×

Analysis of stability for a discrete ratio-dependent predator-prey system. (English) Zbl 1308.39011

Summary: The paper investigated the dynamical behaviors of a two-species discrete ratio-dependent predator-prey system. The local stability of the equilibria is obtained by using the linearization method. Further, a new sufficient condition on the global asymptotic stability of the positive equilibrium is established by using an iteration scheme and the comparison principle of difference equations.

MSC:

39A20 Multiplicative and other generalized difference equations
92D25 Population dynamics (general)
39A30 Stability theory for difference equations
65Q10 Numerical methods for difference equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] N. A. Agiza, E. M. Elabbasy, H. EL-Metwally and A.A. Elsadany, Chaotic dynamics of a discrete pery-predator model with Holling type II, Nonlinear Anal., RWA, 10 (2009), 116–129. · Zbl 1154.37335 · doi:10.1016/j.nonrwa.2007.08.029
[2] M. Basson and M. J. Fogarty, Harvesting in discrete-time predator-prey systems, Math. Biosci., 141 (1997), 41–74. · Zbl 0880.92034 · doi:10.1016/S0025-5564(96)00173-3
[3] C. Celik and O. Duman, Allee effect in a discrete-time predator-prey system, Chaos Solit. Frac., 40 (2009), 1956–1962. · Zbl 1198.34084 · doi:10.1016/j.chaos.2007.09.077
[4] L. Chen, Mathematical Models and Methods in Ecology, Science Press, Beijing (1988). (in Chinese).
[5] X. Chen, Periodicity in a nonlinear discrete predator-prey system with state dependent delays, Nonlinear Anal., RWA, 8 (2007), 435–446. · Zbl 1152.34367 · doi:10.1016/j.nonrwa.2005.12.005
[6] Y. H. Fan and W. T. Li, Permanence for a delayed discrete ratio-dependent predatorprey system with Holling type functional response, J. Math. Anal. Appl., 299 (2004), 357–374. · Zbl 1063.39013 · doi:10.1016/j.jmaa.2004.02.061
[7] M. Fan and K. Wang, Periodic solutions of a discrete time nonautonomous ratiodependent predator-prey system, Math. Comput. Model., 35 (2002), 951–961. · Zbl 1050.39022 · doi:10.1016/S0895-7177(02)00062-6
[8] M. Fazly and M. Hesaaraki, Periodic solutions for discrete time predator-prey system with monotone functional responses, C. R. Acad. Sci. Paris, Ser. I, 345 (2007), 199–202. · Zbl 1122.39005 · doi:10.1016/j.crma.2007.06.021
[9] Z. He and X. Lai, Bifurcation and chaotic behavior of a discrete-time predator-prey system, Nonlinear Anal.: RWA, 12 (2011), 403–417. · Zbl 1202.93038 · doi:10.1016/j.nonrwa.2010.06.026
[10] D. Hu and Z. Zhang, Four positive periodic solutions of a discrete time delayed predator-prey system with nonmonotonic functional response and harvesting, Comp. Math. Appl., 56 (2008), 3015–3022. · Zbl 1165.34400 · doi:10.1016/j.camwa.2008.09.009
[11] H. F. Huo and W. T. Li, Stable periodic solution of the discrete periodic Leslie-Gower predator-prey model, Math. Comput. Model., 40 (2004), 261–269. · Zbl 1067.39008 · doi:10.1016/j.mcm.2004.02.026
[12] H. F. Huo and W. T. Li, Existence and global stability of periodic solutions of a discrete predator-prey system with delays, Appl. Math. Comput., 153 (2004), 337–351 · Zbl 1043.92038 · doi:10.1016/S0096-3003(03)00635-0
[13] V. Hutson and W. Moran, Persistence of species obeying difference equations, J. Math. Biol., 15 (1982), 203–213. · Zbl 0495.92015 · doi:10.1007/BF00275073
[14] V. Hutson and K. Mischaikow, An approach to practical persistence, J. Math. Biol., 37 (1998), 447–466. · Zbl 0927.92030 · doi:10.1007/s002850050137
