Dynamics of a Hyperparasitic System with Prolonged Diapause for Host*


A hyperparasitic system with prolonged diapause for host is investigated. It is assumed that host prolonged diapause occur at larval stage, and parasitoid attack is limited to egg stage before the initiation of host diapause. Such behavior has been reported for many ichneumons. Hyperparasite only attacks the parasitoids that parasitize the hosts. Hyperparasitic system is often used in biological control. The existence and stability of nonnegative fixed points are explored. Numerical simulations are carried out to explore the global dynamics of the system, which demonstrate appropriate prolonged diapause rate and appropriate intrinsic growth rate can stabilize the system. The reasons are explained according to the ecological perspective. Furthermore, many other complexities which include quasi-periodicity, period-doubling bifurcations leading to chaos, chaotic attractor, intermittent and supertransients are observed.

Share and Cite:

L. Zhang and C. Zhang, "Dynamics of a Hyperparasitic System with Prolonged Diapause for Host*," International Journal of Modern Nonlinear Theory and Application, Vol. 2 No. 4, 2013, pp. 201-208. doi: 10.4236/ijmnta.2013.24028.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] J. R. Beddington, C. A. Free and J. H. Lawton, “Dynamic Complexity in Predator-Prey Models Framed in Difference Equations,” Nature, Vol. 255, No. 5503, 1975, pp. 58-60. http://dx.doi.org/10.1038/255058a0
[2] S. Y. Tang and L. S. Chen, “Chaos in Functional Response Host-Parasitoid Esosystem Models,” Chaos, Solitons & Fractals, Vol. 13, No. 4, 2002, pp. 875-884.
[3] C. L. Xu and M. S. Boyce, “Dynamic Complexities in a Mutual Interference Host-Parasitoid Model,” Chaos, Solitons & Fractals, Vol. 24, No. 1, 2005, pp. 175-182.
[4] S. J. Lv and M. Zhao, “The Dynamic Complexity of a Host-Parasitoid Model with a Lower Bound for the Host,” Chaos, Solitons & Fractals, Vol. 36, No. 4, 2008, pp. 911-999. http://dx.doi.org/10.1016/j.chaos.2006.07.020
[5] L. Zhu and M. Zhao, “Dynamic Complexity of a HostParasitoid Ecological Model with the Hassell Growth Function for the Host,” Chaos, Solitons & Fractals, Vol. 39, No. 3, 2009, pp. 1259-1269.
[6] M. Zhao and L. M. Zhang, “Permanence and Chaos in a Host-Parasitoid Model with Prolonged Diapause for the Host,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 12, 2009, pp. 4197-4203.
[7] M. Zhao, H. G. Yu and J. Zhu, “Effects of a Population Floor on the Persistence of Chaos in A Mutual Interference Host-Parasitoid Model,” Chaos, Solitons & Fractals, Vol. 42, No. 2, 2009, pp. 1245-1250.
[8] M. Zhao, L. M. Zhang and J. Zhu, “Dynamics of a Host-Parasitoid Model with Prolonged Diapause for Parasitoid,” Communications in Nonlinear Science and Numerical Simulation, Vol. 16, No. 1, 2011, pp. 455-462.
[9] E. G. Gu, “The Nonlinear Analysis on a Discrete HostParasitoid Model with Pesticidal Interference,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 6, 2009, pp. 2720-2727.
[10] S. Y. Tang, Y. N. Xiao and R. A. Cheke, “Multiple Attractors of Host-Parasitoid Models with Integrated Pest Management Strategies: Eradication, Persistence and Outbreak,” Theoretical Population Biology, Vol. 73, No. 2, 2008, pp. 181-197.
[11] C. A. Cobbold, J. Roland and M. A. Lewis, “The Impact of Parasitoid Emergence Time on Host-Parasitoid Population Dynamics,” Theoretical Population Biology, Vol. 75, No. 2-3, 2009, pp. 201-215.
[12] H. Liu, Z. Z. Li, M. Gao, H. W. Dai and Z. G. Liu, “Dynamics of a Host-Parasitoid Model with Allee Effect for the Host and Parasitoid Aggregation,” Ecological Complexity, Vol. 6, No. 3, 2009, pp. 337-345.
[13] F. Menu, J. Roebuck and M. Viala, “Bet Hedging Diapause Strategies in Stochastic Environment,” The American Naturalist, Vol. 155, No. 6, 2000, pp. 724-734.
[14] G. P. Venture, E. Wajnberg, J. Pizzol and M. L. M. Oliveira, “Diapause in the Egg Parasitoid Trichogramma Cordubensis: Role of Temperature,” Journal of Insect Physiology, Vol. 48, No. 3, 2002, pp. 349-355.
[15] A. Hua, F. S. Xue, H. J. Xiao and X. F. Zhu, “Photoperiodic Counter of Diapause Induction in Pseudopidorus fasciata (Lepidoptera:Zygaenidae),” Journal of Insect Physiology, Vol. 51, No. 12, 2005, pp. 1287-1294.
[16] V. Kostal, “Eco-Physiological Phases of Insect Diapause,” Journal of Insect Physiology, Vol. 52, No. 2, 2006, pp. 113-127.
[17] P. O. Lawrence, “Host-Parasitoid Hormonal Interactions: An Overview,” Journal of Insect Physiology, Vol. 32, No. 4, 1986, pp. 295-298.
[18] R. R. Askew, “Parasitic Insects,” Elsevier, New York 1971.
[19] J. R. Beddington and P. S. Hammond, “On the Dynamics of Host-Parasite-Hyperparasite Interactions,” Journal of Animal Ecology, Vol. 46, No. 3, 1977, pp. 811-821.
[20] P. A. P. Moran, “Some Remarks on Animal Population Dynamics,” Biometrica, Vol. 6, No. 3, 1950, 250-258.
[21] W. E. Ricker, “Stock and Recruitment,” Journal of the Fisheries Research Board of Canada, Vol. 11, No. 5, 1954, pp. 559-623. http://dx.doi.org/10.1139/f54-039
[22] A. J. Nicholson and V. A. Bailey, “The Balances of Animal Populations,” Proceedings of the Zoological Society of London, Vol. 105, No. 3, 1935, pp. 551-598.
[23] J. P. LaSalle, “The Stability and Control of Discrete Processes,” Springer-Verlag, Berlin, 1986.
[24] M. T. Rosenstein, J. J. Collins and C. J. De Luca, “A Practical Method for calculating Largest Lyapunov Exponents from Small Data Sets,” Physica D, Vol. 65, No. 1-2, 1993, pp. 117-134.
[25] R. C. Hilborn, “Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers,” Oxford University Press, New York, 1994.
[26] A. Hastings and K. Higgins, “Persistence of Transients in Spatially Structured Ecological Models,” Science, Vol. 263, No. 5150, 1994, pp. 1133-1136.

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.