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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 280945, 11 pages
http://dx.doi.org/10.1155/2013/280945
Research Article

Filippov Ratio-Dependent Prey-Predator Model with Threshold Policy Control

College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, China

Received 10 June 2013; Revised 23 August 2013; Accepted 2 September 2013

Academic Editor: Hamid Reza Karimi

Copyright © 2013 Xianghong Zhang and Sanyi Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. M. May, “Limit cycles in predator-prey communities,” Science, vol. 177, no. 4052, pp. 900–902, 1972. View at Scopus
  2. Y. Kuang and E. Beretta, “Global qualitative analysis of a ratio-dependent predator-prey system,” Journal of Mathematical Biology, vol. 36, no. 4, pp. 389–406, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. W. Murdoch, C. Briggs, and R. Nisbet, Consumer-Resource Dynamics, Princeton University Press, New York, NY, USA, 2003.
  4. H. I. Freedman, Deterministic Mathematical Models in Population Ecology, vol. 57 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980. View at MathSciNet
  5. G. T. Skalski and J. F. Gilliam, “Functional responses with predator interference: viable alternatives to the Holling type II model,” Ecology, vol. 82, no. 11, pp. 3083–3092, 2001. View at Scopus
  6. R. Arditi and L. R. Ginzburg, “Coupling in predator-prey dynamics: ratio-dependence,” Journal of Theoretical Biology, vol. 139, no. 3, pp. 311–326, 1989. View at Scopus
  7. S.-B. Hsu, T.-W. Hwang, and Y. Kuang, “Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system,” Journal of Mathematical Biology, vol. 42, no. 6, pp. 489–506, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. C. Jost, O. Arino, and R. Arditi, “About deterministic extinction in ratio-dependent predator-prey models,” Bulletin of Mathematical Biology, vol. 61, no. 1, pp. 19–32, 1999. View at Publisher · View at Google Scholar · View at Scopus
  9. D. Xiao and S. Ruan, “Global dynamics of a ratio-dependent predator-prey system,” Journal of Mathematical Biology, vol. 43, no. 3, pp. 268–290, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. Z. Lu, X. Chi, and L. Chen, “Impulsive control strategies in biological control of pesticide,” Theoretical Population Biology, vol. 64, no. 1, pp. 39–47, 2003. View at Publisher · View at Google Scholar · View at Scopus
  11. S. Tang and R. A. Cheke, “Models for integrated pest control and their biological implications,” Mathematical Biosciences, vol. 215, no. 1, pp. 115–125, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. C. Van Lenteren and J. Woets, “Biological and integrated pest control in greenhouses,” Annual Review of Entomology, vol. 33, pp. 239–250, 1988.
  13. B. Dai, H. Su, and D. Hu, “Periodic solution of a delayed ratio-dependent predator-prey model with monotonic functional response and impulse,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 126–134, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. X. Liu, G. Li, and G. Luo, “Positive periodic solution for a two-species ratio-dependent predator-prey system with time delay and impulse,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 715–723, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. G. Jiang, Q. Lu, and L. Qian, “Complex dynamics of a Holling type II prey-predator system with state feedback control,” Chaos, Solitons & Fractals, vol. 31, no. 2, pp. 448–461, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. H. Baek and Y. Lim, “Dynamics of an impulsively controlled Michaelis-Menten type predator-prey system,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 4, pp. 2041–2053, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. T. Zhao, Y. Xiao, and R. J. Smith, “Non-smooth plant disease models with economic thresholds,” Mathematical Biosciences, vol. 241, no. 1, pp. 34–48, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. V. I. Utkin, Sliding Modes in Control and Optimization, Communications and Control Engineering Series, Springer, Berlin, Germany, 1992. View at MathSciNet
  19. M. I. S. Costa, E. Kaszkurewicz, A. Bhaya, and L. Hsu, “Achieving global convergence to an equilibrium population in predator-prey systems by the use of a discontinuous harvesting policy,” Ecological Modelling, vol. 128, no. 2-3, pp. 89–99, 2000. View at Publisher · View at Google Scholar · View at Scopus
  20. B. L. Van De Vrande, D. H. Van Campen, and A. De Kraker, “Approximate analysis of dry-friction-induced stick-slip vibrations by a smoothing procedure,” Nonlinear Dynamics, vol. 19, no. 2, pp. 157–169, 1999. View at Scopus
  21. S. H. Doole and S. J. Hogan, “A piecewise linear suspension bridge model: nonlinear dynamics and orbit continuation,” Dynamics and Stability of Systems, vol. 11, no. 1, pp. 19–47, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. Yu. A. Kuznetsov, S. Rinaldi, and A. Gragnani, “One-parameter bifurcations in planar Filippov systems,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 13, no. 8, pp. 2157–2188, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. M. di Bernardo, C. J. Budd, A. R. Champneys et al., “Bifurcations in nonsmooth dynamical systems,” SIAM Review, vol. 50, no. 4, pp. 629–701, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, vol. 18 of Mathematics and Its Applications (Soviet Series), Kluwer Academic, Dordrecht, The Netherlands, 1988. View at MathSciNet
  25. S. Tang, J. Liang, Y. Xiao, and R. A. Cheke, “Sliding bifurcations of Filippov two stage pest control models with economic thresholds,” SIAM Journal on Applied Mathematics, vol. 72, no. 4, pp. 1061–1080, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. M. I. da Silveira Costa and M. E. M. Meza, “Application of a threshold policy in the management of multispecies fisheries and predator culling,” Mathematical Medicine and Biology, vol. 23, no. 1, pp. 63–75, 2006. View at Publisher · View at Google Scholar · View at Scopus
  27. M. E. Mendoza Meza, A. Bhaya, E. Kaszkurewicz, and M. I. Da Silveira Costa, “Threshold policies control for predator-prey systems using a control Liapunov function approach,” Theoretical Population Biology, vol. 67, no. 4, pp. 273–284, 2005. View at Publisher · View at Google Scholar · View at Scopus
  28. F. Dercole, A. Gragnani, and S. Rinaldi, “Bifurcation analysis of piecewise smooth ecological models,” Theoretical Population Biology, vol. 72, no. 2, pp. 197–213, 2007. View at Publisher · View at Google Scholar · View at Scopus
  29. W. Wang, “Backward bifurcation of an epidemic model with treatment,” Mathematical Biosciences, vol. 201, no. 1-2, pp. 58–71, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. I. Noy-Meir, “Stability of grazing systems: an application of predator-prey graphs,” The Journal of Animal Ecology, vol. 63, no. 2, pp. 459–481, 1975.
  31. R. M. Colombo and V. Křivan, “Selective strategies in food webs,” Mathematical Medicine and Biology, vol. 10, no. 4, pp. 281–291, 1993. View at Publisher · View at Google Scholar · View at Scopus
  32. V. Křivan, “Optimal foraging and predator-prey dynamics,” Theoretical Population Biology, vol. 49, no. 3, pp. 265–290, 1996. View at Publisher · View at Google Scholar · View at Scopus