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
This is a preview of subscription content, log in to check access.
The authors also would like to thank the referee for valuable comments and suggestions which improved the presentation of this paper considerably
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
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
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
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
Issa, T.B, Shen, W.: Uniqueness and stability of coexistence states in two species models with/without chemotaxis on bounded heterogeneous environments, preprint (2017) https://arxiv.org/pdf/1803.04107.pdf