×

A delayed chemostat model with impulsive diffusion and input on nutrients. (English) Zbl 1184.92055

Summary: A chemostat model with delayed response in growth and impulsive diffusion and input on nutrients is considered. Using the discrete dynamical system determined by the stroboscopic map, we obtain a microorganism-extinction periodic solution. Further, it is globally attractive. The permanence condition of the investigated system is also obtained by the theory of impulsive delay differential equations. Finally, numerical analysis is inserted to illustrate the dynamical behaviors of the chemostat system. Our results reveal that the impulsive input amount of nutrients plays an important role on the outcome of the chemostat. Our results provide strategy basis for biochemical reaction management.

MSC:

92D40 Ecology
34K45 Functional-differential equations with impulses
34K60 Qualitative investigation and simulation of models involving functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Baĭnov D, Simeonov P: Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics. Volume 66. Longman Scientific & Technical, Harlow, UK; 1993:x+228. · Zbl 0815.34001
[2] Jiao J, Chen L: Global attractivity of a stage-structure variable coefficients predator-prey system with time delay and impulsive perturbations on predators.International Journal of Biomathematics 2008,1(2):197-208. 10.1142/S1793524508000163 · Zbl 1155.92355 · doi:10.1142/S1793524508000163
[3] Wang L, Liu Z, Hui J, Chen L: Impulsive diffusion in single species model.Chaos, Solitons and Fractals 2007,33(4):1213-1219. 10.1016/j.chaos.2006.01.102 · Zbl 1131.92071 · doi:10.1016/j.chaos.2006.01.102
[4] Lakshmikantham V, Baĭnov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Teaneck, NJ, USA; 1989:xii+273. · doi:10.1142/0906
[5] Benchohra M, Henderson J, Ntouyas S: Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications. Volume 2. Hindawi Publishing Corporation, New York, NY, USA; 2006:xiv+366. · Zbl 1130.34003 · doi:10.1155/9789775945501
[6] Samoĭlenko AM, Perestyuk NA: Impulsive Differential Equations, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises. Volume 14. World Scientific, River Edge, NJ, USA; 1995:x+462.
[7] Zavalishchin ST, Sesekin AN: Dynamic Impulse Systems. Theory and Application, Mathematics and Its Applications. Volume 394. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:xii+256.
[8] Jiao J, Chen L, Cai S: An SEIRS epidemic model with two delays and pulse vaccination.Journal of Systems Science & Complexity 2008,21(2):217-225. 10.1007/s11424-008-9105-y · Zbl 1203.93016 · doi:10.1007/s11424-008-9105-y
[9] Zeng G, Wang F, Nieto JJ: Complexity of a delayed predator-prey model with impulsive harvest and Holling type II functional response.Advances in Complex Systems 2008,11(1):77-97. 10.1142/S0219525908001519 · Zbl 1168.34052 · doi:10.1142/S0219525908001519
[10] Zhang H, Chen L, Nieto JJ: A delayed epidemic model with stage-structure and pulses for pest management strategy.Nonlinear Analysis: Real World Applications 2008,9(4):1714-1726. 10.1016/j.nonrwa.2007.05.004 · Zbl 1154.34394 · doi:10.1016/j.nonrwa.2007.05.004
[11] Wang, W.; Shen, J.; Nieto, JJ, Permanence and periodic solution of predator-prey system with Holling type functional response and impulses, No. 2007, 15 (2007) · Zbl 1146.37370
[12] Jiao J, Chen L: Dynamical analysis of a chemostat model with delayed response in growth and pulse input in polluted environment.Journal of Mathematical Chemistry 2009,46(2):502-513. 10.1007/s10910-008-9474-4 · Zbl 1196.92041 · doi:10.1007/s10910-008-9474-4
[13] Meng X, Li Z, Nieto JJ: Dynamic analysis of Michaelis-Menten chemostat-type competition models with time delay and pulse in a polluted environment.Journal of Mathematical Chemistry 2009,47(1):123-144. · Zbl 1194.92075 · doi:10.1007/s10910-009-9536-2
[14] Jiao J, Yang X, Chen L, Cai S: Effect of delayed response in growth on the dynamics of a chemostat model with impulsive input.Chaos, Solitons and Fractals 2009,42(4):2280-2287. 10.1016/j.chaos.2009.03.132 · Zbl 1198.34134 · doi:10.1016/j.chaos.2009.03.132