[15] Z. Jing and J. Yang, Bifurcation and chaos in discrete-time predator-prey system, Chaos Solit. Frac., 27 (2006), 259–277. · Zbl 1085.92045 · doi:10.1016/j.chaos.2005.03.040
[16] R. Kon, Permanence of discrete-time Kolmogorov systems for two soecies and saturated fixed points, J. Math. Biol., 48 (2004), 57–81. · Zbl 1050.92045 · doi:10.1007/s00285-003-0224-8
[17] X. Liao, S. Zhou and Y. Chen, On permanence and global stability in a general Gilpin-Ayala competition predator-prey discrete system, Appl. Math. Comput., 190 (2007), 500–509. · Zbl 1125.39008 · doi:10.1016/j.amc.2007.01.041
[18] X. Liu, A note on the existence of periodic solution in discrete predator-prey models, Appl. Math. Model., 34 (2010), 2477–2483. · Zbl 1195.39004 · doi:10.1016/j.apm.2009.11.012
[19] X. Liu and D. Xiao, Complex dynamic behaviors of a discrete-time predator-prey system, Chaos Solit. Frac., 32 (2007), 80–94. · Zbl 1130.92056 · doi:10.1016/j.chaos.2005.10.081
[20] Y. Saito, T. Hara and W. Ma, Harmless delays for permanence and impersistence of a Lotka-Volterra discrete predator-prey system, Nonlinear Anal., 50 (2002), 703–715. · Zbl 1005.39013 · doi:10.1016/S0362-546X(01)00778-7
[21] F. J. Solis, Self-limitation in a discrete predator-prey model, Math. Comput. Model., 48 (2008), 191–196. · Zbl 1145.92342 · doi:10.1016/j.mcm.2007.09.006
[22] Z. Teng, Y. Zhang and S. Gao, Permanence criteria for general delayed discrete nonautonomous n-species Kolmogorov systems and its applications, Comp. Math. Appl., 59 (2010), 812–828. · Zbl 1189.39017 · doi:10.1016/j.camwa.2009.10.011
[23] L. Wang and M. Wang, Ordinary Difference Equation, Xinjiang Univ. Press, Xinjiang (1989). (in Chinese) · Zbl 0732.65076
[24] R. Willox, A. Ramani and B. Grammaticos, A discrete-time model for cryptic oscillations in predator-prey systems, Physica D, 238 (2009), 2238–2245. · Zbl 1186.37107 · doi:10.1016/j.physd.2009.09.007
[25] Y. Xia, J. Cao and M. Lin, Discrete-time analogues of predator-prey models with monotonic or nonmonotonic functional responses, Nonlinear Anal.: RWA, 8 (2007), 1079–1095. · Zbl 1127.39038 · doi:10.1016/j.nonrwa.2006.06.007
[26] A. A. Yakubu, Prey dominance in discrete predator-prey systems with a prey refuge, Math. Biosci., 144 (1997), 155–178. · Zbl 0896.92031 · doi:10.1016/S0025-5564(97)00026-6
[27] X. Yang, Uniform persistence and periodic solutions for a discrete predator-prey system with delays, J. Math. Anal. Appl., 316 (2006), 161–177. · Zbl 1107.39017 · doi:10.1016/j.jmaa.2005.04.036
[28] W. Yang and X. Li, Permanence for a delayed discrete ratio-dependent predator-prey model with monotonic functional responses, Nonlinear Anal.: RWA, 10 (2009), 1068–1072. · Zbl 1167.34359 · doi:10.1016/j.nonrwa.2007.11.022
[29] J. Zhang and J. Wang, Periodic solutions for discrete predator-prey systems with the Beddington-DeAngelis functional response, Appl. Math. Letters, 19 (2006), 1361–1366. · Zbl 1140.92325 · doi:10.1016/j.aml.2006.02.004
[30] W. Zhang, D. Zhu and P. Bi, Multiple positive periodic solutions of a delayed discrete predator-prey system with type IV functional responses, Appl. Math. Letters, 20 (2007), 1031–1038. · Zbl 1142.39015 · doi:10.1016/j.aml.2006.11.005
[31] X. Zhang, Q. Zhang and V. Sreeram, Bifurcation analysis and control of a discrete harvested prey-predator system with Beddington-DeAngelis functional respone, J. Frank. Inst., 347 (2010), 1076–1096. · Zbl 1210.92062 · doi:10.1016/j.jfranklin.2010.03.016
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.