Global stability for a model of competition in the chemostat with microbial inputs

Nonlinear Analysis: Real World Applications

Volumn 13, Issue 2, 2012, Pages 582-598

  Robledo, Gonzalo ^a   Grognard, Frédéric ^b   Gouzé, Jean Luc ^b  

^a UNIVERSITY OF CHILE   (Chile)

^b INRIA   (France)

Author keywords

Chemostat; Competitive exclusion; Global stability; Polytopic Lyapunov functions

Indexed keywords

ARBITRARY NUMBER; COMPETITIVE EXCLUSION; DOMINANT SPECIES; GAS EQUILIBRIUM; GLOBAL STABILITY; GLOBALLY ASYMPTOTICALLY STABLE; LYAPUNOV; N SPECIES; NON-TRIVIAL; POSITIVE EQUILIBRIUM;

COMPETITION; LYAPUNOV FUNCTIONS; SOLID SOLUTIONS;

CHEMOSTATS;

EID: 80054884852     PISSN: 14681218     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.nonrwa.2011.07.049     Document Type: Article

References (44)

1. H.L. Smith, and P. Waltman The Theory of the Chemostat. Dynamics of Microbial Competition 1995 Cambridge University Press Cambridge
2. G. Hardin Competitive exclusion principle Science 131 1960 1292 1297
3. S. Dikshitulu, B.C. Baltzis, G.A. Lewandowski, and S. Pavlou Competition between two microbial populations in a sequencing fed-batch reactor: theory, experimental verification, and implications for waste treatment applications Biotechnology and Bioengineering 42 2004 643 656
4. S.R. Hansen, and S.P. Hubell Single nutrient microbial competition: agreement between experimental and forecast outcomes Science 207 1980 1491 1493 (Pubitemid 10119647)
5. E. Postma, A. Kuiper, W.F. Tomasouw, W.A. Scheffers, and J.P. Van Dijken Competition for glucose between the yeasts Saccharomyces cerevisiae and Candida utilis Applied and Environmental Microbiology 55 1989 3124 3220
6. G. Meszéna, M. Gyllenberg, L. Pastor, and J.A. Metz Competitive exclusion and limiting similarity: a unified theory Theoretical Population Biology 69 2006 68 87 (Pubitemid 41797414)
7. C. Neill, T. Daufresne, and C.G. Jones A competitive coexistence principle? Oikos 118 2009 1570 1578
8. J. Arino, S. Pilyugin, and G.S.K. Wolkowicz Considerations on yield, nutrient uptake, cellular growth, and competition in chemostat models The Canadian Applied Mathematics Quarterly 11 2003 107 142
9. S. Pilyugin, and P. Waltman Multiple limit cycles in the chemostat with variable yield Mathematical Biosciences 182 2003 151 166 (Pubitemid 36330981)
10. P. Lenas, and S. Pavlou Coexistence of three competing microbial populations in a chemostat with periodically varying dilution rate Mathematical Biosciences 129 1995 111 142 (Pubitemid 26453740)
11. F. Mazenc, M. Malisoff, and J. Harmand Further results on stabilization of periodic trajectories for a chemostat with two species IEEE Transactions on Automatic Control 53 2008 66 74 (Pubitemid 351314539)
12. G.S.K. Wolkowicz, and X.-Q. Zhao n-species competition in a periodic chemostat Differential Integral Equations 11 1998 465 491
13. S.B. Hsu A competition model for a seasonally fluctuating nutrient Journal of Mathematical Biology 9 1980 115 132 (Pubitemid 10088677)
14. H.L. Smith Competitive coexistence in an oscillating chemostat SIAM Journal on Applied Mathematics 40 1981 498 522
15. P. De Leenheer, and H.L. Smith Feedback control for chemostat models Journal of Mathematical Biology 46 2003 48 70 (Pubitemid 38355703)
16. J.L- Gouzé, and G. Robledo Feedback control for nonmonotone competition models in the chemostat Nonlinear Analysis: Real World Applications 6 2005 671 690 (Pubitemid 40420741)
17. F. Mazenc, and M. Malisoff Remarks on output feedback stabilization of two-species chemostat models Automatica 46 2010 1739 1742
18. P. De Leenheer, and S. Pilyugin Feedback-mediated oscillatory coexistence in the chemostat C. Commault, N. Marchand, Proceedings of the Second Multidisciplinary International Symposium on Positive Systems: Theory and Applications, POSTA 06 Lecture Notes in Control and Information Sciences vol. 341 2006 Springer-Verlag 97 104 (Pubitemid 44597521)
19. W.S. Keeran, P. De Leenheer, and S. Pilyugin Circular and elliptic orbits in a feedback-mediated chemostat Discrete and Continuous Dynamical Systems Series B 7 2007 779 792
20. H. Guo, and S. Zheng A competition model for two resources in un-stirred chemostat Applied Mathematics and Computation 217 2011 6934 6949
