Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Global dynamics of non-smooth Filippov pest-natural enemy system with constant releasing rate

School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, P.R. China

Special Issues: Non-smooth biological dynamical systems and applications

Modelling integrated pest management (IPM) with a threshold control strategy can be achieved with a non-smooth Filippov dynamical system coupled by an untreated subsystem and a treated subsystem which includes chemical and biological tactics. The releasing constant of natural enemies related to biological control generates the complex dynamics. Comprehensive qualitative analyses reveal that the treated subsystem exists with transcritical, saddle-node, Hopf and Bogdanov-Takens bifurcations, for which the threshold conditions and bifurcation curves are provided. Further, by applying techniques of non-smooth dynamical systems including the Filippov convex method and sliding bifurcation techniques, we first obtain the sliding dynamic equation, and then we analyze the existence and stability of regular/virtual equilibria, pseudo-equilibria, boundary equilibria, sliding segments and sliding bifurcations. In particular, if we choose the economic threshold (ET) as the bifurcation parameter, then interesting dynamical behaviors, including boundary equilibrium → pseudo-homoclinic → touching → buckling → crossing bifurcations, occur in succession. It is interesting to note that although the number of pests in the untreated subsystem could increase and exceed the economic injury level (EIL), the final size could be less than ET and stabilizes at a relative low level due to side effects of the pesticide on natural enemies. However, the side effects can be effectively avoided by increasing the releasing constant, which can maintain the number of pests below the EIL always and thus achieve the control purpose.
  Figure/Table
  Supplementary
  Article Metrics

Keywords Filippov system; integrated pest management; bifurcation; sliding dynamics; releasing constant

Citation: Hao Zhou, Xia Wang, Sanyi Tang. Global dynamics of non-smooth Filippov pest-natural enemy system with constant releasing rate. Mathematical Biosciences and Engineering, 2019, 16(6): 7327-7361. doi: 10.3934/mbe.2019366

