×

A stage-structured SI eco-epidemiological model with time delay and impulsive controlling. (English) Zbl 1173.93311

Summary: This paper formulates a robust stage-structured SI eco-epidemiological model with periodic constant pulse releasing of infectious pests with pathogens. The authors show that the conditions for global attractivity of the ‘pest-eradication’ periodic solution and permanence of the system depend on time delay, hence, the authors call it “profitless”. Further, the authors present a pest management strategy in which the pest population is kept under the economic threshold level when the pest population is uniformly persistent. By numerical analysis, the authors also show that constant maturation time delay for the susceptible pests and pulse releasing of the infectious pests can bring obvious effects on the dynamics of system.

MSC:

93A15 Large-scale systems
93A30 Mathematical modelling of systems (MSC2010)
92D30 Epidemiology
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] V. M. Stern, Economic thresholds, Ann. Rev. Entomol, 1973, 18(1): 259–280.
[2] J. van Lenteren, Integrated pest management in protected crops, in Integrated Pest Management (ed. by D. Dent), Chapman & Hall, London, 1995.
[3] J. van Lenteren and J. Woets, Biological and integrated pest control in greenhouses, Ann. Rev. Ent., 1998, 33(1): 239–250.
[4] J. Lü and G. Chen, A new chaotic attractor coined, International Journal of Bifurcation and Chaos, 2002, 12(3): 659–661. · Zbl 1063.34510
[5] J. Lü, G. Chen, D. Cheng, and S. Celikovsky, Bridge the gap between the Lorenz system and the Chen system, International Journal of Bifurcation and Chaos, 2002, 12(12): 2917–2926. · Zbl 1043.37026
[6] L. Falcon, Use of bacteria for microbial control of insetts, Microbial Control of Insects and Mites (ed. by H. D. Burges and N. W. Hussey), Academic Press, New York, 1971.
[7] H. Burges and N. Hussey, Microbial Control of Insects and Mites, Academic Press, New York, 1971.
[8] Y. Tanada, Epizootiology of insect diseases, Biological Control of Insect Pests and Weeds (ed. by P. DeBach), Chapman & Hall, London, 1964.
[9] L. Falcon, Problems associated with the use of arthropod viruses in pest control, Annu. Rev. Entomol., 1976, 21(2): 305–324.
[10] R. Anderson and R. May, Regulation and stability of host-parasite population interactions, I. Regulatory processes, J. Anim. Ecol., 1978, 47(1): 219–247.
[11] W. Aiello and H. Freedman, A time-delay model of single-species growth with stage structure, Math. Biosci., 1990, 101: 139–153. · Zbl 0719.92017
[12] Y. Xiao, L. Chen, and F. Bosch, Dynamical behavior for a stage-structured SIR infectious disease model, Nonlinear Analysis: Real Word Applications, 2002, 3(2): 175–190. · Zbl 1007.92032
[13] X Song and L Chen, Optimal harvesting and stability for a two-species competitive system with stage structure, Math. Biosci., 2001, 170(2): 173–186. · Zbl 1028.34049
[14] S. Gourley and Y. Kuang, A stage-structured predator-prey model and its dependence on through-stage delay and death rate, J. Math. Biol., 2004, 49(2): 188–200. · Zbl 1055.92043
[15] S. Liu and E. Beretta, A stage-structured predator-prey model of beddington-deangelis, Type. Siam J. Appl. Math, 2006, 66(4): 1101–1129. · Zbl 1110.34059
[16] J. Hui and D. Zhu, Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects, Chaos, Solitons and Fractals, 2006, 29(1): 233–251. · Zbl 1095.92067
[17] A. Hastings, Age-dependent predation is not a simple process, I. Continuous time models, Theor. Popul. Biol., 1983, 23(1): 47–62. · Zbl 0507.92016
[18] D. Bainov and P. Simeonov, System with Impulsive Effect: Stability, Theory and Applications, John Wiley and Sons, New York, 1989. · Zbl 0676.34035
[19] V. Lakshmikantham, D. Bainov, and P. Simeonov, Theory of Impulsive Differential Equations, World Scienti. C, Singapore, 1989. · Zbl 0719.34002
[20] S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations, 2003, 188(1): 135–163. · Zbl 1028.34046
[21] S. Zhang and L. Chen, The study of predator-prey system with defensive ability of prey and impulsive perturbations on the predator, Chaos, Solitons and Fractals, 2005, 23(2): 631–643. · Zbl 1081.34041
[22] B. Liu, Y. Zhang, and L. Chen, The dynamical behaviors of a Lotka-Volterra predator-prey model concerning integrated pest management, Nonlinear Analysis: Real World Applications, 2005, 6: 227–243. · Zbl 1082.34039
[23] B. Liu, L. Chen, and Y. Zhang, The dynamics of a prey-dependent consumption model concerning impulsive control strategy, Applied Mathematics and Computation, 2005, 169(1): 305–320. · Zbl 1074.92042
[24] X. Meng, J. Jiao, and L. Chen, The dynamics of an age-structured predator-prey model with disturbing pulse and time delays, Nonlinear Analysis: Real World Applications, 2008, 9(2): 547–561. · Zbl 1142.34054
[25] X. Meng, Z. Song, and L. Chen, A new mathematical model for optimal control strategies of integrated pest management, Journal of Biological Systems, 2007, 15(2): 219–234. · Zbl 1279.92053
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.