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 and Continuous Dynamical Systems, 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), 6324-6328. doi: 10.1073/pnas.78.10.6324.

[2]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.

[3]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. doi: 10.1007/BF00282325.

[4]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151. doi: 10.1016/0022-247X(83)90098-7.

[5]

E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743. doi: 10.1090/S0002-9947-1984-0743741-4.

[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), 337-357. doi: 10.1016/0362-546X(94)E0063-M.

[7]

Y. Du, Positive periodic solutions of a competitor-competitor-mutualist model, Differential Integral Equations, 9 (1996), 1043-1066.

[8]

D. G. Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Differential Equations, 17 (1992), 339-346.

[9]

S. R. Hansen and S. P. Hubbell, Single nutrient microbial competition: Agreement between experimental and theoretical forecast outcomes, Science, 207 (1980), 1491-1493. doi: 10.1126/science.6767274.

[10]

S. B. Hsu, Y. S. Li and P. Waltman, Competition in the presence of a lethal external inhibitor, Math. Biosci., 167 (2000), 177-199. doi: 10.1016/S0025-5564(00)00030-4.

[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), 225-238. doi: 10.1007/BF00178774.

[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), 528-540. doi: 10.1137/0152029.

[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), 1026-1044. doi: 10.1137/0153051.

[14]

S. B. Hsu and P. Waltman, Competition between plasmid-bearing and plasmid-free organisms in selective media, Chem. Engrg. Sci., 52 (1997), 23-35. doi: 10.1016/S0009-2509(96)00385-5.

[15]

S. B. Hsu and P. Waltman, Competition in the chemostat when one competitor produces a toxin, Japan J. Indust. Appl. Math., 15 (1998), 471-490. doi: 10.1007/BF03167323.

[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), 135-155.

[17]

S. B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor, Math. Biosci., 187 (2004), 53-91. doi: 10.1016/j.mbs.2003.07.004.

[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), 731-742. doi: 10.1007/BF00163024.

[19]

B. R. Levin, Frequency-dependent selection in bacterial population, Phil. Trans. R. Soc. Lond., 319 (1988), 459-472. doi: 10.1098/rstb.1988.0059.

[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), 83-93. doi: 10.1016/S0022-5193(86)80226-0.

[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), 989-1009. doi: 10.1142/S0218127406015246.

[22]

H. Nie and J. Wu, Asymptotic behavior of an unstirred chemostat with internal inhibitor, J. Math. Anal. Appl., 334 (2007), 889-908. doi: 10.1016/j.jmaa.2007.01.014.

[23]

H. L. Smith and P. Waltman, "The Theory of the Chemostat," Cambridge University Press, Cambridge, 1995.

[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), 209-229. doi: 10.1137/S0036139903423285.

[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), 1860-1885. doi: 10.1137/050627514.

[26]

J. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835. doi: 10.1016/S0362-546X(98)00250-8.

[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), 300-332.

[28]

S. Zheng and J. Liu, Coexistence solutions for a reaction-diffusion system of un-stirred chemostat model, Appl. Math. Comput., 145 (2003), 579-590. doi: 10.1016/S0096-3003(02)00732-4.

show all references

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), 6324-6328. doi: 10.1073/pnas.78.10.6324.

[2]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.

[3]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. doi: 10.1007/BF00282325.

[4]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151. doi: 10.1016/0022-247X(83)90098-7.

[5]

E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743. doi: 10.1090/S0002-9947-1984-0743741-4.

[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), 337-357. doi: 10.1016/0362-546X(94)E0063-M.

[7]

Y. Du, Positive periodic solutions of a competitor-competitor-mutualist model, Differential Integral Equations, 9 (1996), 1043-1066.

[8]

D. G. Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Differential Equations, 17 (1992), 339-346.

[9]

S. R. Hansen and S. P. Hubbell, Single nutrient microbial competition: Agreement between experimental and theoretical forecast outcomes, Science, 207 (1980), 1491-1493. doi: 10.1126/science.6767274.

[10]

S. B. Hsu, Y. S. Li and P. Waltman, Competition in the presence of a lethal external inhibitor, Math. Biosci., 167 (2000), 177-199. doi: 10.1016/S0025-5564(00)00030-4.

[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), 225-238. doi: 10.1007/BF00178774.

[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), 528-540. doi: 10.1137/0152029.

[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), 1026-1044. doi: 10.1137/0153051.

[14]

S. B. Hsu and P. Waltman, Competition between plasmid-bearing and plasmid-free organisms in selective media, Chem. Engrg. Sci., 52 (1997), 23-35. doi: 10.1016/S0009-2509(96)00385-5.

[15]

S. B. Hsu and P. Waltman, Competition in the chemostat when one competitor produces a toxin, Japan J. Indust. Appl. Math., 15 (1998), 471-490. doi: 10.1007/BF03167323.

[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), 135-155.

[17]

S. B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor, Math. Biosci., 187 (2004), 53-91. doi: 10.1016/j.mbs.2003.07.004.

[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), 731-742. doi: 10.1007/BF00163024.

[19]

B. R. Levin, Frequency-dependent selection in bacterial population, Phil. Trans. R. Soc. Lond., 319 (1988), 459-472. doi: 10.1098/rstb.1988.0059.

[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), 83-93. doi: 10.1016/S0022-5193(86)80226-0.

[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), 989-1009. doi: 10.1142/S0218127406015246.

[22]

H. Nie and J. Wu, Asymptotic behavior of an unstirred chemostat with internal inhibitor, J. Math. Anal. Appl., 334 (2007), 889-908. doi: 10.1016/j.jmaa.2007.01.014.

[23]

H. L. Smith and P. Waltman, "The Theory of the Chemostat," Cambridge University Press, Cambridge, 1995.

[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), 209-229. doi: 10.1137/S0036139903423285.

[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), 1860-1885. doi: 10.1137/050627514.

[26]

J. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835. doi: 10.1016/S0362-546X(98)00250-8.

[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), 300-332.

[28]

S. Zheng and J. Liu, Coexistence solutions for a reaction-diffusion system of un-stirred chemostat model, Appl. Math. Comput., 145 (2003), 579-590. doi: 10.1016/S0096-3003(02)00732-4.

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