler, M.: Stabilization in two-species chemotaxis with a logistic source. Nonlinearity 25, 1413–1425 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    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

Personalised recommendations