Dynamic behaviors of a nonautonomous modified Leslie-Gower predator-prey model with Holling-type III schemes and a prey refuge

Volume 17, Issue 2, pp 266-277 http://dx.doi.org/10.22436/jmcs.017.02.08

Publication Date: June 15, 2017 Submission Date: May 13, 2016

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2106 Downloads
• 3674 Views

Authors

Fengde Chen - College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian, 350002, P. R. China. Qiaoxia Lin - College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian, 350002, P. R. China. Xiangdong Xie - Department of Mathematics, Ningde Normal University, Ningde, Fujian, 352300, P. R. China. Yalong Xue - Department of Mathematics, Ningde Normal University, Ningde, Fujian, 352300, P. R. China.

Abstract

A nonautonomous modified Leslie-Gower predator-prey model with Holling-type III schemes and a prey refuge is proposed and studied in this paper. Sufficient conditions which guarantee the permanence and global stability of the system are obtained, respectively. Our results indicate that the prey refuge has no influence on the persistent property of the system, while it has positive effect on the stability property of the system. Numeric simulations show the feasibility of the main results.

Share and Cite

Keywords

• Predator
• prey
• permanence
• global stability.

MSC

•  34C25
•  34D23
•  92D25
•  34D20
•  34D40

References

• [1] P. Aguirre, E. González-Olivares, E. Sáez, Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, Nonlinear Anal. Real World Appl., 10 (2009), 1401–1416.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] M. A. Aziz-Alaoui, M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie- Gower and Holling-type II schemes, Appl. Math. Lett., 16 (2003), 1069–1075.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] I. Barbălat, Systémes d’équations diffrentielles d’oscillations non linéaires, (French) Rev. Math. Pures Appl., 4 (1959), 267–270.


• [4] L.-J. Chen, F.-D. Chen, Global stability of a Leslie-Gower predator-prey model with feedback controls, Appl. Math. Lett., 22 (2009), 1330–1334.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] L.-J. Chen, F.-D. Chen, Global analysis of a harvested predator-prey model incorporating a constant prey refuge, Int. J. Biomath., 3 (2010), 205–223.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] L.-J. Chen, F.-D. Chen, Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a constant prey refuge, Nonlinear Anal. Real World Appl., 11 (2010), 246–252.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] F.-D. Chen, L.-J. Chen, X.-D. Xie, On a Leslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Anal. Real World Appl., 10 (2009), 2905–2908.
  • View Article
  • MathSciNet
  • Google Scholar


• [8] W.-L. Chen, X.-J. Gong, L. Zhao, H.-Y. Zhang, Dynamics of a nonautonomous discrete Leslie-Gower predator-prey system with a prey refuge, (Chinese) J. Fuzhou Univ. Nat. Sci. Ed., 43 (2015), 6–10.
  • MathSciNet
  • Google Scholar


• [9] F.-D. Chen, Z. Li, Y.-J. Huang, Note on the permanence of a competitive system with infinite delay and feedback controls, Nonlinear Anal. Real World Appl., 8 (2007), 680–687.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] F.-D. Chen, Y.-M. Wu, Z.-Z. Ma, Stability property for the predator-free equilibrium point of predator-prey systems with a class of functional response and prey refuges, Discrete Dyn. Nat. Soc., 2012 (2012 ), 5 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] R. Cressman, J. Garay, A predator-prey refuge system: evolutionary stability in ecological systems, Theor. Popul. Biol., 76 (2009), 248–257.
  • View Article
  • Google Scholar


• [12] Y.-J. Huang, F.-D. Chen, L. Zhong, Stability analysis of a prey-predator model with Holling type III response function incorporating a prey refuge, Appl. Math. Comput., 182 (2006), 672–683.
  • View Article
  • Google Scholar
  • MATH


• [13] L.-L. Ji, C.-Q.Wu, Qualitative analysis of a predator-prey model with constant-rate prey harvesting incorporating a constant prey refuge, Nonlinear Anal. Real World Appl., 11 (2010), 2285–2295.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] T. K. Kar, S. Misra, Influence of prey reserve in a prey-predator fishery, Nonlinear Anal., 65 (2006), 1725–1735.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models , Appl. Math. Lett., 14 (2001), 697–699.
  • View Article
  • MathSciNet
  • Google Scholar


• [16] V. Křivan, On the Gause predator-prey model with a refuge: a fresh look at the history , J. Theoret. Biol., 274 (2011), 67–73.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] P. H. Leslie, Some further notes on the use of matrices in population mathematics, Biometrika, 35 (1948), 213–245.
  • View Article
  • MathSciNet
  • MATH


• [18] P. H. Leslie, A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika, 45 (1958), 16–31.
  • View Article
  • MathSciNet
  • MATH


• [19] Z. Li, M.-A. Han, F.-D. Chen, Global stability of a stage-structured predator-prey model with modified Leslie-Gower and Holling-type II schemes, Int. J. Biomath., 5 (2012), 13 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] Z.-H. Ma, S.-F. Wang, W.-D. Li, The effect of prey refuge in a simple predator-prey model, Ecol. Model., 222 (2011), 3453–3454.


