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Seasonally perturbed prey-predator ecological system with the Beddington-DeAngelis functional response. (English) Zbl 1233.92081
Summary: On the basis of the theories and methods of ecology and ordinary differential equations, a seasonally perturbed prey-predator system with Beddington-DeAngelis functional response is studied analytically and numerically. Mathematical theoretical works have been pursuing the investigation of uniformly persistence, which depicts the threshold expression of some critical parameters. Numerical analyses indicate that seasonality has a strong effect on the dynamical complexity and species biomass using bifurcation diagrams and Poincaré sections. The results show that the seasonality in three different parameters can give rise to rich and complex dynamical behaviors. In addition, the largest Lyapunov exponents are computed. These computations further confirm the existence of chaotic behavior and the accuracy of the numerical simulations. All these results are expected to be of use in the study of the dynamic complexity of ecosystems.

MSC:
92D40Ecology
34C60Qualitative investigation and simulation of models (ODE)
37N25Dynamical systems in biology
65C20Models (numerical methods)
93A30Mathematical modelling of systems
WorldCat.org
Full Text: DOI
References:
[1] S. Gakkhar and R. K. Naji, “Chaos in seasonally perturbed ratio-dependent prey-predator system,” Chaos, Solitons and Fractals, vol. 15, no. 1, pp. 107-118, 2003. · Zbl 1033.92026 · doi:10.1016/S0960-0779(02)00114-5
[2] R. K. Upadhyay and S. R. K. Iyengar, “Effect of seasonality on the dynamics of 2 and 3 species prey-predator systems,” Nonlinear Analysis, vol. 6, no. 3, pp. 509-530, 2005. · Zbl 1072.92058 · doi:10.1016/j.nonrwa.2004.11.001
[3] S. Rinaldi, S. Muratori, and Y. A. Kuznetsov, “Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey communities,” Bulletin of Mathematical Biology, vol. 55, no. 1, pp. 15-35, 1993. · Zbl 0756.92026 · doi:10.1007/BF02460293
[4] H. Baek, “An impulsive two-prey one-predator system with seasonal effects,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 793732, 19 pages, 2009. · Zbl 1190.34055 · doi:10.1155/2009/793732 · eudml:226386
[5] A. Gragnani and S. Rinaldi, “A universal bifurcation diagram for seasonally perturbed predator-prey models,” Bulletin of Mathematical Biology, vol. 57, no. 5, pp. 701-712, 1995. · Zbl 0824.92027 · doi:10.1007/BF02461847
[6] H. G. Yu, S. M. Zhong, R. P. Agarwal, and S. K. Sen, “Effect of seasonality on the dynamical behavior of an ecological system with impulsive control strategy,” Journal of the Franklin Institute, vol. 348, no. 4, pp. 652-670, 2011. · Zbl 1221.34130 · doi:10.1016/j.jfranklin.2011.01.009
[7] T. S. Parker and L. O. Chua, Practical Numerical Algorithms for Chaotic System, Springer, New York, NY, USA, 1989. · Zbl 0692.58001
[8] W. M. Schaffer, “Order and chaos in ecological systems,” Ecology, vol. 66, no. 1, pp. 93-106, 1985.
[9] R. K. Upadhyay and V. Rai, “Crisis-limited chaotic dynamics in ecological systems,” Chaos, Solitons and Fractals, vol. 12, no. 2, pp. 205-218, 2001. · Zbl 0977.92033 · doi:10.1016/S0960-0779(00)00141-7
[10] S. J. Lv and M. Zhao, “The dynamic complexity of a three species food chain model,” Chaos, Solitons and Fractals, vol. 37, no. 5, pp. 1469-1480, 2008. · Zbl 1142.92342 · doi:10.1016/j.chaos.2006.10.057
[11] S. Gakkhar and R. K. Naji, “Order and chaos in a food web consisting of a predator and two independent preys,” Communications in Nonlinear Science and Numerical Simulation, vol. 10, no. 2, pp. 105-120, 2005. · Zbl 1063.34038 · doi:10.1016/S1007-5704(03)00120-5
[12] F. Y. Wang, C. P. Hao, and L. S. Chen, “Bifurcation and chaos in a Monod-Haldene type food chain chemostat with pulsed input and washout,” Chaos, Solitons and Fractals, vol. 32, no. 1, pp. 181-194, 2007. · Zbl 1130.92058 · doi:10.1016/j.chaos.2005.10.083
[13] Y. K. Xue and X. F. Duan, “The dynamic complexity of a Holling type-IV predator-prey system with stage structure and double delays,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 509871, 19 pages, 2011. · Zbl 1213.37128 · doi:10.1155/2011/509871 · eudml:225144
[14] M. Zhao and S. J. Lv, “Chaos in a three-species food chain model with a Beddington-DeAngelis functional response,” Chaos, Solitons and Fractals, vol. 40, no. 5, pp. 2305-2316, 2009. · Zbl 1198.37139 · doi:10.1016/j.chaos.2007.10.025
[15] S. Gakkhar and R. K. Naji, “Seasonally perturbed prey-predator system with predator-dependent functional response,” Chaos, Solitons and Fractals, vol. 18, no. 5, pp. 1075-1083, 2003. · Zbl 1068.92045 · doi:10.1016/S0960-0779(03)00075-4
[16] T. W. Hwang, “Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response,” Journal of Mathematical Analysis and Applications, vol. 290, no. 1, pp. 113-122, 2004. · Zbl 1086.34028 · doi:10.1016/j.jmaa.2003.09.073
[17] N. Zhang, F. D. Chen, Q. Q. Su, and T. Wu, “Dynamic behaviors of a harvesting Leslie-Gower predator-prey model,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 473949, 14 pages, 2011. · Zbl 1213.37129 · doi:10.1155/2011/473949 · eudml:231594
[18] V. K\vrivan and J. Eisner, “The effect of the Holling type II functional response on apparent competition,” Theoretical Population Biology, vol. 70, no. 4, pp. 421-430, 2006. · Zbl 1112.92063 · doi:10.1016/j.tpb.2006.07.004
[19] X. Li and W. Yang, “Permanence of a discrete predator-prey systems with Beddington-deAngelis functional response and feedback controls,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 149267, 8 pages, 2008. · Zbl 1147.39007 · doi:10.1155/2008/149267 · eudml:129292
[20] R. X. Wu and L. Li, “Permanence and global attractivity of discrete predator-prey system with hassell-varley type functional response,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 323065, 17 pages, 2009. · Zbl 1178.39006 · doi:10.1155/2009/323065 · eudml:226989
[21] L. J. Chen, J. Y. Xu, and Z. Li, “Permanence and global attractivity of a delayed discrete predator-prey system with general holling-type functional response and feedback controls,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 629620, 17 pages, 2008. · Zbl 1161.39017 · doi:10.1155/2008/629620 · eudml:129899
[22] J. R. Beddington, “Mutual interference between parasites or predator and its effect on searching efficiency,” Journal of Animal Ecology, vol. 44, no. 1, pp. 331-340, 1975.
