Chaos in a three-species food chain model with a Beddington-DeAngelis functional response. (English) Zbl 1198.37139

Summary: This paper investigates a three-species food chain model with a Beddington-DeAngelis functional response, both analytically and through numerical simulations. First the equilibrium states of the system are identified and their stability analyzed analytically. The results of simulations demonstrate chaotic long-term behavior over a broad range of parameters. The existence of a strange attractor and computation of the largest Lyapunov exponent also demonstrate the presence of chaotic dynamics in the model.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.


37N25 Dynamical systems in biology
92D25 Population dynamics (general)
Full Text: DOI


[1] Kuang, Y.; Beretta, E., Global quantities analysis of a ratio-dependent predator-prey system, J Math Biol, 36, 389-406 (1998) · Zbl 0895.92032
[2] Tang, S. Y.; Chen, L. S., Chaos in functional response host-parasitoid ecosystem models, Chaos Solitons & Fractals, 13, 875-884 (2002) · Zbl 1022.92042
[3] Lv, S.; Zhao, M., The dynamic complexity of a host-parasitoid model with a lower bound for the host, Chaos Solitons & Fractals, 36, 911-919 (2008)
[4] Hui, J.; Zhu, D., Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects, Chaos Solitons & Fractals, 29, 233-251 (2006) · Zbl 1095.92067
[5] Sun, M.; Tian, L.; Fu, Y.; Qian, W., Dynamics and adaptive synchronization of the energy resource system, Chaos Solitons & Fractals, 31, 879-888 (2007) · Zbl 1149.34032
[6] Letellier, C.; Aziz-Alaoui, M. A., Analysis of the dynamics of a realistic ecological model, Chaos Solitons & Fractals, 13, 95-107 (2002) · Zbl 0977.92029
[7] Sheu, L. J.; Chen, H. K.; CHen, J. H.; Tam, L. M., Chaos in a new system with fractional order, Chaos Solitons & Fractals, 312, 1203-1212 (2007)
[8] Hastings, A.; Powell, T., Chaos in a three-species food chain, Ecology, 72, 896-903 (1991)
[9] Gakkhar, S.; Naji, R. K., Order and chaos in predator to prey ratio-dependent food chain, Chaos Solitons & Fractals, 18, 229-239 (2003) · Zbl 1068.92044
[10] Gakkhar, S.; Naji, R. K., Chaos in three species ratio dependent food chain, Chaos Solitons & Fractals, 14, 771-778 (2002) · Zbl 0994.92037
[11] Sabin, G. C.W.; Summer, D., Chaos in a periodically forced predator-prey ecosystem model, Math Biosci, 113, 91-113 (1993) · Zbl 0767.92028
[12] Vandermmeer, J.; Maruca, S., Indirect effects with a keystone predator: coexistence and chaos, Theor Popul Biol, 54, 38-43 (1998) · Zbl 0941.92033
[13] Hui, J.; Chen, L., Dynamic complexities in a periodically pulsed ratio-dependent predator-prey ecosystem modeled on a chemostat, Chaos Solitons & Fractals, 29, 407-416 (2006) · Zbl 1095.92066
[14] Rai, V.; Kumar Upadhyay, R., Chaotic population dynamics and biology of the top-predator, Chaos Solitons & Fractals, 21, 1195-1204 (2004) · Zbl 1057.92056
[15] Peet, A. B.; Deutsch, P. A.; Peacock-Lopez, E., Complex dynamics in a three-level trophic system with intraspecies interaction, J Theor Biol, 232, 491-503 (2005) · Zbl 1442.92195
[16] Kumar Upadhyay, R.; Iyengar, S. R.K., Effect of seasonality on the dynamics of 2 and 3 species prey-predator systems, Nonlinear Anal Real World Appl, 6, 509-530 (2005) · Zbl 1072.92058
[17] Lv, S.; Zhao, M., The dynamic complexity of a three species food chain model, Chaos Solitons & Fractals, 37, 1469-1480 (2008) · Zbl 1142.92342
[18] Li, Z.; Wang, W.; Wang, H., The dynamics of a Beddington-type system with impulsive control strategy, Chaos Solitons & Fractals, 29, 1229-1239 (2006) · Zbl 1142.34305
[19] Cantrell, R. S.; Consner, C., On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, J Math Anal Appl, 257, 206-222 (2001) · Zbl 0991.34046
[20] Gakkhar, S.; Naji, R. K., Seasonality perturbed prey-predator system with predator-dependent functional response, Chaos Solitons & Fractals, 18, 1075-1083 (2003) · Zbl 1068.92045
[21] Naji, R. K.; Balasim, A. T., Dynamical behavior of a three species food chain model with Beddington-DeAngelis functional response, Chaos Solitons & Fractals, 32, 1853-1866 (2007) · Zbl 1195.92061
[22] Hwang, T. W., Global analysis of the predator-prey system with Beddington-DeAngelis functional response, J Math Anal Appl, 281, 395-401 (2003) · Zbl 1033.34052
[23] Hwang, T. W., Uniqueness of limit cycles of the predator-prey system with the Beddington-DeAngelis functional response, J Math Anal Appl, 290, 113-122 (2004) · Zbl 1086.34028
[24] Zhang, S.; Tan, D.; Chen, L., Chaotic behavior of a chemostat model with Beddington-DeAngelis functional response and periodically impulsive invasion, Chaos Solitons & Fractals, 29, 474-482 (2006) · Zbl 1121.92070
[25] Beddington, J. R., Mutual interference between parasites or predators and its effect on searching efficiency, J Animal Ecol, 44, 331-340 (1975)
[26] Qiwu, C.; Lawson, G. J., Study on models of single populations: an expansion of the logistic and exponential equations, J Theor Biol, 98, 645-659 (1982)
[27] Rosenstein, M. T.; Collins, J. J.; De Luca, C. J., A practical method for calculating largest Lyapunov exponents from small data sets, Physica D, 65, 117-134 (1993) · Zbl 0779.58030
[28] Sprott, J. C., Chaos and time-series analysis (2003), Oxford University Press: Oxford University Press Oxford, p. 116-7
[29] Grond, F.; Diebner, H. H.; Sahle, S.; Mathias, A., A robust, locally interpretable algorithm for Lyapunov exponents, Chaos Solitons & Fractals, 16, 841-852 (2003)
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