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)
Dynamical analysis of plant disease models with cultural control strategies and economic thresholds. (English) Zbl 1183.92060
Summary: Plant disease models including impulsive cultural control strategies were developed and analyzed. Sufficient conditions under which the infected plant free periodic solution with fixed moments is globally stable are obtained. For the model with an economic threshold (ET) of infected plants, detailed investigations imply that the number of healthy plants either goes to extinction or tends to infinity, and the maximum value of infected plants is always less than the given ET. In order to prevent the healthy plant population going to extinction, we further propose a bi-threshold-value model, which has richer dynamical behavior including order 1-k or order k-1 periodic solutions with $k\geq 1$. Under certain parameter spaces, the infected plant free periodic solution is globally stable for the bi-threshold-value model. The modeling methods and analytical analysis presented can serve as an integrating measure to identify, evaluate and design appropriate plant disease control strategies.

92C80Plant biology
93C95Applications of control theory
34A37Differential equations with impulses
34C25Periodic solutions of ODE
34D20Stability of ODE
93C15Control systems governed by ODE
Full Text: DOI
[1] Bainov, D. D.; Simenov, P. S.: Systems with impulsive effect, (1982)
[2] Chan, M. S.; Jeger, M. J.: An analytical model of plant virus disease dynamics with roguing and replanting, J. appl. Ecol. 31, 413-427 (1994)
[3] Cho, J. J.; Custer, D. M.; Brommonschenkel, S. H.; Tanksley, S. D.: Conventional breeding: host-plant resistance and the use of molecular markers to develop resistance to tomato spotted wilt virus in vegetables, Acta hortic. 431, 367-378 (1996)
[4] 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, 329-359 (1996) · Zbl 0863.65008
[5] Fishman, S.; Marcus, R.; Talpaz, H.; Bar-Joseph, M.; Oren, Y.; Salomon, R.; Zohar, M.: Epidemiological and economic models for spread and control of citrus tristeza virus disease, Phytoparasitica 11, 39-49 (1983)
[6] Fishman, S.; Marcus, R.: A model for spread of plant disease with periodic removals, J. math. Biol. 21, 149-158 (1984) · Zbl 0548.92015 · doi:10.1007/BF00277667
[7] Gibson, R. W.; Aritua, V.: The perspective of sweet potato chlorotic stunt virus in sweet potato production in africa, a review, Afric. crop sci. J. 10, 281-310 (2002)
[8] Gibson, R. W.; Aritua, V.; Byamukama, E.; Mpembe, I.; Kayongo, J.: Control strategies for sweet potato virus disease in africa, Virus res. 100, 115-122 (2004)
[9] Holt, J.; Chancellor, T. C. B.: A model of plant disease epidemics in asynchronously-planted cropping systems, Plant pathol. 46, 490-501 (1997)
[10] Holt, J.; Jeger, M. J.; Thresh, J. M.; Otim-Nape, G. W.: An epidemiological model incorporating vector population dynamics applied to african cassava mosaic virus disease, J. appl. Ecol. 34, 793-806 (1997)
[11] Jeger, M. J.; Holt, J.; Den Bosch, F. Van; Madden, L. V.: Epidemiology of insect-transmitted plant viruses: modelling disease dynamics and control interventions, Physiol. entomol., 291-304 (2004)
[12] Jones, R. A. C.: Determining threshold levels for seed-borne virus infection in seed stocks, Virus res. 71, 171-183 (2000)
[13] Jones, R. A. C.: Developing integrated disease management strategies against non-persistently aphid-borne viruses: a model programme, Integr. pest manage. Rev. 6, 15-46 (2001)
[14] Jones, R. A. C.: Using epidemiological information to develop effective integrated virus disease management strategies, Virus res. 100, 5-30 (2004)
[15] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations, vol. 6, (1989) · Zbl 0719.34002
[16] 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 · doi:10.1007/s002850100121
[17] Tang, S. Y.; Chen, L. S.: Multiple attractors in stage-structured population models with birth pulses, Bull. math. Biol. 65, 479-495 (2003)
[18] Tang, S. Y.; Chen, L. S.: The effect of seasonal harvesting on stage-structured population models, J. math. Biol. 48, 357-374 (2004) · Zbl 1058.92051 · doi:10.1007/s00285-003-0243-5
[19] Tang, S. Y.; Cheke, R. A.: State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, J. math. Biol. 50, 257-292 (2005) · Zbl 1080.92067 · doi:10.1007/s00285-004-0290-6
[20] Tang, S. Y.; Xiao, Y. N.; Clancy, D.: New modeling approach concerning integrated disease control and cost-effectivity. Nonlinear analysis, Tma 63, 439-471 (2005) · Zbl 1078.92059 · doi:10.1016/j.na.2005.05.029
[21] 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, 181-197 (2008) · Zbl 1208.92093 · doi:10.1016/j.tpb.2007.12.001
[22] Tang, S. Y.; Cheke, R. A.: Models for integrated pest control and their biological implications, Math. biosci. 215, 115-125 (2008) · Zbl 1156.92046 · doi:10.1016/j.mbs.2008.06.008
[23] Thresh, J. M.; Cooter, R. J.: Strategies for controlling cassava mosaic disease in africa, Plant pathol. 54, 587-614 (2005)
[24] Den Bosch, F. Van; Jeger, M. J.; Gilligan, C. A.: Disease control and its selection for damaging plant virus strains in vegetatively propagated staple food crops: a theoretical assessment, Proc. R. Soc. B 274, 11-18 (2007)
[25] Van Lenteren, J. C.; Woets, J.: Biological and integrated pest control in greenhouses, Ann. rev. Entomol. 33, 239-250 (1988)
[26] J.C. Van Lenteren, Integrated pest management in protected crops, in: D. Dent (Ed.), Integrated Pest Management, Chapman & Hall, London, 1995, pp. 311 -- 320.
[27] Zhang, X. S.; Holt, J.: Mathematical models of cross protection in the epidemiology of plant-virus diseases, Phytopathology, 924-934 (2004)