[23] D. L. DeAngelis, R. A. Goldstein, and R. V. Neill, “A model for trophic interaction,” Ecology, vol. 56, no. 4, pp. 881-892, 1975.
[24] H. Baek, “Species extinction and permanence of an impulsively controlled two-prey one-predator system with seasonal effects,” BioSystems, vol. 98, no. 1, pp. 7-18, 2009. · doi:10.1016/j.biosystems.2009.06.008
[25] S. Gakkhar and R. K. Naji, “Seasonally perturbed prey-predator system with predator-dependent functional response,” Chaos, Solitons and Fractals, vol. 18, no. 5, pp. 1075-1083, 2003. · Zbl 1068.92045 · doi:10.1016/S0960-0779(03)00075-4
[26] K. Wang, “Permanence and global asymptotical stability of a predatorprey model with mutual interference,” Nonlinear Analysis: Real World Applications, vol. 12, no. 2, pp. 1062-1071, 2011. · Zbl 1213.34067 · doi:10.1016/j.nonrwa.2010.08.028
[27] F. D. Chen, “On a nonlinear nonautonomous predator-prey model with diffusion and distributed delay,” Journal of Computational and Applied Mathematics, vol. 180, no. 1, pp. 33-49, 2005. · Zbl 1061.92058 · doi:10.1016/j.cam.2004.10.001
[28] R. Barrio, “Sensitivity tools vs. Poincaré sections,” Chaos, Solitons and Fractals, vol. 25, no. 3, pp. 711-726, 2005. · Zbl 1092.37531 · doi:10.1016/j.chaos.2004.11.092
[29] C. Masoller, A. C. S. Schifino, and L. Romanelli, “Characterization of strange attractors of lorenz model of general circulation of the atmosphere,” Chaos, Solitons and Fractals, vol. 6, pp. 357-366, 1995. · Zbl 0905.58023 · doi:10.1016/0960-0779(95)80041-E
[30] 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, pp. 4197-4203, 2009. · doi:10.1016/j.cnsns.2009.02.014
[31] M. Hagmüller and G. Kubin, “Poincaré pitch marks,” Speech Communication, vol. 48, no. 12, pp. 1650-1665, 2006. · doi:10.1016/j.specom.2006.07.008
[32] H. G. Yu, M. Zhao, S. J. Lv, and L. L. Zhu, “Dynamic complexities in a parasitoid-host-parasitoid ecological model,” Chaos, Solitons and Fractals, vol. 39, no. 1, pp. 39-48, 2009. · Zbl 1197.37127 · doi:10.1016/j.chaos.2007.01.149
[33] J. G. Sportt, Chaos and Time-Series Analysis, Oxford University Press, New York, NY, USA, 2003.
[34] 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, pp. 117-134, 1993. · Zbl 0779.58030 · doi:10.1016/0167-2789(93)90009-P
[35] S. J. Lv and M. Zhao, “The dynamic complexity of a host-parasitoid model with a lower bound for the host,” Chaos, Solitons and Fractals, vol. 36, no. 4, pp. 911-919, 2008. · doi:10.1016/j.chaos.2006.07.020
[36] Q. Jia, “Hyperchaos generated from the Lorenz chaotic system and its control,” Physics Letters, Section A, vol. 366, no. 3, pp. 217-222, 2007. · Zbl 1203.93086 · doi:10.1016/j.physleta.2007.02.024
[37] 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, pp. 455-462, 2011. · Zbl 1221.37208 · doi:10.1016/j.cnsns.2010.03.011
[38] 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 and Fractals, vol. 42, no. 2, pp. 1245-1250, 2009. · doi:10.1016/j.chaos.2009.03.027
[39] F. Grond, H. H. Diebner, S. Sahle, A. Mathias, S. Fischer, and O. E. Rossler, “A robust, locally interpretable algorithm for Lyapunov exponents,” Chaos, Solitons and Fractals, vol. 16, no. 5, pp. 841-852, 2003. · doi:10.1016/S0960-0779(02)00479-4
[40] L. L. Zhu and M. Zhao, “Dynamic complexity of a host-parasitoid ecological model with the Hassell growth function for the host,” Chaos, Solitons and Fractals, vol. 39, no. 1, pp. 1259-1269, 2009. · Zbl 1197.37133 · doi:10.1016/j.chaos.2007.10.023