zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Chaotic behavior of a chemostat model with Beddington-DeAngelis functional response and periodically impulsive invasion. (English) Zbl 1121.92070
Summary: We consider a predator-prey model with Beddington-DeAngelis functional response in a periodic pulsed chemostat. We discuss the boundness of the system and the stability of prey and predator-eradication periodic solutions of the system. Further, using numerical simulations, we show that this impulsive system with periodically pulsed substrate displays a series of complex phenomena, which include (1) period-doubling cascades, (2) period-halfing cascades, (3) chaos and (4) periodic windows.

MSC:
92D40Ecology
34A37Differential equations with impulses
34D05Asymptotic stability of ODE
34C25Periodic solutions of ODE
WorldCat.org
Full Text: DOI
References:
[1] Kot, M.; Sayler, G. S.; Schultz, T. W.: Complex dynamics in a model microbial system. Bull math biol 54, 619-648 (1992) · Zbl 0761.92041
[2] Pavlou, S.; Kevrekids, I. G.: Microbial predation in a periodically operated chemostat: a global study of the interaction between natural and externally and externally imposed frequencies. Math biosci 108, 1-55 (1992) · Zbl 0729.92522
[3] Alessandra, G.; Oscar, D. F.; Sergio, R.: Food chains in the chemostat: relationships between mean yield and complex dynamics. Bull math biol 60, 703-719 (1998) · Zbl 0934.92033
[4] Deanglis, D. L.; Goldstein, R. A.; O’neill, R. V.: A model for trophic interaction. Ecology 56, 881-892 (1975)
[5] Beddington, J. R.: Mutual interference between parasites or predators and its effect on searching efficiency. J animal ecol 44, 331-340 (1975)
[6] Ruxton, G.; Gurney, W. S. C.; Deroos, A.: Interference and generation cycles. Theoret population biol 42, 235-253 (1992) · Zbl 0768.92025
[7] Cosner, C.; Deangelis, D. L.; Ault, J. S.; Olson, J. S.: Effects of spatial grouping on the functional response of predators. Theret population biol 56, 65-75 (1999) · Zbl 0928.92031
[8] Harrision, G. W.: Comparing predator-prey models to luckinbill’s experiment with didinium and paramecium. Ecology 76, 357-369 (1995)
[9] Bainov, D. D.; Simeonov, D. D.: Impulsive differential equations: periodic solutions and applications. (1993) · Zbl 0815.34001
[10] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. C.: Theory of impulsive differential equations. (1989) · Zbl 0719.34002
[11] Tang, S. Y.; Chen, L. S.: Density-dependent birth rate, birth pulse and their population dynamic consequences. J math biol 44, 185-199 (2002) · Zbl 0990.92033
[12] Roberts, M. G.; Kao, R. R.: The dynamics of an infectious disease in a population with birth pulses. Math biosci 149, 23-36 (1998) · Zbl 0928.92027
[13] Shulgin, B.; Stone, L.; Agur, Z.: Pulse vaccination strategy in the SIR epidemic model. Bull math biol 60, 1-26 (1998) · Zbl 0941.92026
[14] D’onofrio, A.: Stability properties of pulse vaccination strategy in SEIR epidemic model. Math comput modell 26, 59-72 (1997)
[15] Panetta, J. C.: A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competitive environment. Bull math biol 58, 425-447 (1996) · Zbl 0859.92014
[16] Lakmeche, A.; Arino, O.: Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment. Dyn continuous discrete impulsive syst 7, 165-187 (2000) · Zbl 1011.34031
[17] Ballinger, G.; Liu, X.: Permanence of population growth models with impulsive effects. Math comput modell 26, 59-72 (1997) · Zbl 1185.34014
[18] Funasaki, E.; Kot, M.: Invasion and chaos in a periodically pulsed mass-action chemostat. Theoret population biol, 203-224 (1997) · Zbl 0782.92020
[19] Tang, S. Y.; Chen, L. S.: Quasipeiodic solutions and chaos in a periodically forced predator-prey model with age structure for predator. Int J bifurc chaos 13, No. 4, 973-980 (2003) · Zbl 1063.37586
[20] Zhang, S. W.; Chen, L. S.: A Holling II functional response food chain model with impulsive perturbations. Chaos, solitons & fractals 24, 1269-1278 (2005) · Zbl 1086.34043
[21] Tang, S. Y.; Chen, L. S.: Chaos in functional response host-parasitoid ecosystem models. Chaos, solitons & fractals 13, 875-884 (2002) · Zbl 1022.92042
[22] May, R. M.: Biological population with nonoverlapping generations: stable points, stable cycles, and chaos. Science 186, 645-657 (1974)
[23] May, R. M.; Oster, G. F.: Bifurcations and dynamic complexity in simple ecological models. Am nature 110, 573-599 (1976)
[24] Collet, P.; Eeckmann, J. P.: Iterated maps of the interval as dynamical systems. (1980)
[25] Venkatesan, A.; Parthasarathy, S.; Lakshmanan, M.: Occurrence of multiple period-doubling bifurcation route to chaos in periodically pulsed chaotic dynamical systems. Chaos, solitons & fractals 18, 891-898 (2003) · Zbl 1073.37038
[26] Neubert, M. G.; Caswell, H.: Density-dependent vital rates and their population dynamic consequences. J math biol 41, 103-121 (2000) · Zbl 0956.92029
[27] Wikan, A.: From chaos to chaos. An analysis of a discrete age-structured prey-predator model. J math biol 43, 471-500 (2001) · Zbl 0996.92031