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March  2016, 21(2): 621-639. doi: 10.3934/dcdsb.2016.21.621

## Competition between two similar species in the unstirred chemostat

 1 College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710119 2 Institute for Mathematical Sciences, Renmin University of China, Haidian District, Beijing, 100872

Received  January 2015 Revised  May 2015 Published  November 2015

This paper deals with the competition between two similar species in the unstirred chemostat. Due to the strict competition of the unstirred chemostat model, the global dynamics of the system is attained by analyzing the equilibria and their stability. It turns out that the dynamics of the system essentially depends upon certain function of the growth rate. Moreover, one of the semi-trivial stationary solutions or the unique coexistence steady state is a global attractor under certain conditions. Biologically, the results indicate that it is possible for the mutant to force the extinction of resident species or to coexist with it.
Citation: Hua Nie, Yuan Lou, Jianhua Wu. Competition between two similar species in the unstirred chemostat. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 621-639. doi: 10.3934/dcdsb.2016.21.621
##### References:
 [1] F. Castella and S. Madec, Coexistence phenomena and global bifurcation structure in a chemostat-like model with species-dependent diffusion rates,, J. Math. Biol., 68 (2014), 377. doi: 10.1007/s00285-012-0633-7. [2] L. Dung and H. L. Smith, A parabolic system modeling microbial competition in an unmixed bio-reactor,, J. Differential Equations, 130 (1996), 59. doi: 10.1006/jdeq.1996.0132. [3] G. Guo, J. H. Wu and Y. Wang, Bifurcation from a double eigenvalue in the unstirred chemostat,, Appl. Anal., 92 (2013), 1449. doi: 10.1080/00036811.2012.683786. [4] P. Hess, Periodic Parabolic Boundary Value Problems and Positivity,, Longman Scientific & Technical, (1991). [5] S. B. Hsu, H. L. Smith and P. Waltman, Dynamics of competition in the unstirred Chemostat,, Canad. Appl. Math. Quart., 2 (1994), 461. [6] S. B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an unstirred Chemostat,, SIAM J. Appl. Math., 53 (1993), 1026. doi: 10.1137/0153051. [7] V. Hutson, Y. Lou, K. Mischaikow and P. Poláčik, Competing species near a degenerate limit,, SIAM J. Math. Anal., 35 (2003), 453. doi: 10.1137/S0036141002402189. [8] Y. Lou, S. Martinez and P. Poláčik, Loops and branches of coexistence states in a Lotka-Volterra competition model,, J. Differential Equations, 230 (2006), 720. doi: 10.1016/j.jde.2006.04.005. [9] H. Nie and J. H. Wu, Positive solutions of a competition model for two resources in the unstirred chemostat,, J. Math. Anal. Appl., 355 (2009), 231. doi: 10.1016/j.jmaa.2009.01.045. [10] H. Nie and J. H. Wu, Uniqueness and stability for coexistence solutions of the unstirred chemostat model,, Appl. Anal., 89 (2010), 1141. doi: 10.1080/00036811003717954. [11] H. Nie and J. H. Wu, The effect of toxins on the plasmid-bearing and plasmid-free model in the unstirred chemostat,, Discrete Contin. Dyn. Syst., 32 (2012), 303. doi: 10.3934/dcds.2012.32.303. [12] H. Nie and J. H. Wu, Multiple coexistence solutions to the unstirred chemostat model with plasmid and toxin,, European J. Appl. Math., 25 (2014), 481. doi: 10.1017/S0956792514000096. [13] H. Nie, W. Xie and J. H. Wu, Uniqueness of positive steady state solutions to the unstirred chemostat model with external inhibitor,, Comm. Pure Appl. Anal., 12 (2013), 1279. doi: 10.3934/cpaa.2013.12.1279. [14] T. Kato, Perturbation Theory of Linear Operators,, Springer, (1966). [15] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Mathematical Surveys and Monographs 41, (1995). [16] H. L. Smith and P. Waltman, The Theory of the Chemostat,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511530043. [17] J. W. H. So and P. Waltman, A nonlinear boundary value problem arising from competition in the chemostat,, Appl. Math. Comput., 32 (1989), 169. doi: 10.1016/0096-3003(89)90092-1. [18] H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755. doi: 10.1007/BF00173267. [19] J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat,, Nonlinear Anal., 39 (2000), 817. doi: 10.1016/S0362-546X(98)00250-8. [20] J. H. Wu, H. Nie and G. S. K. Wolkowicz, The effect of inhibitor on the plasmid-bearing and plasmid-free model in the unstirred chemostat,, SIAM J. Math. Anal., 38 (2007), 1860. doi: 10.1137/050627514. [21] J. H. Wu and G. S. K. Wolkowicz, A system of resource-based growth models with two resources in the un-stirred chemostat,, J. Differential Equations, 172 (2001), 300. doi: 10.1006/jdeq.2000.3870.

