# American Institute of Mathematical Sciences

January  2012, 32(1): 303-329. doi: 10.3934/dcds.2012.32.303

## The effect of toxins on the plasmid-bearing and plasmid-free model in the unstirred chemostat

 1 College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710062, China 2 College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710119, China

Received  July 2010 Revised  June 2011 Published  September 2011

This paper deals with an unstirred chemostat model of competition between plasmid-bearing and plasmid-free organisms when the plasmid-bearing organism produces toxins. The toxins are lethal to the plasmid-free organism, which leads to the conservation principle cannot be applied, and the resulting dynamical system is described by three nonlinear partial differential equations and is not monotone. First, the existence and multiplicity of the positive steady-state solutions are determined by bifurcation theory and degree theory. Second, the effects of the toxins are considered by perturbation technique. The results show that if the parameter $r$, which measures the effect of the toxins, is sufficiently large, this model has at least two positive solutions provided that the maximal growth rate $a$ of $u$ lies in a certain range; and has only a unique asymptotically stable positive solution when $a$ belongs to another range.
Citation: Hua Nie, Jianhua Wu. The effect of toxins on the plasmid-bearing and plasmid-free model in the unstirred chemostat. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 303-329. doi: 10.3934/dcds.2012.32.303
##### References:
 [1] L. Chao and B. R. Levin, Structured habitats and the evolution of anti-competitor toxins in bacteria,, Proc. Natl Acad. Sci., 75 (1981), 6324.  doi: 10.1073/pnas.78.10.6324.  Google Scholar [2] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar [3] M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability,, Arch. Rational Mech. Anal., 52 (1973), 161.  doi: 10.1007/BF00282325.  Google Scholar [4] E. N. Dancer, On the indices of fixed points of mappings in cones and applications,, J. Math. Anal. Appl., 91 (1983), 131.  doi: 10.1016/0022-247X(83)90098-7.  Google Scholar [5] E. N. Dancer, On positive solutions of some pairs of differential equations,, Trans. Amer. Math. Soc., 284 (1984), 729.  doi: 10.1090/S0002-9947-1984-0743741-4.  Google Scholar [6] E. N. Dancer and Y. Du, Positive solutions for a three-species competition system with diffusion, Part I, General existence results,, Nonlinear Anal., 24 (1995), 337.  doi: 10.1016/0362-546X(94)E0063-M.  Google Scholar [7] Y. Du, Positive periodic solutions of a competitor-competitor-mutualist model,, Differential Integral Equations, 9 (1996), 1043.   Google Scholar [8] D. G. Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation,, Comm. Partial Differential Equations, 17 (1992), 339.   Google Scholar [9] S. R. Hansen and S. P. Hubbell, Single nutrient microbial competition: Agreement between experimental and theoretical forecast outcomes,, Science, 207 (1980), 1491.  doi: 10.1126/science.6767274.  Google Scholar [10] S. B. Hsu, Y. S. Li and P. Waltman, Competition in the presence of a lethal external inhibitor,, Math. Biosci., 167 (2000), 177.  