Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Nonautonomous Dynamical Systems

formerly Nonautonomous and Stochastic Dynamical Systems

Editor-in-Chief: Diagana, Toka

Managing Editor: Cánovas, Jose


Mathematical Citation Quotient (MCQ) 2017: 0.71

Open Access
Online
ISSN
2353-0626
See all formats and pricing
More options …

Dynamical analysis of a predator-prey interaction model with time delay and prey refuge

Jai Prakash Tripathi
  • Department of Mathematics, Central University of Rajasthan, NH-8, Bandarsindri, Kishangarh-305801, Distt.-Ajmer, Rajasthan, India
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Swati Tyagi / Syed Abbas
Published Online: 2018-11-10 | DOI: https://doi.org/10.1515/msds-2018-0011

Abstract

In this paper, we study a predator-prey model with prey refuge and delay. We investigate the combined role of prey refuge and delay on the dynamical behaviour of the delayed system by incorporating discrete type gestation delay of predator. It is found that Hopf bifurcation occurs when the delay parameter τ crosses some critical value. In particular, it is shown that the conditions obtained for the Hopf bifurcation behaviour are sufficient but not necessary and the prey reserve is unable to stabilize the unstable interior equilibrium due to Hopf bifurcation. In particular, the direction and stability of bifurcating periodic solutions are determined by applying normal form theory and center manifold theorem for functional differential equations. Mathematically, we analyze the effect of increase or decrease of prey reserve on the equilibrium states of prey and predator species. At the end, we perform some numerical simulations to substantiate our analytical findings.

Keywords: Time delay; Stability; Center manifold theorem; Hopf bifurcation; functional response

References

  • [1] A.J. Lotka, Elements of Mathematical Biology, Dover, New York, 1956.Google Scholar

  • [2] J. smith, Models in ecology, Cambridge University Press, Cambridge, 1974.Google Scholar

  • [3] A.A. Berryman, The origin and evolution of predator-prey theory, Ecological Society of America, 73 (1992), 1530-1535.Google Scholar

  • [4] R.D. Parshad, A. Basheer, D. Jana, J.P. Tripathi, Do prey handling predators really matter: Subtle effects of a Crowley-Martinfunctional response. Chaos, Solitons & Fractals, 103 (2017), 410-421.Google Scholar

  • [5] K. Gopalsamy, Stability and oscillation in delay differential equations of population dynamics Kluwer Academic, Dordrecht,1992.Google Scholar

  • [6] Y. Kuang, Delay differential equations with applications in population dynamics, Academic Press, Boston, 1993.Google Scholar

  • [7] S. Liu, E. Beretta, A stage-structured predator-prey model of Bedington-DeAngelis type, SIAM J. Appl. Math. 66 (2006),1101-1129.CrossrefGoogle Scholar

  • [8] E. Bereta, Y. Kuang, Global analysis in some delayed ratio dependent predator-prey systems, Nonlinear Analysis: Theory,Method & Applications, 32(3) (1998), 381-408.Google Scholar

  • [9] M. Annik, S. Ruan, Predator-prey models with delay and prey harvesting, J. Math. Biol. 43(3) (2001), 247-267.CrossrefGoogle Scholar

  • [10] S. Liu, E. Beretta, D. Breda, Predator-prey model of Bedington-DeAngelis type withmaturation and gestation delays, NonlinAnal: RWA, 11 (2010), 4072-4091.Google Scholar

  • [11] A.Y. Morozov, M. Banerjee, S.V. Petrovskii, Long-term transients and complex dynamics of a stage-structured populationwith time delay and the Allee effect, J. Theor. Biol. 396 (2016), 116-124.Google Scholar

  • [12] R. Agrawal, D. Jana, R.K. Upadhyay, V.S.H. Rao,Complex dynamics of sexually reproductive generalist predator and gestationdelay in a food chain model: double Hopf-bifurcation to chaos, J. Appl. Math. Comput. 55 (2017), 513-547.Google Scholar

  • [13] J.P. Tripathi, S.S. Meghwani, M. Thakur, S. Abbas, A modified Leslie-Gower predator-prey interaction model and parameteridentifiability, Commun. Nonlinear Sci. Numer. Simulat. 54 (2018), 331-346.CrossrefGoogle Scholar

  • [14] A.F. Nindjin, M.A. Aziz-Alaoui, M. Cadivel, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type IIschemes with time delay, Nonlinear Anal.: RWA, 7 (2006), 1104-1118.Google Scholar

  • [15] H.L. Smith, An introduction to delay differential equations with applications to the life sciences, Springer, New York, 2011.Google Scholar

  • [16] Y. Li, C. Li, Stability and Hopf bifurcation analysis on a delayed Leslie-Gower predator-prey system incorporating a preyrefuge, Appl. Math. Comput. 219 (2013), 4576-4589.Google Scholar

  • [17] D. Jana, R. Agrawal, R.K. Upadhyay, Dynamics of generalist predator in a stochastic environment: effect of delayed growthand prey refuge, App. Math. Comput. 268 (2015), 1072-1094.Google Scholar

  • [18] J.P. Tripathi, S. Abbas, M. Thakur, A density dependent delayed predator-prey model with Beddington-DeAngelis type functionresponse incorporating a prey refuge, Commun. Nonlin. Sci. Numer. Simulat. 22 (2015), 427-450.Google Scholar

