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)
Dynamics of an impulsively controlled Michaelis-Menten type predator-prey system. (English) Zbl 1221.34031
Summary: We study a predator–prey system with a Michaelis–Menten functional response and impulsive perturbations which contain chemical and biological control terms. By applying the Floquet theory, we establish conditions for the existence and stability of prey-free solutions of the system. We also show the existence of a positive periodic solution of the system by using the bifurcation theorem and find a sufficient condition that makes the system permanent. Moreover, numerical results on impulsive perturbations show that the system we consider can give birth to various kinds of dynamical behaviors.
34A37Differential equations with impulses
34D20Stability of ODE
92D25Population dynamics (general)
93C15Control systems governed by ODE
[1]Andrews, J. F.: A mathematical model for the continuous culture of macroorganisms untilizing inhibitory substrates, Biotechnol bioeng 10, 707-723 (1968)
[2]Arditi, R.; Ginzburg, L. R.: Coupling in predator – prey dynamics: ratio-dependence, J theor biol 139, 311-326 (1989)
[3]Baek, H.; Kim, S. D.; Kim, P.: Permanence and stability of a ivlev-type predator – prey system with impulsive control strategy, Math comput model 50, 1385-1393 (2009) · Zbl 1185.34067 · doi:10.1016/j.mcm.2009.07.007
[4]Baek, H.: Extinction and permanence of a three species Lotka – Volterra system with impulsive control strategies, Discrete dyn nat soc 2008 (2008) · Zbl 1167.34350 · doi:10.1155/2008/752403
[5]Bainov DD, Simeonov PS. Impulsive differential equations: periodic solutions and applications of Pitman monographs and surveys in pure and applied mathematics, vol. 66. Harlo, UK: Longman Science amp; Technical; 1993.
[6]Brauer, F.; Castillo-Chávez, C.: Mathematical models in population biology and epidemiology, Texts in applied mathematics 30 (2001)
[7]Chen, L.; Jing, Z.: The existence and uniqueness of predator – prey differential equations, Chin sci bull 9, 521-523 (1984)
[8]Collings, J. B.: The effects of the functional response on the bifurcation behavior of a mite predator – prey interaction model, J math biol 36, 149-168 (1997) · Zbl 0890.92021 · doi:10.1007/s002850050095
[9]Cushing, J. M.: Periodic time-dependent predator – prey systems, SIAM J appl math 32, 82-95 (1977) · Zbl 0348.34031 · doi:10.1137/0132006
[10]Freedman HI. Deterministic mathematical models in population ecology. New York: Marcel Dekker; 1980.
[11]Hsu, S. -B.; Huang, T. -W.: Global stability for a class of predator – prey systems, SIAM J appl math 55, No. 3, 763-783 (1995) · Zbl 0832.34035 · doi:10.1137/S0036139993253201
[12]Hwang, T. -W.: Uniqueness of limit cycles of the predator – prey system with beddington – deangelis functional response, J math anal appl 290, 113-122 (2004) · Zbl 1086.34028 · doi:10.1016/j.jmaa.2003.09.073
[13]Jiang, G.; Lu, Q.; Qian, L.: Complex dynamics of a Holling type II prey – predator system with state feedback control, Chaos soliton fract 31, 448-461 (2007) · Zbl 1203.34071 · doi:10.1016/j.chaos.2005.09.077
[14]Lakmeche, A.; Arino, O.: Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dyn contin discrete impuls syst 7, 265-287 (2000) · Zbl 1011.34031
[15]Lakshmikantham, V.; Bainov, D.; Simeonov, P.: Theory of impulsive differential equations, (1989) · Zbl 0718.34011
