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.
Similar content being viewed by others
References
Ahmad, S.: Convergence and ultimate bounds of solutions of the nonautonomous Volterra–Lotka competition equations. J. Math. Anal. Appl. 127(2), 377–387 (1987)
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)
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). https://doi.org/10.1093/imamat/hxw036
Fu, S., Ma, R.: Existence of a global coexistence state for periodic competition diffusion systems. Nonlin. Anal. 28, 1265–1271 (1977)
Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer, Berlin (1977)
Herrero, M.A., Velzquez, J.J.L.: Finite-time aggregation into a single point in a reaction–diffusion system. Nonlinearity 10, 1739–1754 (1997)
Hetzer, G., Shen, W.: Convergence in almost periodic competition diffusion systems. J. Math. Anal. Appl. 262, 307–338 (2001)
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)
Hillen, T., Painter, K.J.: A users guide to PDE models for chemotaxis. Math. Biol. 58, 183–217 (2009)
Horstmann, D.: From 1970 until present: The Keller–Segel model in chemotaxis and its consequences I. Jber DMW 105, 103–165 (2003)
Isenbach, M.: Chemotaxis. Imperial College Press, London (2004)
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)
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)
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
Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)
Keller, E.F., Segel, L.A.: A model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971)
Lauffenburger, D.A.: Quantitative studies of bacterial chemotaxis and microbial population dynamics. Microb. Ecol. 22(1991), 175–85 (1991)
Negreanu, M., Tello, J.I.: On a competitive system under chemotaxis effects with non-local terms. Nonlinearity 26, 1083–1103 (2013)
Stinner, C., Tello, J.I., Winkler, W.: Competive exclusion in a two-species chemotaxis. J. Math. Biol. 68, 1607–1626 (2014)
Tao, Y., Winkler, M.: Persistence of mass in a chemotaxis system with logistic source. J. Differ. Equ. 259(11), 6142–6161 (2015)
Tello, J.I., Winkler, M.: Stabilization in two-species chemotaxis with a logistic source. Nonlinearity 25, 1413–1425 (2012)
Winkler, M.: Finite time blow-up in th higher-dimensional parabolic-parabolic Keller–Segel system. J. Math. Pures Appl. 100, 748–767 (2013)
Acknowledgements
The authors also would like to thank the referee for valuable comments and suggestions which improved the presentation of this paper considerably
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Issa, T.B., Shen, W. Persistence, Coexistence and Extinction in Two Species Chemotaxis Models on Bounded Heterogeneous Environments. J Dyn Diff Equat 31, 1839–1871 (2019). https://doi.org/10.1007/s10884-018-9686-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-018-9686-7
Keywords
- Global existence
- Classical solutions
- Persistence
- Coexistence states
- Entire solutions
- Periodic solutions
- Almost periodic solutions
- Steady state solutions
- Extinction
- Comparison principle