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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 280945, 11 pages
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.
- R. M. May, “Limit cycles in predator-prey communities,” Science, vol. 177, no. 4052, pp. 900–902, 1972.
- 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.
- W. Murdoch, C. Briggs, and R. Nisbet, Consumer-Resource Dynamics, Princeton University Press, New York, NY, USA, 2003.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- J. C. Van Lenteren and J. Woets, “Biological and integrated pest control in greenhouses,” Annual Review of Entomology, vol. 33, pp. 239–250, 1988.
- 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.
- 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.
- 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.
- 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.
- 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.
- V. I. Utkin, Sliding Modes in Control and Optimization, Communications and Control Engineering Series, Springer, Berlin, Germany, 1992.
- 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.
- 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.
- 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.
- 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.
- 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.
- A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, vol. 18 of Mathematics and Its Applications (Soviet Series), Kluwer Academic, Dordrecht, The Netherlands, 1988.
- 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.
- 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.
- 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.
- 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.
- W. Wang, “Backward bifurcation of an epidemic model with treatment,” Mathematical Biosciences, vol. 201, no. 1-2, pp. 58–71, 2006.
- 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.
- R. M. Colombo and V. Křivan, “Selective strategies in food webs,” Mathematical Medicine and Biology, vol. 10, no. 4, pp. 281–291, 1993.
- V. Křivan, “Optimal foraging and predator-prey dynamics,” Theoretical Population Biology, vol. 49, no. 3, pp. 265–290, 1996.