[16]Li, Z.; Wang, W.; Wang, H.: The dynamics of a beddington-type system with impulsive control strategy, Chaos soliton fract 29, 1229-1239 (2006) · Zbl 1142.34305 · doi:10.1016/j.chaos.2005.08.195
[17]Liu, X.; Chen, L.: Complex dynamics of Holling type II Lotka – Volterra predator – prey system with impulsive perturbations on the predator, Chaos soliton fract 16, 311-320 (2003) · Zbl 1085.34529 · doi:10.1016/S0960-0779(02)00408-3
[18]Liu, B.; Teng, Z.; Chen, L.: Analsis of a predator – prey model with Holling II functional response concerning impulsive control strategy, J comput appl math 193, No. 1, 347-362 (2006) · Zbl 1089.92060 · doi:10.1016/j.cam.2005.06.023
[19]Liu, B.; Zhang, Y.; Chen, L.: Dynamic complexities in a Lotka – Volterra predator – prey model concerning impulsive control strategy, Int J bifurcat chaos 15, No. 2, 517-531 (2005) · Zbl 1080.34026 · doi:10.1142/S0218127405012338
[20]Liu, B.; Zhang, Y.; Chen, L.: The dynamical behaviors of a Lotka – Volterra predator – prey model concerning integrated pest mangement, Nonlinearity anal 6, 227-243 (2005) · Zbl 1082.34039 · doi:10.1016/j.nonrwa.2004.08.001
[21]Lu, Z.; Chi, X.; Chen, L.: Impulsive control strategies in biological control and pesticide, Theor popul biol 64, 39-47 (2003) · Zbl 1100.92071 · doi:10.1016/S0040-5809(03)00048-0
[22]May, R.: Limit cycles in predator prey communities, Science 177, 900-902 (1972)
[23]Nieto, J. J.; O’regan, D.: Variational approach to impulsive differential equations, Nonlinear anal: real world appl 10, 680-690 (2009) · Zbl 1167.34318 · doi:10.1016/j.nonrwa.2007.10.022
[24]Rafikov, M.; Balthazar, J. M.; Von Bremen, H. F.: Mathematical modeling and control of population systems: applications in biological pest control, Appl math comput 200, 557-573 (2008) · Zbl 1139.92022 · doi:10.1016/j.amc.2007.11.036
[25]Saez, E.; Gonzalez-Olivares, E.: Dynamics of a predator – prey model, SIAM J appl math 59, No. 5, 1867-1878 (1999) · Zbl 0934.92027 · doi:10.1137/S0036139997318457
[26]Samoilenko, A. M.; Perestyuk, N. A.: Impulsive differential equations, (1995) · Zbl 0837.34003
[27]Skalski, G. T.; Gilliam, J. F.: Functional response with predator interference: viable alternatives to the Holling type II model, Ecology 82, No. 11, 3083-3092 (2001)
[28]Tang, S.; Cheke, R. A.: State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, J math biol 50, 257-292 (2005) · Zbl 1080.92067 · doi:10.1007/s00285-004-0290-6
[29]Tang, S.; Xiao, Y.; Chen, L.; Cheke, R. A.: Integrated pest management models and their dynamical behaviour, Bull math biol 67, 115-135 (2005)
[30]Wang, W. B.; Shen, J. H.; Nieto, J. J.: Permanence periodic solution of predator prey system with Holling type functional response and impulses, Discrete dyn nat soc (2007) · Zbl 1146.37370 · doi:10.1155/2007/81756
[31]Wang, H.; Wang, W.: The dynamical complexity of a ivlev-type prey – predator system with impulsive effect, Chaos soliton fract (2007)
[32]Zavalishchin ST, Sesekin AN. Dynamic impulse systems. Theory and applications. Mathematics and its applications, vol. 394. Dordrecht: Kluwer Publishes Group; 1997. · Zbl 0880.46031
[33]Zhang, H.; Chen, L.; Nieto, J. J.: A delayed epidemic model with stage-structure and pulses for pest management strategy, Nonlinear anal: real world appl. 9, 1714-1726 (2008) · Zbl 1154.34394 · doi:10.1016/j.nonrwa.2007.05.004