doi: 10.1016/S0025-5564(00)00030-4.  Google Scholar [11] S. B. Hsu, T. K. Luo and P. Waltman, Competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor,, J. Math. Biol., 34 (1995), 225.  doi: 10.1007/BF00178774.  Google Scholar [12] S. B. Hsu and P. Waltman, Analysis of a model of two competitors in a chemostat with an external inhibitor,, SIAM J. Appl. Math., 52 (1992), 528.  doi: 10.1137/0152029.  Google Scholar [13] S. B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an un-stirred chemostat,, SIAM J. Appl. Math., 53 (1993), 1026.  doi: 10.1137/0153051.  Google Scholar [14] S. B. Hsu and P. Waltman, Competition between plasmid-bearing and plasmid-free organisms in selective media,, Chem. Engrg. Sci., 52 (1997), 23.  doi: 10.1016/S0009-2509(96)00385-5.  Google Scholar [15] S. B. Hsu and P. Waltman, Competition in the chemostat when one competitor produces a toxin,, Japan J. Indust. Appl. Math., 15 (1998), 471.  doi: 10.1007/BF03167323.  Google Scholar [16] S. B. Hsu and P. Waltman, A model of the effect of anti-competitor toxins on plasmid-bearing, plasmid-free compettion,, Taiwanese J. Math., 6 (2002), 135.   Google Scholar [17] S. B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor,, Math. Biosci., 187 (2004), 53.  doi: 10.1016/j.mbs.2003.07.004.  Google Scholar [18] S. B. Hsu, P. Waltman and G. S. K. Wolkowicz, Global analysis of a model of plasmid-bearing, plasmid-free competition in the chemostat,, J. Math. Biol., 32 (1994), 731.  doi: 10.1007/BF00163024.  Google Scholar [19] B. R. Levin, Frequency-dependent selection in bacterial population,, Phil. Trans. R. Soc. Lond., 319 (1988), 459.  doi: 10.1098/rstb.1988.0059.  Google Scholar [20] R. E. Lenski and S. Hattingh, Coexistence of two competitors on one resource and one inhibitor: A chemostat model based on bacteria and antibiotics,, J. Theoret. Biol., 122 (1986), 83.  doi: 10.1016/S0022-5193(86)80226-0.  Google Scholar [21] H. Nie and J. Wu, A system of reaction-diffusion equations in the unstirred chemostat with an inhibitor,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 989.  doi: 10.1142/S0218127406015246.  Google Scholar [22] H. Nie and J. Wu, Asymptotic behavior of an unstirred chemostat with internal inhibitor,, J. Math. Anal. Appl., 334 (2007), 889.  doi: 10.1016/j.jmaa.2007.01.014.  Google Scholar [23] H. L. Smith and P. Waltman, "The Theory of the Chemostat,", Cambridge University Press, (1995).   Google Scholar [24] J. Wu, H. Nie and G. S. K. Wolkowicz, A mathematical model of competition for two essential resources in the unstirred chemostat,, SIAM J. Appl. Math., 65 (2004), 209.  doi: 10.1137/S0036139903423285.  Google Scholar [25] J. 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.  Google Scholar [26] J. 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.  Google Scholar [27] J. 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.   Google Scholar [28] S. Zheng and J. Liu, Coexistence solutions for a reaction-diffusion system of un-stirred chemostat model,, Appl. Math. Comput., 145 (2003), 579.  doi: 10.1016/S0096-3003(02)00732-4.  Google Scholar

