×

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.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34A37 Ordinary differential equations with impulses
92D25 Population dynamics (general)
34C23 Bifurcation theory for ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34H05 Control problems involving ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bainov, D.; Simeonov, P., Impulsive Differential Equations: Periodic Solutions and Applications (1993), Longman/John Wiley: Longman/John Wiley New York, NY · 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), Springer-Verlag: Springer-Verlag New York, NY · Zbl 0487.47039
[4] Georgescu, P.; Moroşanu, G., Pest regulation by means of impulsive controls, Appl. Math. Comput., 190, 790-803 (2007) · Zbl 1117.93006
[5] Hethcote, H. W.; van den Driessche, P., 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), Elsevier/North-Holland: Elsevier/North-Holland New York, NY · 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) · Zbl 1334.92410
[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.; van den Bosch, R.; 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.