21. S.B. Hsu, and P. Waltman On a system of reactiondiffusion equations arising from competition in an unstirred chemostat SIAM Journal on Applied Mathematics 53 1993 1026 1044
22. H. Nie, and J. Wu Coexistence of an unstirred chemostat model with BeddingtonDe Angelis functional response and inhibitor Nonlinear Analysis: Real World Applications 11 2010 3639 3652 2010
23. S. Pilyugin, and P. Waltman Competition in the unstirred chemostat with periodic input and washout SIAM Journal on Applied Mathematics 59 1989 1157 1177
24. J. Hofbauer, and J.W.H. So Competition in the gradostat: the global stability problem Nonlinear Analysis: Theory Methods Applications 22 1994 1017 1031
25. S. Pavlou Microbial competition in bioreactors Chemical Industry and Chemical Engineering Quarterly 12 2006 71 81
26. P. De Leenheer, D. Angeli, and E.D. Sontag Crowding effects promote coexistence in the chemostat Journal of Mathematical Analysis and Applications 319 2006 48 60 (Pubitemid 43506845)
27. B. Haegeman, and A. Rapaport How flocculation can explain coexistence in the chemostat Journal of Biological Dynamics 2 2008 1 13
28. B. Li, and H.L. Smith Competition for essential resources: a brief review Dynamical Systems and its Applications in Biology vol. 36 2003 Fields Institute Communications 213 227
29. X. Meng, Z. Li, and J.J. Nieto Dynamic analysis of MichaelisMenten chemostat-type competition models with time delay and pulse in a polluted environment Journal of Mathematical Chemistry 47 2010 123 144
30. X. Meng, Q. Gao, and Z. Li The effects of delayed growth response on the dynamic behaviors of the Monod type chemostat model with impulsive input nutrient concentration Nonlinear Analysis: Real World Applications. 11 2010 4476 4486
31. H. Zhang, P. Georgescu, J.J. Nieto, and L. Chen Impulsive perturbation and bifurcation of solutions for a model of chemostat with variable yield Applied Mathematics and Mechanics 30 2009 933 944
32. C. Lobry, F. Mazenc, and A. Rapaport Persistence in ecological models of competition for a single resource Comptes Rendus Mathematique 340 2005 199 204 (Pubitemid 40176118)
33. F. Grognard, F. Mazenc, and A. Rapaport Polytopic Lyapunov functions for persistence analysis of competing species Discrete and Continuous Dynamical Systems Series B 8 2007 73 93
34. T. Kuniya Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model Nonlinear Analysis: Real World Applications 12 2011 2640 2655
35. L. Wang, L. Chen, and J.J. Nieto The dynamics of an epidemic model for pest control with impulsive effect Nonlinear Analysis: Real World Applications 11 2010 1374 1386
36. H. Zhang, L. Chen, and J.J Nieto A delayed epidemic model with stage-structure and pulses for pest management strategy Nonlinear Analysis: Real World Applications 9 2008 1714 1726 (Pubitemid 351685810)
37. M. Vidyasagar Decomposition techniques for large-scale systems with nonadditive interactions: stability and stabilizability IEEE Transactions on Automatic Control AC-21 1980 773 779 (Pubitemid 11437550)
38. O. Bernard, G. Malara, and A. Sciandra The effects of a controlled fluctuating nutrient environment on continous cultures of phytoplankton monitored by computers Journal of Experimental Marine Biology and Ecology 197 1996 263 278 (Pubitemid 26350330)
39. I. Vatcheva, O. Bernard, H. De Jong, and N. Mars Experiment selection for the discrimination of semi-quantitative models of dynamical systems Artificial Intelligence 170 2006 472 506 (Pubitemid 43262062)
40. J.H. Connel Diversity in tropical rain forests and coral reefs Science 199 1978 1302 1310
41. S.B. Hsu, and T. Hsu Competitive exclusion of microbial species for a single-nutrient with internal storage SIAM Journal on Applied Mathematics 68 2008 1600 1617
42. P. Masci, F. Grognard, E. Benot, O. Bernard, Competition between diverse types of microorganisms: exclusion and coexistence (submitted for publication).
43. J. Hofbauer A unified approach to persistence Acta Applicandae Mathematicae 14 1989 14 22
44. S. Chareyron, Stabilité des Systmes dynamiques Non-Réguliers, Application aux Robots Marcheurs. Ph.D. Thesis, Institut National Polytechnique de Grenoble, Grenoble, 2005.