Characterization of positive solutions of the unstirred chemostat with an inhibitor

Nonlinear Analysis: Real World Applications

Volumn 9, Issue 3, 2008, Pages 1078-1089

  Nie, Hua ^a   Zhang, Hongwei ^a   Wu, Jianhua ^a  

^a SHAANXI NORMAL UNIVERSITY   (China)

Author keywords

Chemostat; Fixed point index; Monotone method

Indexed keywords

ENZYME INHIBITION; MATHEMATICAL MODELS; NONLINEAR ANALYSIS; NUMERICAL METHODS;

FIXED POINT INDEX; MONOTONE METHODS; POSITIVE SOLUTIONS; SPECIFIC INITIAL FUNCTION;

CHEMOSTATS;

EID: 38949100056     PISSN: 14681218     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.nonrwa.2007.01.018     Document Type: Article

References (23)

1. Amann H. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18 (1976) 620-709
2. Chao L., and Levin B.R. Structured habitats and the evolution of anti-competitor toxins in bacteria. Proc. Natl. Acad. Sci. 75 (1981) 6324-6328
3. Crandall M.G., and Rabinowitz P.H. Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Rat. Mech. Anal. 52 (1973) 161-181
4. Dancer E.N. On positive solutions of some pairs of differential equations. Trans. Amer. Math. Soc. 284 (1984) 729-743
5. Hsu S.B., Hubbell S.P., and Waltman P. A mathematical model for single nutrient competition in continuous cultures of micro-organisms. SIAM J. Appl. Math. 32 (1977) 366-382
6. Hsu S.B., Smith H.L., and Waltman P. Dynamics of competition in the un-stirred chemostat. Canad. Appl. Math. Quart. 4 (1994) 461-483
7. Hsu S.B., and Waltman P. Analysis of a model of two competitors in a Chemostat with an external inhibitor. SIAM J. Appl. Math. 52 2 (1992) 528-540
8. Hsu S.B., and Waltman P. On a system of reaction-diffusion equations arising from competition in an un-stirred Chemostat. SIAM J. Appl. Math. 53 (1993) 1026-1044
9. Hsu S.B., and Waltman P. A survey of mathematical models of competition with an inhibitor. Math. Biosci. 187 (2004) 53-91
10. Lenski R.E., and Hattingh S. 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
11. Levin B.R. Frequency-dependent selection in bacterial populations. Phil. Trans. Roy. Soc. London 319 (1988) 459-472
12. Nie H., and Wu J.H. A system of reaction-diffusion equations in the unstirred chemostat with an inhibitor. Int. J. Bifurc. Chaos 16 4 (2006) 989-1009
13. Pao C.V. Nonlinear Parabolic and Elliptic Equations (1992), Plenum, New York
14. Ruan W.H., and Pao C.V. Positive steady-state solutions of a competing reaction-diffusion system. J. Differential Equations 117 (1995) 411-427
15. Schuler M.L., and Kargi F. Bioprocess Engineering Basic Concepts (1992), Prentice-Hall, Englewood Cliffs, NJ
16. Smith H.L. Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. Mathematical Surveys and Monographs vol. 41 (1995), American Mathematical Society, Providence, RI
17. Smith H.L., and Waltman P. The Theory of the Chemostat: Dynamics of Microbial Competition (1995), Cambridge University, Cambridge
18. Smoller J. Shock Waves and Reaction-Diffusion Equations (1999), Springer, New York
19. Wang M.X. Nonlinear Parabolic Equation (1993), Science Press, Beijing (in Chinese)
20. Wu J.H. Global bifurcation of coexistence state for the competition model in the chemostat. Nonlinear Anal. 39 (2000) 817-835
21. Wu J.H., Nie H., and Wolkowicz G.S.K. A mathematical model of competition for two essential resources in the un-stirred chemostat. SIAM J. Appl. Math. 65 (2004) 209-229
22. Wu J.H., and Wolkowicz G.S.K. A system of resource-based growth models with two resources in the un-stirred chemostat. J. Differential Equations 172 (2001) 300-332
23. J.H. Wu, H. Nie, 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.