show all references

##### References:
 [1] F. Castella and S. Madec, Coexistence phenomena and global bifurcation structure in a chemostat-like model with species-dependent diffusion rates,, J. Math. Biol., 68 (2014), 377. doi: 10.1007/s00285-012-0633-7. [2] L. Dung and H. L. Smith, A parabolic system modeling microbial competition in an unmixed bio-reactor,, J. Differential Equations, 130 (1996), 59. doi: 10.1006/jdeq.1996.0132. [3] G. Guo, J. H. Wu and Y. Wang, Bifurcation from a double eigenvalue in the unstirred chemostat,, Appl. Anal., 92 (2013), 1449. doi: 10.1080/00036811.2012.683786. [4] P. Hess, Periodic Parabolic Boundary Value Problems and Positivity,, Longman Scientific & Technical, (1991). [5] S. B. Hsu, H. L. Smith and P. Waltman, Dynamics of competition in the unstirred Chemostat,, Canad. Appl. Math. Quart., 2 (1994), 461. [6] S. B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an unstirred Chemostat,, SIAM J. Appl. Math., 53 (1993), 1026. doi: 10.1137/0153051. [7] V. Hutson, Y. Lou, K. Mischaikow and P. Poláčik, Competing species near a degenerate limit,, SIAM J. Math. Anal., 35 (2003), 453. doi: 10.1137/S0036141002402189. [8] Y. Lou, S. Martinez and P. Poláčik, Loops and branches of coexistence states in a Lotka-Volterra competition model,, J. Differential Equations, 230 (2006), 720. doi: 10.1016/j.jde.2006.04.005. [9] H. Nie and J. H. Wu, Positive solutions of a competition model for two resources in the unstirred chemostat,, J. Math. Anal. Appl., 355 (2009), 231. doi: 10.1016/j.jmaa.2009.01.045. [10] H. Nie and J. H. Wu, Uniqueness and stability for coexistence solutions of the unstirred chemostat model,, Appl. Anal., 89 (2010), 1141. doi: 10.1080/00036811003717954. [11] H. Nie and J. H. Wu, The effect of toxins on the plasmid-bearing and plasmid-free model in the unstirred chemostat,, Discrete Contin. Dyn. Syst., 32 (2012), 303. doi: 10.3934/dcds.2012.32.303. [12] H. Nie and J. H. Wu, Multiple coexistence solutions to the unstirred chemostat model with plasmid and toxin,, European J. Appl. Math., 25 (2014), 481. doi: 10.1017/S0956792514000096. [13] H. Nie, W. Xie and J. H. Wu, Uniqueness of positive steady state solutions to the unstirred chemostat model with external inhibitor,, Comm. Pure Appl. Anal., 12 (2013), 1279. doi: 10.3934/cpaa.2013.12.1279. [14] T. Kato, Perturbation Theory of Linear Operators,, Springer, (1966). [15] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Mathematical Surveys and Monographs 41, (1995). [16] H. L. Smith and P. Waltman, The Theory of the Chemostat,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511530043. [17] J. W. H. So and P. Waltman, A nonlinear boundary value problem arising from competition in the chemostat,, Appl. Math. Comput., 32 (1989), 169. doi: 10.1016/0096-3003(89)90092-1. [18] H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755. doi: 10.1007/BF00173267. [19] J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat,, Nonlinear Anal., 39 (2000), 817. doi: 10.1016/S0362-546X(98)00250-8. [20] J. H. Wu, H. Nie and G. S. K. Wolkowicz, The effect of inhibitor on the plasmid-bearing and plasmid-free model in the unstirred chemostat,, SIAM J. Math. Anal., 38 (2007), 1860. doi: 10.1137/050627514. [21] J. H. Wu and G. S. K. Wolkowicz, A system of resource-based growth models with two resources in the un-stirred chemostat,, J. Differential Equations, 172 (2001), 300. doi: 10.1006/jdeq.2000.3870.
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