• [21] A. F. Nindjin, M. A. Aziz-Alaoui, M. Cadivel, Analysis of a predator-prey model with modified Leslie-Gower and Hollingtype II schemes with time delay , Nonlinear Anal. Real World Appl., 7 (2007), 1104–1118.
  • View Article
  • Google Scholar


• [22] Y.-D. Tao, X. Wang, X.-Y. Song, Effect of prey refuge on a harvested predator-prey model with generalized functional response, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1052–1059.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] Y.-M. Wu, F.-D. Chen, W.-L. Chen, Y.-H. Lin, Dynamic behaviors of a nonautonomous discrete predator-prey system incorporating a prey refuge and Holling type II functional response, Discrete Dyn. Nat. Soc., 2012 (2012 ), 14 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] Y.-M.Wu, F.-D. Chen, Z.-Z. Ma, Extinction of predator species in a nonautonomous predator-prey system with prey refuge , (Chinese) Appl. Math. J. Chinese Univ. Ser. A, 27 (2012), 359–365.
  • MathSciNet
  • Google Scholar


• [25] Y.-M. Wu, F.-D. Chen, Z.-Z. Ma, Y.-H. Lin, Permanence and extinction of a nonautonomous predator-prey system incorporating prey refuge and Rosenzweig functional response , (Chinese) J. Biomath., 29 (2014), 727–731.
  • MathSciNet


• [26] Y.-M. Wu, W.-L. Chen, H.-Y. Zhang, On permanence and global attractivity of a nonautonomous modified Leslie-Gower predator-prey system incorporating a prey refuge, J. Minjiang Univ., 5 (2012), 13–16.
  • Google Scholar


• [27] R.-X. Wu, L. Lin, Permanence and global attractivity of the discrete predator-prey system with Hassell-Varley-Holling III type functional response, Discrete Dyn. Nat. Soc., 2013 (2013 ), 9 pages.
  • View Article
  • MathSciNet
  • MATH


• [28] R.-X. Wu, L. Zhao, Q.-X. Lin, Stability analysis of a two species amensalism model with Holling II functional response and a cover for the first species, J. Nonlinear Funct. Anal., 2016 (2016 ), 15 pages.


• [29] X.-D. Xie, Y.-L. Xue, J.-H. Chen, T.-T. Li, Permanence and global attractivity of a nonautonomous modified Leslie-Gower predator-prey model with Holling-type II schemes and a prey refuge, Adv. Difference Equ., 2016 (2016 ), 11 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [30] R. Yafia, F. El Adnani, H. T. Alaoui, Limit cycle and numerical similations for small and large delays in a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Nonlinear Anal. Real World Appl., 9 (2006), 2055–2067.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [31] W.-S. Yang, Y.-Q. Li, Dynamics of a diffusive predator-prey model with modified Leslie-Gower and Holling-type III schemes, Comput. Math. Appl., 65 (2013), 1727–1737.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [32] R.-Z. Yang, C.-R. Zhang, Dynamics in a diffusive predator-prey system with a constant prey refuge and delay, Nonlinear Anal. Real World Appl., 31 (2016), 1–22.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [33] R.-Z. Yang, C.-R. Zhang, The effect of prey refuge and time delay on a diffusive predatorprey system with hyperbolic mortality, Complexity, 21 (2016), 446–459.
  • View Article
  • Google Scholar


• [34] S.-B. Yu, Global asymptotic stability of a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Discrete Dyn. Nat. Soc., 2012 (2012 ), 8 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [35] S.-B. Yu, Global stability of a modified Leslie-Gower model with Beddington-DeAngelis functional response, Adv. Difference Equ., 2014 (2014 ), 14 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [36] Q. Yue, Dynamics of a modified Leslie-Gower predator-prey model with Holling-type II schemes and a prey refuge, Springerplus, 5 (2016), 12 pages.
  • View Article
  • Google Scholar


• [37] N. Zhang, F.-D. Chen, Q.-Q. Su, T. Wu, Dynamic behaviors of a harvesting Leslie-Gower predator-prey model, Discrete Dyn. Nat. Soc., 2011 (2011 ), 14 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [38] K.-J. Zhuang, Z.-H. Wen, Dynamical behaviors in a discrete predator-prey model with a prey refuge, Int. J. Comput. Math. Sci., 5 (2011), 195–197.
  • MathSciNet
  • Google Scholar