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)
The study of predator--prey system with defensive ability of prey and impulsive perturbations on the predator. (English) Zbl 1081.34041
A predator-prey system with nonmonotonic functional response and impulsive perturbations on the predator is considered. By using Floquet’s theorem and the method of small amplitude perturbation, a locally asymptotically stable prey-eradication periodic solution is established when the impulsive period is less than some critical value. Otherwise, if the impulsive period is larger than the critical value, the system is permanent. Further, using numerical simulations, the influences of the impulsive perturbations on the inherent oscillation are investigated. With increasing impulsive value, the system displays a series of complex phenomena, which include (1) quasi-periodic oscillating, (2) period-doubling, (3) period-halfing, (4) nonunique dynamics (meaning that several attractors coexist), (5) attractor crisis and (6) chaotic bands with periodic windows.

34C60Qualitative investigation and simulation of models (ODE)
92D25Population dynamics (general)
34A37Differential equations with impulses
34C25Periodic solutions of ODE
34D05Asymptotic stability of ODE
34D20Stability of ODE
34C23Bifurcation (ODE)
34C28Complex behavior, chaotic systems (ODE)
Full Text: DOI
[1] Holling, C. S.: The functional response of predator to prey density and its role in mimicry and population regulation. Mem. ent. Sec. can. 45, 1-60 (1965)
[2] Andrews, J. F.: A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotechnol. bioeng. 10, 707-723 (1968)
[3] Sugie, J.; Howell, J. A.: Kinetics of phenol oxidation by washed cell. Biotechnol. bioeng. 23, 2039-2049 (1980)
[4] Tener, J. S.: Muskoxen. (1965)
[5] Ruan, S.; Xiao, D.: Global analysis in a predator--prey system with nonmonotonic functional response. SIAM J. Appl. math. 61, 1445-1472 (2001) · Zbl 0986.34045
[6] 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
[7] Brauer, F.; Soudack, A. C.: Constant-rate stocking of predator--prey systems. J. math. Biol. 11, 1-14 (1981) · Zbl 0448.92020
[8] Croft, B. A.: Arthropod biological control agents and pesticides. (1990)
[9] Debach, P.; Rosen, D.: Biological control by natural enemies. (1991)
[10] Van Lenteren, J. C.: Measures of success in biological of anthrop pods by augmentation of natural enemies. Measures of success in biological control, 77-89 (2000)
[11] Luff, M. L.: The potential of predators for pest control. Agri. ecos. Environ. 10, 159-181 (1983)
[12] Van Lenteren, J. C.; Woets, J.: Biological and integrated pest control in greenhouses. Ann. rev. Ent. 33, 239-250 (1988)
[13] 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
[14] 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
[15] Shulgin, B.; Stone, L.; Agur, Z.: Pulse vaccination strategy in the SIR epidemic model. Bull. math. Biol. 60, 1-26 (1998) · Zbl 0941.92026
[16] D’onofrio, A.: Stability properties of pulse vaccination strategy in SEIR epidemic model. Math. comput. Modell. 26, 59-72 (1997)
[17] 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
[18] 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
[19] Ballinger, G.; Liu, X.: Permanence of population growth models with impulsive effects. Math. comput. Modell. 26, 59-72 (1997) · Zbl 1185.34014
[20] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. C.: Theory of impulsive differential equations. (1989) · Zbl 0719.34002
[21] Bainov, D. D.; Simeonov, D. D.: Impulsive differential equations:periodic solutions and applications. (1993) · Zbl 0815.34001
[22] Davies, B.: Exploring chaos, theory and experiment. (1999) · Zbl 0959.37001
[23] Grebogi, C.; Ott, E.; Yorke, J. A.: Crises,sudden changes in chaotic attractors and chaotic transients. Physick D 7, 181-200 (1983) · Zbl 0561.58029
[24] May, R. M.: Biological population with nonoverlapping generations: stable points, stable cycles, and chaos. Science 186, 645-657 (1974)
[25] May, R. M.; Oster, G. F.: Bifurcations and dynamic complexity in simple ecological models. Am. nature 110, 573-599 (1976)
[26] Collet, P.; Eeckmann, J. P.: Iterated maps of the inter val as dynamical systems. (1980)
[27] 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
[28] 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
[29] Gakkhar, S.; Naji, R. K.: Chaos in seasonally perturbed ratio-dependent prey--predator system. Chaos, solitons & fractals 11, 107-118 (2000) · Zbl 1033.92026
[30] Neubert, M. G.; Caswell, H.: Density-dependent vital rates and their population dynamic consequences. J. math. Biol. 41, 103-121 (2000) · Zbl 0956.92029
[31] 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
[32] Tang, S. Y.; Chen, L. S.: Chaos in functional response host--parasitoid ecosystem models. Chaos, solitons & fractals 13, 875-884 (2002) · Zbl 1022.92042