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)
Bifurcation of nontrivial periodic solutions for an impulsively controlled pest management model. (English) Zbl 1151.34037
Summary: This paper investigates the onset of nontrivial periodic solutions for an integrated pest management model which is subject to pulsed biological and chemical controls. The biological control consists in the periodic release of infective individuals, while the chemical control consists in periodic pesticide spraying. It is assumed that both controls are used with the same periodicity, although not simultaneously. To model the spread of the disease which is propagated through the release of infective individuals, an unspecified force of infection is employed. The problem of finding nontrivial periodic solutions is reduced to showing the existence of nontrivial fixed points for the associated stroboscopic mapping of time snapshot equal to the common period of controls. The latter problem is in turn treated via a projection method. It is then shown that once a threshold condition is reached, a stable nontrivial periodic solution emerges via a supercritical bifurcation.

34C60Qualitative investigation and simulation of models (ODE)
34A37Differential equations with impulses
92D25Population dynamics (general)
34C23Bifurcation (ODE)
37G15Bifurcations of limit cycles and periodic orbits
34H05ODE in connection with control problems
34C25Periodic solutions of ODE
Full Text: DOI
[1] Bainov, D.; Simeonov, P.: Impulsive differential equations: periodic solutions and applications. (1993) · Zbl 0815.34001
[2] Capasso, V.; Serio, G.: A generalization of kermack -- mckendrick deterministic epidemic model. Math. biosci. 42, 43-61 (1978) · Zbl 0398.92026
[3] Chow, S. N.; Hale, J.: Methods of bifurcation theory. (1982) · Zbl 0487.47039
[4] Georgescu, P.; Moro┼čanu, G.: Pest regulation by means of impulsive controls. Appl. math. Comput. 190, 790-803 (2007) · Zbl 1117.93006
[5] Hethcote, H. W.; Den Driessche, P. Van: Some epidemiological models with nonlinear incidence. J. math. Biol. 29, 271-287 (1991) · Zbl 0722.92015
[6] Iooss, G.: Bifurcation of maps and applications. (1979) · Zbl 0408.58019
[7] Korobeinikov, A.: Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull. math. Biol. 30, 615-626 (2006)
[8] Korobeinikov, A.; Maini, P. K.: Nonlinear incidence and stability of infectious disease models. Math. med. Biol. 22, 113-128 (2005) · Zbl 1076.92048
[9] Lakmeche, A.; Arino, O.: Bifurcation of nontrivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment. Dyn. continum. Discrete impul. Syst. ser. A: math. Anal. 7, 265-287 (2000) · Zbl 1011.34031
[10] Lakmeche, A.; Arino, O.: Nonlinear mathematical model of pulsed-therapy of heterogeneous tumors. Nonlinear anal. Real world appl. 2, 455-465 (2001) · Zbl 0982.92016
[11] Liu, W. M.; Hethcote, H. W.; Levin, S. A.: Dynamical behavior of epidemiological models with nonlinear incidence rates. J. math. Biol. 25, 359-380 (1987) · Zbl 0621.92014
[12] Lu, Z.; Chi, X.; Chen, L.: Impulsive control strategies in biological control of pesticide. Theor. pop. Biol. 64, 39-47 (2003) · Zbl 1100.92071
[13] Lu, Z.; Chi, X.; Chen, L.: The effect of constant and pulse vaccination of a SIR epidemic model with horizontal and vertical transmission. Math. comput. Modell. 36, 1039-1057 (2002) · Zbl 1023.92026
[14] Panetta, J. C.: A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competition environment. Bull. math. Biol. 58, 425-447 (1996) · Zbl 0859.92014
[15] Ruan, S.; Wang, W.: Dynamical behavior of an epidemic model with a nonlinear incidence rate. J. diff. Eq. 188, 135-163 (2003) · Zbl 1028.34046
[16] Stern, V. M.; Smith, R. F.; Den Bosch, R. Van; Hagen, K. S.: The integrated control concept. Hilgardia 29, 81-101 (1959)
[17] Xiao, D.; Ruan, S.: Global analysis of an epidemic model with nonmonotone incidence rate. Math. biosci. 208, 419-429 (2007) · Zbl 1119.92042