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##### References:
 [1] L. Chao and B. R. Levin, Structured habitats and the evolution of anti-competitor toxins in bacteria,, Proc. Natl Acad. Sci., 75 (1981), 6324.  doi: 10.1073/pnas.78.10.6324.  Google Scholar [2] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar [3] M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability,, Arch. Rational Mech. Anal., 52 (1973), 161.  doi: 10.1007/BF00282325.  Google Scholar [4] E. N. Dancer, On the indices of fixed points of mappings in cones and applications,, J. Math. Anal. Appl., 91 (1983), 131.  doi: 10.1016/0022-247X(83)90098-7.  Google Scholar [5] E. N. Dancer, On positive solutions of some pairs of differential equations,, Trans. Amer. Math. Soc., 284 (1984), 729.  doi: 10.1090/S0002-9947-1984-0743741-4.  Google Scholar [6] E. N. Dancer and Y. Du, Positive solutions for a three-species competition system with diffusion, Part I, General existence results,, Nonlinear Anal., 24 (1995), 337.  doi: 10.1016/0362-546X(94)E0063-M.  Google Scholar [7] Y. Du, Positive periodic solutions of a competitor-competitor-mutualist model,, Differential Integral Equations, 9 (1996), 1043.   Google Scholar [8] D. G. Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation,, Comm. Partial Differential Equations, 17 (1992), 339.   Google Scholar [9] S. R. Hansen and S. P. Hubbell, Single nutrient microbial competition: Agreement between experimental and theoretical forecast outcomes,, Science, 207 (1980), 1491.  doi: 10.1126/science.6767274.  Google Scholar [10] S. B. Hsu, Y. S. Li and P. Waltman, Competition in the presence of a lethal external inhibitor,, Math. Biosci., 167 (2000), 177.  doi: 10.1016/S0025-5564(00)00030-4.  Google Scholar [11] S. B. Hsu, T. K. Luo and P. Waltman, Competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor,, J. Math. Biol., 34 (1995), 225.  doi: 10.1007/BF00178774.  Google Scholar [12] S. B. Hsu and P. Waltman, Analysis of a model of two competitors in a chemostat with an external inhibitor,, SIAM J. Appl. Math., 52 (1992), 528.  doi: 10.1137/0152029.  Google Scholar [13] S. B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an un-stirred chemostat,, SIAM J. Appl. Math., 53 (1993), 1026.  doi: 10.1137/0153051.  Google Scholar [14] S. B. Hsu and P. Waltman, Competition between plasmid-bearing and plasmid-free organisms in selective media,, Chem. Engrg. Sci., 52 (1997), 23.  doi: 10.1016/S0009-2509(96)00385-5.  Google Scholar [15] S. B. Hsu and P. Waltman, Competition in the chemostat when one competitor produces a toxin,, Japan J. Indust. Appl. Math., 15 (1998), 471.  doi: 10.1007/BF03167323.  Google Scholar [16] S. B. Hsu and P. Waltman, A model of the effect of anti-competitor toxins on plasmid-bearing, plasmid-free compettion,, Taiwanese J. Math., 6 (2002), 135.   Google Scholar [17] S. B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor,, Math. Biosci., 187 (2004), 53.  doi: 10.1016/j.mbs.2003.07.004.  Google Scholar [18] S. B. Hsu, P. Waltman and G. S. K. Wolkowicz, Global analysis of a model of plasmid-bearing, plasmid-free competition in the chemostat,, J. Math. Biol., 32 (1994), 731.  doi: 10.1007/BF00163024.  Google Scholar [19] B. R. Levin, Frequency-dependent selection in bacterial population,, Phil. Trans. R. Soc. Lond., 319 (1988), 459.  doi: 10.1098/rstb.1988.0059.  Google Scholar [20] R. E. Lenski and S. Hattingh, Coexistence of two competitors on one resource and one inhibitor: A chemostat model based on bacteria and antibiotics,, J. Theoret. Biol., 122 (1986), 83.  doi: 10.1016/S0022-5193(86)80226-0.  Google Scholar [21] H. Nie and J. Wu, A system of reaction-diffusion equations in the unstirred chemostat with an inhibitor,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 989.  doi: 10.1142/S0218127406015246.  Google Scholar [22] H. Nie and J. Wu, Asymptotic behavior of an unstirred chemostat with internal inhibitor,, J. Math. Anal. Appl., 334 (2007), 889.  doi: 10.1016/j.jmaa.2007.01.014.  Google Scholar [23] H. L. Smith and P. Waltman, "The Theory of the Chemostat,", Cambridge University Press, (1995).   Google Scholar [24] J. Wu, H. Nie and G. S. K. Wolkowicz, A mathematical model of competition for two essential resources in the unstirred chemostat,, SIAM J. Appl. Math., 65 (2004), 209.  doi: 10.1137/S0036139903423285.  Google Scholar [25] J. 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.  Google Scholar [26] J. 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.  Google Scholar [27] J. 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.   Google Scholar [28] S. Zheng and J. Liu, Coexistence solutions for a reaction-diffusion system of un-stirred chemostat model,, Appl. Math. Comput., 145 (2003), 579.  doi: 10.1016/S0096-3003(02)00732-4.  Google Scholar
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