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 effects of impulsive releasing methods of natural enemies on pest control and dynamical complexity. (English) Zbl 1238.93044
Summary: In the paper, different releasing methods including constant releasing and proportional to the predator population are considered and analyzed. The effects of these releasing methods of natural enemies on dynamical behavior are investigated. We firstly take into account the model with an impulsive effect at fixed moments, and the results imply that under some conditions the pest may serve to extinction. Several types of attractors can coexist, with switch-like transitions among their attractors showing that varying dosages and frequencies of insecticide applications and the numbers of natural enemies released are crucial. Secondly, the model with unfixed moments is further investigated. Different periodic solutions also exist and the maximum amplitude of the host is always less than the economic threshold. Comparing the results obtained for the two models concludes that the proportional releasing predator has strong effects on the dynamical behavior.
93C15Control systems governed by ODE
93C95Applications of control theory
93A15Large scale systems
[1]Van Lenteren, J. C.: Measures of success in biological control of anthropoids by augmentation of natural enemies, Measures of success in biological control, 77 (2000)
[2]Van Lenteren, J. C.: Integrated pest management in protected crops, Integrated pest management, 311 (1995)
[3]Van Lenteren, J. C.; Woets, J.: Biological and integrated pest control in greenhouses, Ann. rev. Ent. 33, 239 (1988)
[4]Tang, S. Y.; Cheke, R. A.: Models for integrated pest control and their biological implications, Math. biosci. 215, 115 (2008) · Zbl 1156.92046 · doi:10.1016/j.mbs.2008.06.008
[5]Tang, S. Y.; Cheke, R. A.: State-dependent impulsive models of integrated pest management(IPM) strategies and their dynamic consequences, J. math. Biol. 50, 257 (2005) · Zbl 1080.92067 · doi:10.1007/s00285-004-0290-6
[6]Tang, S. Y.; Chen, L. S.: Modelling and analysis of integrated pest management strategy, Dis. cont. Dyn. sys. B 4, 761 (2004) · Zbl 1114.92074 · doi:10.3934/dcdsb.2004.4.759
[7]Tang, S. Y.; Xiao, Y. N.; Cheke, R. A.: Multiple attractors of host-parasitoid models with integrated pest management strategies: eradication, persistence and outbreak, Theor. popul. Biol. 73, 181 (2008) · Zbl 1208.92093 · doi:10.1016/j.tpb.2007.12.001
[8]M.P. Hoffmann, A.C. Frodsham, Natural enemies of vegetable insect pests, Cooperative Extension, Cornell University, Ithaca, NY, 1993.
[9]Stinner, R. E.: Efficacy of inundative releases, A. res. Ent. 22, 515 (1977)
[10]Summers, T. E.; King, E. G.; Martin, D. F.; Jackson, R. D.: Biological control of diatraea sacchanalis in Florida by periodic releases of lixophaga diatraeae, Entomophaga 4, 359 (1976)
[11]Lotka, A. J.: Undamped oscillations derived from the law of mass action, J. amer. Chem. soc. 42, 1595 (1920)
[12]Volterra, V.: Variations and fluctuations of a number of individuals in animal species living together, Animal ecology, 409 (1931)
[13]Grebogi, C.; Ott, E.; Yorke, J. A.: Crises, sudden changes in chaotic attractors and chaotic transients, Physica D 7, 181 (1983) · Zbl 0561.58029 · doi:10.1016/0167-2789(83)90126-4
[14]Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J.; Knuth, D. E.: On the Lambert W function, Adv. comput. Math. 5, 329 (1996)