zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Effect of delayed response in growth on the dynamics of a chemostat model with impulsive input. (English) Zbl 1198.34134
Summary: A chemostat model with delayed response in growth and impulsive perturbations on the substrate is considered. Using the discrete dynamical system determined by the stroboscopic map, we obtain a microorganism-extinction periodic solution, further, the globally attractive condition of the microorganism-extinction periodic solution is obtained. By the use of the theory on delay functional and impulsive differential equation, we also obtain the permanent condition of the investigated system. Our results indicate that the discrete time delay has influence to the dynamics behaviors of the investigated system, and provide tactical basis for the experimenters to control the outcome of the chemostat. Furthermore, numerical analysis is inserted to illuminate the dynamics of the system affected by the discrete time delay. Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

34K12Growth, boundedness, comparison of solutions of functional-differential equations
34K45Functional-differential equations with impulses
Full Text: DOI
[1] Bulert, G. L.; Hsu, S. B.; Waltman, P.: A mathematical model of the chemostat with periodic washout rate, SIAM J appl math 45, 435-449 (1985) · Zbl 0584.92027 · doi:10.1137/0145025
[2] Bainov, D.; Simeonov, P.: Impulsive differential equations: periodic solutions and applications, Longman 66 (1993) · Zbl 0815.34001
[3] Caltagirone, L. E.; Doutt, R. L.: Global behavior of an SEIRS epidemic model with delays. The history of the vedalia beetle importation to California and its impact on the development of biological control, Ann rev entomol 34, 1-16 (1989)
[4] Ellermeyer, S. F.: Competition in the chemostat: global asymptotic behavior of a model with delayed response in growth, SIAM J appl math 54, 456-465 (1994) · Zbl 0794.92023 · doi:10.1137/S003613999222522X
[5] Funasaki, E.; Kot, M.: Invasion and chaos in a periodically pulsed mass-action chemostat, Theor popul biol 44, 203-224 (1993) · Zbl 0782.92020 · doi:10.1006/tpbi.1993.1026
[6] Freedman, H. I.; So, J. W. H.; Waltman, P.: Chemostat competition with delays, Biomedicial modelling and simulation, 171-173 (1989)
[7] Gao, Shujing; Chen, Lansun; Nieto, Juan J.; Torres, Angela: Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine 24, 6037-6045 (2006)
[8] Hale, J. K.; Somolinas, A. M.: Competition for fluctuating nutrient, J math biol 18, 255-280 (1983) · Zbl 0525.92024 · doi:10.1007/BF00276091
[9] Hsu, S. B.; Hubbell, S. P.; Waltman, P.: A mathematical theory for single nutrient competition in continuous cultures of microorganisms, SIAM J appl math 32, 366-383 (1977) · Zbl 0354.92033 · doi:10.1137/0132030
[10] Hsu, S. B.; Waltman, P.; Ellermeyer, S. F.: A remark on the global asymptotic stability of a dynamical system modeling two species competition, Hiroshima math J 24, 435-445 (1994) · Zbl 0806.92016
[11] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P.: Theory of impulsive differential equations, (1989) · Zbl 0718.34011
[12] Wolkowicz, R. J. Smith. G. S. K.: Analysis of a model of the nutrient driven self-cycling fermentation process, Dyn contin discrete impul syst ser B 11, 239-265 (2004) · Zbl 1069.34121
[13] Smith, H.; Waltman, P.: Theory of chemostat, (1995)
[14] Smith, H.; Waltman, P.: Perturbation of a globally stable steady state, Proc AMS 127, No. 2, 447-453 (1999) · Zbl 0924.58087 · doi:10.1090/S0002-9939-99-04768-1
[15] Wolkowicz, G. S. K.; Zhao, X. Q.: N-species competition in a periodic chemostat, Diff integr eq 11, 465-491 (1998) · Zbl 1005.92027
[16] Yang, Kuang: Delay differential equation with application in population dynamics, (1993) · Zbl 0777.34002
[17] Sean, Ellermeyer; Jerald, Hendrix; Nariman, Ghoochan: A theoretical and empirical investigation of delayed growth response in the continuous culture of bacteria, J theor biol 222, No. 4, 485-494 (2003)
[18] Wang, Lin; Wolkowicz, Gail S. K.: A delayed chemostat model with general nonmonotone response functions and differential removal rates, J math anal appl 321, 452-468 (2006) · Zbl 1092.92048 · doi:10.1016/j.jmaa.2005.08.014
[19] Monod, J.: La technique de culture continue; theorie et applications, Ann inst pasteur 79, 390-410 (1950)
[20] Chen, L. S.; Chen, J.: Nonlinear biological dynamic systems, (1993)
[21] Sun, S. L.; Chen, L. S.: Dynamic behaviors of monod type chemostat model with impulsive perturbation on the nutrient concentration, J math chem 42, No. 4, 837-847 (2007) · Zbl 1217.34125 · doi:10.1007/s10910-006-9144-3
[22] El-Owaidy, H. M.; Ismail, M.: Asymptotic behaviour of the chemostat model with delayed response in growth, Chaos, solitons and fractals 13, No. 4, 787-795 (2002) · Zbl 1022.92041 · doi:10.1016/S0960-0779(01)00055-8
[23] Sanling, Yuan; Maoan, Han; Zhien, Ma: Competition in the chemostat: convergence of a model with delayed response in growth, Chaos, solitons and fractals 17, No. 4, 659-667 (2003) · Zbl 1036.92037 · doi:10.1016/S0960-0779(02)00478-2
[24] Fu, Guifang; Ma, Wanbiao: Hopf bifurcations of a variable yield chemostat model with inhibitory exponential substrate uptake, Chaos, solitons and fractals 30, No. 4, 845-850 (2006) · Zbl 1142.34334 · doi:10.1016/j.chaos.2005.05.056
[25] El-Sheikh, M. M. A.; Mahrouf, S. A. A.: Stability and bifurcation of a simple food chain in a chemostat with removal rates, Chaos, solitons and fractals 23, No. 4, 1475-1489 (2005) · Zbl 1062.92068 · doi:10.1016/j.chaos.2004.06.079
[26] Jiao, Jianjun; Chen, Lansun: Global attractivity of a stage-structured variable coefficients predator -- prey system with time delay and impulsive perturbations on predators, Int J biomath 1, No. 2, 197-208 (2008) · Zbl 1155.92355 · doi:10.1142/S1793524508000163
[27] Meng, Xinzhu; Chen, Lansun: Permanence and global stability in an impulsive Lotka-Volterra N-species competitive system with both discrete delays and continuous delays, Int J biomath 1, No. 2, 179-197 (2008) · Zbl 1155.92356 · doi:10.1142/S1793524508000151
[28] Song, Xinyu; Guo, Hongjian: Globali stability of a stage-structured predator -- prey system, Int J biomath 1, No. 3, 313-326 (2008) · Zbl 1173.34043 · doi:10.1142/S1793524508000266
[29] Lu, Zhiqi; Wu, Jingjing: Global stability of a chemostat model with delayed response in growth and a lethal external inhibitor, Int J biomath 1, No. 4, 503-520 (2008) · Zbl 1156.92041 · doi:10.1142/S1793524508000436