[15] Tagashira O: Permanent coexistence in chemostat models with delayed feedback control.Nonlinear Analysis: Real World Applications 2009,10(3):1443-1452. 10.1016/j.nonrwa.2008.01.015 · Zbl 1162.34334 · doi:10.1016/j.nonrwa.2008.01.015
[16] Smith HL, Waltman P: The Theory of the Chemostat. Dynamics of Microbial Competition, Cambridge Studies in Mathematical Biology. Volume 13. Cambridge University Press, Cambridge, UK; 1995:xvi+313. · Zbl 1139.92029 · doi:10.1017/CBO9780511530043
[17] Smith HL, Waltman P: Perturbation of a globally stable steady state.Proceedings of the American Mathematical Society 1999,127(2):447-453. 10.1090/S0002-9939-99-04768-1 · Zbl 0924.58087 · doi:10.1090/S0002-9939-99-04768-1
[18] Kuang Y: Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering. Volume 191. Academic Press, Boston, Mass, USA; 1993:xii+398. · Zbl 0777.34002
[19] Freedman, HI; So, JW-H; Waltman, P., Chemostat competition with time delays, No. 5, 171-173 (1989), Basel, Switzerland
[20] Ellermeyer SF: Competition in the chemostat: global asymptotic behavior of a model with delayed response in growth.SIAM Journal on Applied Mathematics 1994,54(2):456-465. 10.1137/S003613999222522X · Zbl 0794.92023 · doi:10.1137/S003613999222522X
[21] Caltagirone LE, Doutt RL: The history of the vedalia beetle importation to California and its impact on the development of biological control.Annual Review of Entomology 1989, 34: 1-16. 10.1146/annurev.en.34.010189.000245 · doi:10.1146/annurev.en.34.010189.000245
[22] Hsu S-B, Waltman P, Ellermeyer SF: A remark on the global asymptotic stability of a dynamical system modeling two species competition.Hiroshima Mathematical Journal 1994,24(2):435-445. · Zbl 0806.92016
[23] Hale JK, Somolinos AS: Competition for fluctuating nutrient.Journal of Mathematical Biology 1983,18(3):255-280. 10.1007/BF00276091 · Zbl 0525.92024 · doi:10.1007/BF00276091
[24] Hsu SB, Hubbell S, Waltman P: A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms.SIAM Journal on Applied Mathematics 1977,32(2):366-383. 10.1137/0132030 · Zbl 0354.92033 · doi:10.1137/0132030
[25] Wolkowicz GSK, Zhao X-Q:[InlineEquation not available: see fulltext.]-species competition in a periodic chemostat.Differential and Integral Equations 1998,11(3):465-491. · Zbl 1005.92027
[26] Ellermeyer S, Hendrix J, Ghoochan N: A theoretical and empirical investigation of delayed growth response in the continuous culture of bacteria.Journal of Theoretical Biology 2003,222(4):485-494. 10.1016/S0022-5193(03)00063-8 · Zbl 1464.92153 · doi:10.1016/S0022-5193(03)00063-8
[27] Bush AW, Cook AE: The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater.Journal of Theoretical Biology 1975,63(2):385-395. · doi:10.1016/0022-5193(76)90041-2
[28] Levin SA, Segel LA: Hypothesis for origin of planktonic patchiness.Nature 1976,259(5545):659. · doi:10.1038/259659a0
[29] Mimura M: Asymptotic behaviors of a parabolic system related to a planktonic prey and predator model.SIAM Journal on Applied Mathematics 1979,37(3):499-512. 10.1137/0137039 · Zbl 0426.35017 · doi:10.1137/0137039
[30] Okubo A: Diffusion and Ecological Problems: Mathematical Models, Biomathematics. Volume 10. Springer, Berlin, Germany; 1980:xiii+254.
[31] Freedman HI, Ruan S: On reaction-diffusion systems of zooplankton-phytoplankton-nutrient models.Differential Equations and Dynamical Systems 1994,2(1):49-64. · Zbl 0868.92026
[32] Ruan S: Turing instability and travelling waves in diffusive plankton models with delayed nutrient recycling.IMA Journal of Applied Mathematics 1998,61(1):15-32. 10.1093/imamat/61.1.15 · Zbl 0911.92028 · doi:10.1093/imamat/61.1.15
[33] Funasaki E, Kot M: Invasion and chaos in a periodically pulsed mass-action chemostat.Theoretical Population Biology 1993,44(2):203-224. 10.1006/tpbi.1993.1026 · Zbl 0782.92020 · doi:10.1006/tpbi.1993.1026
[34] Smith RJ, Wolkowicz GSK: Analysis of a model of the nutrient driven self-cycling fermentation process.Dynamics of Continuous, Discrete & Impulsive Systems. Series B 2004,11(3):239-265. · Zbl 1069.34121
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.