×

Rich dynamic of a stage-structured prey-predator model with cannibalism and periodic attacking rate. (English) Zbl 1222.37105

Summary: The dynamic behavior of a stage-structure prey–predator model with cannibalism for prey and periodic attacking rate for predator is investigated. Firstly, the permanence, locally and globally asymptotic stability analyses of the model with constant attacking rate are explored. After that, sufficient conditions for the permanence of the corresponding nonautonomous system with periodic attacking rate are obtained. Furthermore, numerical simulations are presented to illustrate the effects of periodic attacking rate. Simulation results show that the system with periodic attacking rate shows a rich behaviors, including period-doubling and period-having bifurcations, chaos and windows of periodicity.

MSC:

37N25 Dynamical systems in biology
92D25 Population dynamics (general)
34D05 Asymptotic properties of solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Cui, J.; Takeuchi, Y., A predator-prey system with a stage structure for the prey, Math Comp Model, 44, 1126-1132 (2006) · Zbl 1132.92340
[2] Yang, W. S.; Li, X. P.; Bai, Z. J., Permanence of periodic Holling type-IV predator-prey system with stage structure for prey, Math Comput Model, 48, 677-684 (2008) · Zbl 1156.34327
[3] Chen, F. D.; You, M. S., Permanence, extinction and periodic solution of the predator-prey system with Beddington-DeAngelis functional response and stage structure for prey, Nonlinear Anal RWA, 9, 207-221 (2008) · Zbl 1142.34051
[4] Liu, S. Q.; Zhang, J. H., Coexistence and stability of predator-prey model with Beddington-DeAngelis functional response and stage structure, J Math Anal Appl, 342, 446-460 (2008) · Zbl 1146.34057
[5] Huang, C. Y.; Zhao, M.; Zhao, L. C., Permanence of periodic predator-prey system with two predators and stage structure for prey, Nonliner Anal RWA (2009)
[6] Zhang, L. M., Stage-structured predator-prey system with efficiency of resources utilization, Sci Tech Engine, 8, 6082-6084 (2008), [in Chinese]
[7] Wang, F. Y.; Pang, G. P., The global stability of a delayed predator-prey system with two stage-structure, Chaos, Solitons Fractals, 40, 778-785 (2009) · Zbl 1197.34100
[8] Gao, S. J.; Chen, L. S.; Teng, Z. D., Hopf bifurcation and global stability for a delayed predator-prey system with stage structure for predator, Appl Math Comput, 202, 721-729 (2008) · Zbl 1151.34067
[9] Xu, R.; Chaplain, M. A.J.; Davidson, F. A., Persistence and global stability of a ratio-dependent predator-prey model with stage structure, Appl Math Comput, 158, 729-744 (2004) · Zbl 1058.92053
[10] Fessehaye, Y.; Kabir, A.; Bovenhuis, H.; Komen, H., Prediction of cannibalism in juvenile Oreochromis niloticus based on predaror to prey weight ratio, and effects of age and stocking density, Aquac, 255, 314-322 (2006)
[11] Smith, C.; Reay, P., Cannibalism in teleost fish, Rev Fish Biol Fish, 1, 41-64 (1991)
[12] Smith, C.; Wootton, R. J., The costs of parental care in teleost fishes, Rev Fish Biol Fish, 5, 7-22 (1995)
[13] Rickers, S.; Schen, s., Cannibalism in Paradosa palustris (Araneae, Lycosidae): effects of alternative prey, habitat structure, and density, Basic Appl Ecol, 6, 471-478 (2005)
[14] Anthony, C. D., Kinship influences cannibalism in the wolf spider, Pardosa milvina, J Insect Behav, 16, 23-36 (2003)
[15] Claessen, D.; De, R.; Persson, Am, Population dynamic theory of size-dependent cannibalism, Proc R Soc, 271, 333-340 (2004)
[16] Armsby, M.; Tisch, N., Intraguild predation and cannibalism in a size-structured community of marine amphipods, J Exp Mar Biol Ecol, 333, 286-295 (2006)
[17] Magnusson, K. G., Destabilizing effect of cannibalism on a structured predator-prey system, Math Biosci, 155, 61-75 (1999) · Zbl 0943.92030
[18] Kaewmanee, C.; Tang, I. M., Cannibalism in an age-structured predator-prey system, Ecol Mod, 167, 220-231 (2003)
[19] Wu, H.; Cheng, X.; Zou, Y., Predation on Myzus persicae by propylaea japonica adults with different extents of starvation, Chin J Appl Ecol, 11, 749-752 (2000)
[20] Zou, Y.; Chen, G.; Meng, Q.; Wang, G.; Geng, J., A study on predation of Coccinella septempunctata on different ranges of starvation, Chin Acta Ecol Sin, 19, 143-147 (1999)
[21] Gopalsamy, K., Stability and oscillation in delay differential equations of population dynamics (1992), Kluwer Academic: Kluwer Academic Dordrecht/Norwell, MA · Zbl 0752.34039
[22] Gakkhar, S.; Singh, B., Dynamics of modified Leslie-Gower-type prey-predator model with seasonally varying parameters, Chaos, Solitons Fractals, 27, 1239-1255 (2006) · Zbl 1094.92059
[23] Gakkhar, S.; Negi, K.; Sahani, S. K., Effects of seasonal growth on ratio dependent delayed prey-predator system, Commun Nonlinear Sci Numer Simul, 14, 850-862 (2009) · Zbl 1221.34187
[24] Gakkhar, S.; Sahani, S. K.; Negi, K., Effects of seasonal growth on delayed prey-predator model, Chaos, Solitons Fractals, 39, 230-239 (2009) · Zbl 1197.34134
[25] Gao, M.; Shi, H. H.; Li, Z. Z., Chaos in a seasonally and periodically forced phytoplankton-zooplankton system, Nonlinear Anal RWA, 10, 1643-1650 (2009) · Zbl 1160.92340
[26] Fan, X. M.; Wang, Z. G.; Luo, Z. J., Persistence and periodic solutions for nonautonomous predator-prey system with functional responses and invest rate, Math Pract Theory, 38, 123-130 (2008), [in Chinese] · Zbl 1175.92062
[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] Yang, Z. H., Nonlinear Bifurcation: Theory and Computation (2007), Science: Science Bei Jing, p. 180-1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.