A Volterra model with mutual interference concerning integrated pest management is proposed and analyzed. By using Floquet theorem and small amplitude perturbation method and comparison theorem, we show the existence of a globally asymptotically stable pest-eradication periodic solution. Further, we prove that when the stability of pest-eradication periodic solution is lost, the system is permanent and there exists a locally stable positive periodic solution which arises from the pest-eradication periodic solution by bifurcation theory. When the unique positive periodic solution loses its stability, numerical simulation shows there is a characteristic sequence of bifurcations, leading to a chaotic dynamics. Finally, we compare the validity of integrated pest management (IPM) strategy with classical methods and conclude IPM strategy is more effective than classical methods.

Classification: 34A37, 92D25

Keywords: integrated pest management (IPM), mutual interference, permanence, bifurcation, chaos

@article{M2AN_2004__38_1_143_0, author = {Zhang, Yujuan and Liu, Bing and Chen, Lansun}, title = {Dynamical behavior of Volterra model with mutual interference concerning IPM}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {38}, number = {1}, year = {2004}, pages = {143-155}, doi = {10.1051/m2an:2004007}, zbl = {1081.34042}, mrnumber = {2073934}, language = {en}, url = {http://www.numdam.org/item/M2AN_2004__38_1_143_0} }

Zhang, Yujuan; Liu, Bing; Chen, Lansun. Dynamical behavior of Volterra model with mutual interference concerning IPM. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 1, pp. 143-155. doi : 10.1051/m2an:2004007. http://www.numdam.org/item/M2AN_2004__38_1_143_0/

[1] Impulsive differential equations: periodic solutions and applications. Wiley, New York (1993). | MR 1266625 | Zbl 0815.34001

and ,[2] Pathogen incidence and their potential as microbial control agents in IPM of maize stemborers in West Africa. Biocontrol 44 (1999) 301-327.

, , and ,[3] Antimonotonicity: inevitable reversals of period-doubling cascades. Phys. Lett. A 162 (1992) 249-254.

, and ,[4] The area of discovery of two insect parasites, Nasonia vitripennis (Walker) and Trichogramma evanescens Westwood, in an artificial environment. Can. Ent. 93 (1961) 475-481.

,[5] A two-component model of host-parasitoid interactions: determination of the size of inundative releases of parasitoids in biological pest control. Math. Biosci. 169 (2001) 207-216. | Zbl 0966.92026

, , et al.,[6] Crises, sudden changes in chaotic attractors and chaotic transients. Physica D 7 (1983) 181-200. | Zbl 0561.58029

, and ,[7] Parasite behavior as a factor contributing to the stability of insect host-parasite interactions, in Dynamics of Population, P.J. den Boer and G.R. Gradwell Eds., Proc. Adv. Study Inst. Oosterbeek (1970).

,[8] Mutual interference between searching insect parasites. J. Anim. Ecol. 40 (1971) 473-486.

,[9] New inductive population model for insect parasites and its bearing on biological control. Nature 223 (1969) 1133-1136.

, ,[10] Insect parasite responses in the development of population models. J. Anim. Ecol. 41 (1972) 661-676.

and ,[11] Persistence of transients in spatially structured ecological models. Since 263 (1994) 1133-1136.

and ,[12] The natural enemy component in natural control and the theory of biological control, in Biological Control, C.B. Huffaker Ed., Proc. A.A.A.S. Symp. Plenum Press, New York (1969).

, and ,[13] Integrated Pest Management for Walnuts, University of California Statewide Integrated Pest Management Project, in Division of Agriculture and Natural Resources, Second Edition, M.L. Flint Ed., University of California, Oakland, CA, publication 3270 (1987).

[14] Theory of impulsive differential equations. World Scientific, Singapore (1989). | MR 1082551 | Zbl 0719.34002

, and ,[15] Bifurcation of non trival periodic solutions of impulsive differential equations arising chemotherapeutic treatment. Dynam. Contin. Discrete Impuls. Systems 7 (2000) 205-287. | Zbl 1011.34031

and ,[16] The potential of predators for pest control. Agri. Ecos. Environ. 10 (1983) 159-181.

,[17] Random search and insect population models. J. Anim. Ecol. 41 (1972) 369-383.

,[18] University of California, Division of Agriculture and Natural Resources, Integrated Pest Management for Alfafa hay. Publications, Division of Agriculture and Nature Resources, University of Califania, Oakland, CA, publication 3312 (1981).

[19] Integrated pest management in protected crops, in Integrated pest management, D. Dent Ed., Chapman & Hall, London (1995).

,[20] Stability of Volterra model with mutual interference. J. Jiangxi Norm. Univ. 2 (1986) 42-45. | Zbl 0632.92016

,