Dynamics of a delay differential equation model of phage growth in two-stage chemostat
Authors:
Huiyan Zhu and Yang Luo
Journal:
Quart. Appl. Math. 70 (2012), 299-310
MSC (2010):
Primary 92B05, 92C17; Secondary 92D25
DOI:
https://doi.org/10.1090/S0033-569X-2012-01266-5
Published electronically:
February 14, 2012
MathSciNet review:
2953105
Full-text PDF Free Access
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Abstract: In this paper, we consider a mathematical model for phage growth in a two-stage chemostat, taking account of the delay from infection to lysis. We give sufficient conditions for the globally asymptotic stability of noninfected equilibrium and obtain sufficient conditions for the local stability of infected equilibrium. We also perform some numerical simulations which illustrate the theoretical results obtained. The formula for $\mathcal R_0$ shows that decreasing the delay and/or increasing the burst size will increase $\mathcal R_0$.
References
- Edoardo Beretta and Yang Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal. 33 (2002), no. 5, 1144–1165. MR 1897706, DOI https://doi.org/10.1137/S0036141000376086
- J.J. Bull, J. Millstein, J. Orcutt, H.A. Wichman, Evolutionary feedback mediated through population density, illustrated with viruses in chemostats,The American Naturalist 167(2006)39-51.
- Stavros Busenberg and Kenneth Cooke, Vertically transmitted diseases, Biomathematics, vol. 23, Springer-Verlag, Berlin, 1993. Models and dynamics. MR 1206227
- C. Castillo-Chavez, H.R.Thieme, Asymptotically autonomous epidemic models, In: O. Arino et al. (Eds.) Mathematical Population Dynamics: Analysis of Heterogeneity, I. Theory of Epidemics. Wuerz, Winnepeg, Canada, 1995.
- H. I. Freedman and Yuan Tong Xu, Models of competition in the chemostat with instantaneous and delayed nutrient recycling, J. Math. Biol. 31 (1993), no. 5, 513–527. MR 1229447, DOI https://doi.org/10.1007/BF00173890
- Jack K. Hale and Sjoerd M. Verduyn Lunel, Introduction to functional-differential equations, Applied Mathematical Sciences, vol. 99, Springer-Verlag, New York, 1993. MR 1243878
- Shigui Ruan and Gail S. K. Wolkowicz, Bifurcation analysis of a chemostat model with a distributed delay, J. Math. Anal. Appl. 204 (1996), no. 3, 786–812. MR 1422773, DOI https://doi.org/10.1006/jmaa.1996.0468
- Hal L. Smith and Paul Waltman, The theory of the chemostat, Cambridge Studies in Mathematical Biology, vol. 13, Cambridge University Press, Cambridge, 1995. Dynamics of microbial competition. MR 1315301
- Gail S. K. Wolkowicz and Huaxing Xia, Global asymptotic behavior of a chemostat model with discrete delays, SIAM J. Appl. Math. 57 (1997), no. 4, 1019–1043. MR 1462050, DOI https://doi.org/10.1137/S0036139995287314
- Gail S. K. Wolkowicz, Huaxing Xia, and Shigui Ruan, Competition in the chemostat: a distributed delay model and its global asymptotic behavior, SIAM J. Appl. Math. 57 (1997), no. 5, 1281–1310. MR 1470924, DOI https://doi.org/10.1137/S0036139995289842
- Huiyan Zhu and Xingfu Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete Contin. Dyn. Syst. Ser. B 12 (2009), no. 2, 511–524. MR 2525152, DOI https://doi.org/10.3934/dcdsb.2009.12.511
- Huiyan Zhu, Xingfu Zou, Impact of delays in cell infection and virus production on HIV-1 dynamics, Mathematical Medicine and Biology: A Journal of the IMA., 25(2008)99-112.
References
- E. Beretta,Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal. 33(2002)1144-1165. MR 1897706 (2003d:34144)
- J.J. Bull, J. Millstein, J. Orcutt, H.A. Wichman, Evolutionary feedback mediated through population density, illustrated with viruses in chemostats,The American Naturalist 167(2006)39-51.
- S. Busenberg, K.L. Cooke, Vertically Transmitted Diseases, Models and Dynamics (Biomathematics. 23). Springer, New York, 1993. MR 1206227 (93m:92020)
- C. Castillo-Chavez, H.R.Thieme, Asymptotically autonomous epidemic models, In: O. Arino et al. (Eds.) Mathematical Population Dynamics: Analysis of Heterogeneity, I. Theory of Epidemics. Wuerz, Winnepeg, Canada, 1995.
- H. I. Freedman, Y. T. Xu, Models of competition in the chemostat with instantaneous and delayed nutrient recycling, J. Math. Biol., 31(1993)513-527. MR 1229447 (94f:92009)
- J. Hale, S.M.Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Vol.99. Springer-Verlag, New York, 1993. MR 1243878 (94m:34169)
- S. Ruan, G.S.K.Wolkowicz, Bifurcation analysis of a chemostat model with discrete delays, J. Math. Anal. Appl., 204(1996)188-192. MR 1422773 (97k:92024)
- H.L. Smith, P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition. Cambridge University Press, Cambridge, 1995. MR 1315301 (96e:92016)
- G.S.K. Wolkowicz, H. Xia. Global asymptotic behavior of a chemostat model with discrete delays, SIAM J. Appl. Math., 57(1997)1019-1043. MR 1462050 (98f:92010)
- G.S.K. Wolkowicz, H. Xia, S. Ruan, Competition in the chemostat: a distributed delay model and its global asymptotic behavior, SIAM J. Appl. Math., 57(1997)1281-1310. MR 1470924 (98f:92011)
- Huiyan Zhu, Xingfu Zou, Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay, Discrete and Continuous Dynamical Systems - Series B, 12(2009)513-526. MR 2525152 (2010g:92093)
- Huiyan Zhu, Xingfu Zou, Impact of delays in cell infection and virus production on HIV-1 dynamics, Mathematical Medicine and Biology: A Journal of the IMA., 25(2008)99-112.
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Additional Information
Huiyan Zhu
Affiliation:
College of Mathematics-Physics, University of South China, Hengyang, Hunan 421001, People’s Republic of China
Email:
zhuhuiyan@126.com
Yang Luo
Affiliation:
College of Computer Science and Technology, University of South China, Hengyang, Hunan 421001, People’s Republic of China
Email:
luoyang@usc.edu.cn
Keywords:
Dynamics,
delay differential equation,
stability,
phage,
two-stage chemostat
Received by editor(s):
September 10, 2010
Published electronically:
February 14, 2012
Additional Notes:
Research supported by Hunan Provincial Natural Science Foundation of China (No. 11JJ9001), the Natural Science Foundation of China (No. 11071060), and the project of Hunan Provincial Science & Technology Department (No. 2010GK3013).
Article copyright:
© Copyright 2012
Brown University
The copyright for this article reverts to public domain 28 years after publication.
