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)
A Holling II functional response food chain model with impulsive perturbations. (English) Zbl 1086.34043
A food chain system with Holling-type II functional response and periodic constant impulsive perturbations is considered. Conditions for the extinction of the predator as a pest are given. Local stability of predator eradication periodic solution is studied. Influences of the impulsive perturbation on the inherent oscillation is studied numerically in the positive octant. Rich dynamic behavior appeared. The dynamic behavior is found to be sensitive to the parameter values and to the initial values. The reviewer’s opinion is that despite its rich mathematical content, the language of this paper is poor.

34C60Qualitative investigation and simulation of models (ODE)
34A37Differential equations with impulses
37N25Dynamical systems in biology
34C25Periodic solutions of ODE
34D20Stability of ODE
92D25Population dynamics (general)
Full Text: DOI
[1] Van Lenteren, J. C.; Woets, J.: Biological and integrated pest control in greenhouses. Ann ann ent 33, 239-250 (1988)
[2] Van Lenteren, J. C.: Measures of success in biological of anthropoids by augmentation of natural enemies. Measures of success in biological control, 77-89 (2000)
[3] Brauer, F.; Soudack, A. C.: Coexistence properties of some predator-prey systems under constant rate harvesting and stocking. J math biol 12, 101-114 (1981) · Zbl 0482.92015
[4] Brauer, F.; Soudack, A. C.: Constant-rate stocking of predator-prey systems. J math biol 11, 1-14 (1981) · Zbl 0448.92020
[5] Laksmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations. (1989)
[6] Bainov D, Simeonor P. Impulsive differential equations: periodic solutions and applications. Pitman Monographs and Surreys in Pure and Applied Mathematics. 1993;66
[7] Shulgin, B.; Stone, L.; Agur, I.: Pulse vaccination strategy in the SIR epidemic model. Bull math biol 60, 1-26 (1998) · Zbl 0941.92026
[8] Ballinger, G.; Liu, X.: Permanence of population growth models with impulsive effects. Math compu modell 26, 59-72 (1997) · Zbl 1185.34014
[9] Funasaki, E.; Kot, M.: Invasion and chaos in a periodically pulsed mass-action chemostat. Theor popul biol 44, 203-224 (1993) · Zbl 0782.92020
[10] Lakmeche, A.; Arino, O.: Bifurcation of non-trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment. Dynamics of continuous. Discrete impulsive syst 7, 265-287 (2000) · Zbl 1011.34031
[11] Panetta, J. C.: A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competition environment. Bull math biol 58, 425-447 (1996) · Zbl 0859.92014
[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] 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
[14] Klebanoff, A.; Hastings, A.: Chaos in three species food chains. J math biol 32, 427-451 (1994) · Zbl 0823.92030
[15] Hsu, S. B.; Hwang, T. W.; Kuang, Y.: A ratio-dependent food chain model and its applications to biological control. Math biosci 181, 55-83 (2003) · Zbl 1036.92033
[16] Gakkhar, S.; Naji, M. A.: Order and chaos in predator to prey ratio-dependent food chain. Chaos, solitons & fractal. 18, 229-239 (2003) · Zbl 1068.92044
[17] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. C.: Theory of impulsive differential equations. (1989) · Zbl 0719.34002
[18] May, R. M.: Biological population with nonoverlapping generations: stable points, stable cycles, and chaos. Science 186, 645-657 (1974)
[19] May, R. M.; Oster, G. F.: Bifurcations and dynamic complexity in simple ecological models. Am nature 110, 573-599 (1976)
[20] Collet, P.; Eeckmann, J. P.: Iterated maps of the inter val as dynamical systems. (1980)
[21] 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
[22] Vandermeer, J.; Stone, L.; Blasius, B.: Categories of chaos and fractal basin boundaries in forced predator-prey models. Chaos, solitons & fractals 12, 265-276 (2001) · Zbl 0976.92033
[23] Neubert, M. G.; Caswell, H.: Density-dependent vital rates and their population dynamic consequences. J math biol 41, 103-121 (2000) · Zbl 0956.92029
[24] 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