References

  • 1. M. E. M. Meza, A. Bhaya and E. Kaszkurewicz, Threshold policies control for predator-prey systems using a control Liapunov function approach, Theor. Popul. Biol., 67 (2005), 273–284.
  • 2. M. I. D. S. Costa and M. E. M. Meza, Application of a threshold policy in the management of multispecies fisheries and predator culling, Math. Med. Biol., 23 (2006), 63–75.
  • 3. S. C. M. Da, Harvesting induced fluctuations: insights from a threshold management policy, Math. Biosci., 205 (2007), 77–82.
  • 4. S. Y. Tang and R. A. Cheke, Models for integrated pest control and their biological implications, Math. Biosci., 215 (2008), 115–125.
  • 5. S. Y. Tang and L. S. Chen, Modelling and analysis of integrated pest management strategy, Discrete Contin. Dyn-B, 4 (2012), 759–768.
  • 6. S. Y. Tang and R. A. Cheke, State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, J. Math. Biol., 50 (2005), 257–292.
  • 7. G. Y. Tang, W. J. Qin and S. Y. Tang, Complex dynamics and switching transients in periodically forced Filippov prey-predator system, Chaos, Solitons & Fractals, 61 (2014), 13–23.
  • 8. A. L. Wang, Y. N. Xiao and R. Smith, Using non-smooth models to determine thresholds for microbial pest management, J. Math. Biol., 78 (2019), 1389–1424.
  • 9. M. I. D. S. Costa and L. D. B. Faria, Integrated pest management: theoretical insights from a threshold policy, Neotrop. Entomol., 39 (2010), 1–4.
  • 10. L. M. Wang, L. S. Chen and J. J. Nieto, The dynamics of an epidemic model for pest control with impulsive effect, Nonlinear Anal-Real, 3 (2010), 1374–1386.
  • 11. R. Mu, A. R. Wei and Y. P. Yang, Global dynamics and sliding motion in A(H7N9) epidemic models with limited resources and Filippov control, J. Math. Anal. Appl., 477 (2019), 1296–1317.
  • 12. M. Biák, T. Hanus and D. Janovská, Some applications of Filippov's dynamical systems, J. Comput. Appl. Math., (2013), 132–143.
  • 13. F. Dercole, A. Gragnani, Y. A. Kuznetsov, et al., Numerical sliding bifurcation analysis: an application to a relay control system, IEEE T. Circuits-I, 50 (2003), 1058–1063.
  • 14. F. Dercole and Y. A. Kuznetsov, SlideCont: An auto97 driver for bifurcation analysis of Filippov systems, Acm T. Math. Software, 31 (2005), 95–119.
  • 15. Y. X. Chen and L. H. Huang, A Filippov system describing the effect of prey refuge use on a ratio-dependent predator-prey model, J. Math. Anal. Appl., 2 (2015), 817–837.
  • 16. J. W. Qin, X. W. Tan, X. T. Shi, et al., Dynamics and bifurcation analysis of a Filippov predator-prey ecosystem in a seasonally fluctuating environment, Int. J. Bifurcat. Chaos, 29 (2019).
  • 17. S. Y. Tang, G. Y. Tang and W. J. Qin, Codimension-1 sliding bifurcations of a Filippov pest growth model with threshold policy, Int. J. Bifurcat. Chaos, 24 (2014), 1450122.
  • 18. S. Y. Tang, J. H. Liang, Y. N. Xiao, et al., Sliding bifurcations of Filippov two stage pest control models with economic thresholds, SIAM J. Appl. Math., 72 (2012), 1061–1080.
  • 19. J. H. Frank, M. P. Hoffman and A. C. Frodsham, Natural enemies of vegetable insect pests, Fla. Entomol., 76 (1993), 531.
  • 20. D. J. Greathead, Natural enemies of tropical locusts and grasshoppers: their impact and potential as biological control agents, Biological Control of Locusts & Grasshoppers, (1992), 105–121.
  • 21. J. C. Van Lenteren and J. Woets, Biological and integrated pest control in greenhouses, Ann. Rev. Entomol., 33 (1988), 239–269.
  • 22. J. C. Van Lenteren, Measures of success in biological control of arthropods by augmentation of natural enemies, Measures of Success in Biological Control, 3 (2000), 77–89.
  • 23. Y. A. Kuznetsov, S. Rinalai and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Int. J. Bifurcat. Chaos, 13 (2003), 2157–2188.
  • 24. F. Dercole, A. Gragnani and S. Rinaldi, Bifurcation analysis of piecewise smooth ecological models, Theor. Popul. Biol., 72 (2007), 197–213.
  • 25. K. Gupta and S. Gakkhar, The Filippov approach for predator-prey system involving mixed type of functional responses, Differ. Eq. Dyn. Syst., (2016), 1–21.
  • 26. M. Antali and G. Stepan, Sliding and crossing dynamics in extended Filippov systems, SIAM J. Appl. Dyn. Syst., 17 (2018), 823–858.
  • 27. R. Cristiano, T. Carvalho, D. J. Tonon, et al., Hopf and homoclinic bifurcations on the sliding vector field of switching systems in R 3 : A case study in power electronics, Physica D, 347 (2017), 12–20.
  • 28. L. S. Chen and Z. J. Jing, The existence and uniqueness of the limit cycle of the differential equations in the predator-prey system, Chinese Sci. Bull., 7 (1986), 73–80.
  • 29. M. A. Aziz-Alaoui and M. D. Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Appl. Math. Lett., 16 (2003), 1069–1075.
  • 30. Z. F. Zhang, T. R. Ding, W. Z. Huang, et al., Qualitative Theory of Differential Equations, Science Press, Beijing, 1985.
  • 31. D. M. Xiao and S. G. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, Fields Institute Communications, 21 (1999), 493–506.
  • 32. A. F. Filippov, Differential equations with discontinuous righthand sides, Kulwer Academic Publishers, Dordrecht, The Netherlands, 1988.
  • 33. S. Y. Tang and Y. N. Xiao, Biological Dynamics of Single Species, Science Press, Beijing, 2008.
  • 34. Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edition, Springer, New York, 1998.
  • 35. L. Perko, Differential Equations and Dynamical Systems 3rd edition, Springer, New York, 1996.
  • 36. H. J. Cai, Y. J. Gong and S. G. Ruan, Bifurcation analysis in a predator-prey model with constant- yield predator harvesting, Discrete Conti. Dyn- (DCDS-B), 18 (2014), 2101–2121.
  • 37. S. R. Lubkin, Simulating, analyzing, and animating dynamical systems: A guide to XPPAUT for researchers and students by Bard Ermentrout, SIAM Rev., 45 (2003), 150–152.
  • 38. J. H. Liang, S. Y. Tang, J. J. Nieto, et al., Analytical methods for detecting pesticide switches with evolution of pesticide resistance, Math. Biosci., 245 (2013), 249–257.
  • 39. J. H. Liang, S. Y. Tang, R. A. Cheke, et al., Adaptive Release of Natural Enemies in a Pest-Natural Enemy System with Pesticide Resistance, B. Math. Biol., 75 (2013), 2167–2195.
  • 40. S. S. Ge, C. Wang and T. H. Lee, Adaptive backstepping control of a class of chaotic systems, Int. J. Bifurcat. Chaos, 10 (2000), 1149–1156.
  • 41. C. Y. Gong, Y. M. Li and X. H. Sun, Adaptive backstepping control of the biomathematical model of muscular blood vessel, J. Biomath., 22 (2007), 503–508.

 

Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved