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)
Complex dynamics of a Holling type II prey-predator system with state feedback control. (English) Zbl 1203.34071
Summary: The complex dynamics of a Holling type II prey-predator system with impulsive state feedback control is studied in both theoretical and numerical ways. Sufficient conditions for the existence and stability of positive periodic solutions are obtained by using the Poincaré map and the analogue of the Poincaré criterion. A qualitative analysis shows that the positive periodic solution bifurcates from a solution through a fold bifurcation. The bifurcation diagrams, Lyapunov exponents, and phase portraits are illustrated by an example, in which the chaotic solutions appear via a cascade of period-doubling bifurcations. The superiority of the state feedback control strategy is also discussed.

34C60Qualitative investigation and simulation of models (ODE)
34C28Complex behavior, chaotic systems (ODE)
92D25Population dynamics (general)
34A37Differential equations with impulses
93B52Feedback control
34C25Periodic solutions of ODE
Full Text: DOI
[1] Guckenheimer, J.; Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. (1983) · Zbl 0515.34001
[2] Kuznetsov, Y. A.: Elements of applied bifurcation theory. (1995) · Zbl 0829.58029
[3] Ivanov, Alexander P.: Bifurcations in impact systems. Chaos, solitons & fractals 7, 1615-1634 (1996) · Zbl 1080.37570
[4] Leine, R. I.; Van Campen, D. H.; Van De Vrande, B. L.: Bifurcations in nonlinear discontinuous systems. Nonlinear dynam 23, 105-164 (2000) · Zbl 0980.70018
[5] Lsksmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations. (1989)
[6] Bainov, D. D.; Simeonov, P. S.: Impulsive differential equations: periodic solutions and applications. (1993) · Zbl 0815.34001
[7] Simeonov, P. E.; Bainov, D. D.: Orbital stability of periodic solutions of autonomous systems with impulse effect. Int J syst sci 19, 2562-2585 (1988) · Zbl 0669.34044
[8] Shulgin, B.; Stone, L.; Agur, Z.: Theoretical examination of pulse vaccination policy in the SIR epidemic model. Math comput model 31, 207-215 (2000) · Zbl 1043.92527
[9] Lu, Z.; Chi, X.; Chen, L.: The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission. Math comput model 36, 1039-1057 (2002) · Zbl 1023.92026
[10] D’onofrio, A.: Stability properties of pulse vaccination strategy in SEIR epidemic model. Math biosci 179, 57-72 (2002) · Zbl 0991.92025
[11] Ballinger, G.; Liu, X.: Permanence of population growth models with impulsive effects. Math comput model 26, 59-72 (1997) · Zbl 1185.34014
[12] Lakmeche, A.; Arino, O.: Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment. Dyn contin discrete impulsive syst 7, 265-287 (2000) · Zbl 1011.34031
[13] Liu, X. Z.; Rohlf, K.: Impulsive control of a Lotka -- Volterra system. IMA J math contr inform 15, 269-284 (1998) · Zbl 0949.93069
[14] Liu, X.; Chen, L.: Complex dynamics of Holling type II lotaka -- Volterra predator -- prey system with impulsive perturbations on the predator. Chaos, solitons & fractals 16, 311-320 (2003) · Zbl 1085.34529
[15] Tang, S. Y.; Chen, L. S.: Density-dependent birth rate, birth pulses and their population dynamic consequences. J math biol 44, 185-199 (2002) · Zbl 0990.92033
[16] Liu, B.; Zhang, Y.; Chen, L.: Dynamic complexities of a Holling I predator -- prey model concerning periodic biological and chemical control. Chaos, solitons & fractals 22, 123-134 (2004) · Zbl 1058.92047
[17] Zhang, S.; Dong, L.; Chen, L.: The study of predator -- prey system with defensive ability of prey and impulsive perturbations on the predator. Chaos, solitons & fractals 23, 631-643 (2005) · Zbl 1081.34041
[18] Wang, F. Y.; Zhang, S. W.; Chen, L. S.: Permanence and complexity of a three species food chain with impulsive effect on predator. Int J nonlinear sci numer simulat 6, 169-180 (2005)
[19] El-Shahed, M.: Application of he’s homotopy perturbation method to Volterra’s integro-differential equation. Int J nonlinear sci numer simulat 6, 163-168 (2005)
[20] Van Lenteren, J. C.: Integrated pest management in protected crops. Integrated pest management, 311-320 (1995)
[21] Holling, C. S.: The functional response of predator to prey density and its role in mimicry and population regulation. Mem ent soc Canada 45, 1-60 (1965)