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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
Article

Abstract

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.

Keywords

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 

Notes

Acknowledgements

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

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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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