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Dynamics of a model of allelopathy and bacteriocin with a single mutation. (English) Zbl 1209.34061
Summary: We discuss a model of allelopathy and bacteriocin in the chemostat with a wild-type organism and a single mutant. Dynamical properties of this model show the basic competition between two microorganisms. A qualitative analysis about the boundary equilibrium, a state at that both microorganisms vanish, is carried out. The existence and uniqueness of the interior equilibrium is proved. Its dynamical properties are given by using the index theory of equilibria. We further discuss its bifurcations. Our results are demonstrated by numerical simulations.

34C60 Qualitative investigation and simulation of ordinary differential equation models
92D25 Population dynamics (general)
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[1] Neu, H.C., The crisis in antibiotic resistance, Science, 257, 1064-1073, (1992)
[2] Levy, S.B.; Marshall, B., Antibacterial resistance worldwide: causes, challenges and responses, Nat. med., 10, S122-S129, (2004)
[3] Riley, M.A.; Wertz, J.E., Bacteriocins: evolution, ecology, and application, Annu. rev. microbiol., 56, 117-137, (2002)
[4] Walker, E.S.; Levy, F., Genetic trends in a population evolving antibiotic resistance, Evolution, 55, 1110-1122, (2001)
[5] Riley, M.A., Molecular mechanisms of bacteriocin evolution, Annu. rev. genet., 32, 255-278, (1998)
[6] Durrett, R.; Levin, S.A., Allelopathy in spatially distributed populations, J. theor. biol., 185, 165-171, (1997)
[7] Iwasa, Y.; Nakamaru, M.; Levin, S.A., Allelopathy of bacteria in a lattice population: competition between colicin-sensitive and colicin-producing strains, Evol. ecol., 12, 785-802, (1998)
[8] Abell, M.; Braselton, J.; Braselton, L., A model of allelopathy in the context of bacteriocin production, Appl. math. comput., 183, 916-931, (2006) · Zbl 1104.92056
[9] Chao, L.; Levin, B.R., Structured habitats and the evolution of anticompetitor in bacteria, Proc. natl. acad. sci. USA, 78, 6324-6328, (1981)
[10] Frank, S.A., Spatial polymorphism of bacteriocins and other allelopathic traits, Evolution ecol., 8, 369-386, (1994)
[11] Smith, H.L.; Waltman, P., The theory of the chemostat: dynamics of microbial competition, (1992), Cambridge University Press Cambridge
[12] Herbert, D.; Elsworth, R.; Telling, R.C., The continuous culture of bacteria: a theoretical and experimental study, J. gen. microbiol., 14, 601-622, (1956)
[13] Hsu, S.-B.; Hubbell, S.; Waltman, P., A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM J. appl. math., 32, 366-383, (1977) · Zbl 0354.92033
[14] Hsu, S.-B.; Waltman, P.; Wolkowicz, G.S.K., Global analysis of a model of plasmid-bearing, plasmid-free competition in a chemostat, J. math. biol., 32, 731-742, (1994) · Zbl 0802.92027
[15] Monod, J., La technique de la culture continuous: theorie et applications, Ann. inst. pasteur, 79, 390-410, (1950)
[16] Novick, A.; Szilard, L., Description of the chemostat, Science, 112, 215-216, (1950)
[17] Taylor, P.A.; Williams, P.J.L., Theoretical studies on the coexistence of competing species under continuous-flow conditions, Can. J. microbiol., 21, 90-98, (1975)
[18] Hsu, S.-B.; Waltman, P., A survey of mathematical model of competition with an inhibitor, Math. biosci., 87, 53-91, (2004) · Zbl 1034.92034
[19] Ruan, S., The dynamics of chemostat models, J. central China normal univ. natur. sci., 31, 377-397, (1997), (in Chinese) · Zbl 0908.92034
[20] Thieme, H.R., Convergence results and a poincaré-bendixon trichotomy for asymptotically autonomous differential equations, J. math. biol., 30, 755-763, (1992) · Zbl 0761.34039
[21] Arnol’d, V.I., Ordinary differential equations, (1992), Springer-Verlag Berlin, Heidelberg, Transl. Roger Cooke
[22] Zhang, Z.-F.; Ding, T.-R.; Huang, W.-Z.; Dong, Z.-X., ()
[23] Bendixson, I., Sur LES courbes définies par des équations différentielles, Acta math., 24, 1-88, (1901) · JFM 31.0328.03
[24] Carr, J., Applications of center manifold theory, (1981), Springer-Verlag New-York
[25] Chow, S.N.; Li, C.Z.; Wang, D., Normal forms and bifurcation of planar vector fields, (1994), Cambridge University Press
[26] Lenski, R.E.; Hattingh, S.E., Coexistence of two competitors on one resource and one inhibitor: a chemostat model based on bacteria and antibiotics, J. theor. biol., 122, 83-93, (1986)
[27] Peck, S.L., Antibiotic and insecticide resistance modeling is it time to start talking?, Trends microbiol., 9, 286-292, (2001)
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