  • [19] J.P. Tripathi, S. Tyagi, S. Abbas, Global analysis of a delayed density dependent predator-prey model with Crowley-Martinfunctional response, Commun. Nonlin. Sci. Simulat. 30 (2016), 45-69.Google Scholar

  • [20] S. Abbas, M. Banerjee, N. Hungerbuhler, Existence, uniqueness and stability analysis of allelopathic stimulatory phytoplanktonmodel, J. Math. Anal. Appl. 367 (2010), 249-259.Google Scholar

  • [21] T.K. Kar, Stability analysis of a predator-prey model incorporating a prey refuge, Commun. Nonlinear Sci. Numer. Simulat.10 (2006), 681-691.Google Scholar

  • [22] Y. Huang, F. Chen, L. Zhong, Stability analysis of a predator-prey model with Holling type III response function incorporatinga prey refuge, Appl. Math. Comput. 182 (2006), 672-683.Google Scholar

  • [23] B. Dubey, P. Chandra, P. Sinha, A model for fishery resource with reserve area, Nonlinear Analysis: RWA, 4 (2003), 625-637.Google Scholar

  • [24] D. Mukherjee, Persistence in a generalized prey-predator model with prey reserve, International Journal of Nonlinear Science,14 (2012), 160-165.Google Scholar

  • [25] B. Dubey, A prey-predator model with a reserved area, Nonlinear Analysis: Modelling and Control, 12(4) (2007), 479-494.Google Scholar

  • [26] T.K. Kar, S. Misra, Influence of a prey reserve in a predator-prey dishery, Nonlinear Analysis, 65 (2006), 1725-1735.Google Scholar

  • [27] J.P. Tripathi, S. Abbas, M. Thakur, Dynamical analysis of a prey-predator model with Beddington-DeAngelis type functionresponse incorporating a prey refuge, Nonlinear Dyn. 80 (2015), 177-196.Google Scholar

  • [28] Y Du., J Shi, A diffusive predator-prey model with protection zone, J. Differ. Equat. 229 (2006), 63-91.Google Scholar

  • [29] Y. Lv, R. Ruan, Y. Pei, A prey-predator model with harvesting for fishery resource with prey reserve area, Appl. Math. Model.37 (2013), 3048-3062.Web of ScienceGoogle Scholar

  • [30] A. Sih, Prey refuges and predator-prey stability, Theor. Pop. Biol. 31 (1987), 1-12.Google Scholar

  • [31] J.N. McNair, Stability effects of prey refuges with entry-exit dynamics, J. Theor. Biol. 125 (1987), 449-464.Google Scholar

  • [32] J.B. Collings, Bifurcation and stability analysis of a temperature-dependent mite predator-prey interaction model incorporatingprey refuge, Bull. Math. Biol. 50 (1995), 379-409.Google Scholar

  • [33] E.G. Olivers, R.R. Jiliberto, Dynamic consequences of prey refuges in a simple model system: more prey, fewer predatorsand enhances stability, Ecol. Model. 166 (2003), 135-146.Google Scholar

  • [34] T.K. Kar, Stability analysis of a predator-prey model incorporating a prey refuge, Commun. Nonlin. Sci. Numer. Simulat. 10(2005), 681-691.Google Scholar

  • [35] J.P. Tripathi, S. Abbas, M. Thakur, A density dependent delayed predator-prey model with Beddington-DeAngelis type FunctionResponse incorporating a prey refuge, Commun. Nonlin. Sci. Numer. Simulat. 22 (2015), 427-450.Google Scholar

  • [36] W. Ko, K. Ryu, Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a preyrefuge, J. Differential Equations, 231 (2006). 534-550.Google Scholar

  • [37] L. Ji C.Wu, Qualitative analysis of a predator-prey model with constant-rate prey harvesting incorporating prey refuge, NonlinearAnalysis.: RWA. 11 (2010), 2285-2295.Google Scholar

  • [38] Z. Ma, S. Wang, W. Li, Z. Li, The effect of prey refuge in a patchy predator-prey system, Math. Biosci. 243 (2013), 126-230.Web of ScienceGoogle Scholar

  • [39] M.A. Aziz-Alaoui, M.D. Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower andHolling-type II schemes, Appl. Math. Lett. 16 (2003), 1069-1075.Google Scholar

  • [40] B.D. Hassard, N.D. Kazarinoff, Y.H. Wan, Theory and application of Hopf bifurcation, CUP Archive, 1981.Google Scholar

  • [41] R. Yuan, W. Jiang, Y. Wang, Saddle-node-Hopf bifurcation in a modified Leslie-Gower predator-prey model with time-delayand prey harvesting, J. Math. Anal. Appl. 422(2) (2015), 1072-1090.Google Scholar

  • [42] J.K. Hale, S.M. Verduyn, Introduction to functional differential equations, Springer Science and Business Media, 1993.Google Scholar

About the article

Received: 2017-03-13

Accepted: 2018-10-10

Published Online: 2018-11-10

Published in Print: 2018-10-01


Citation Information: Nonautonomous Dynamical Systems, Volume 5, Issue 1, Pages 138–151, ISSN (Online) 2353-0626, DOI: https://doi.org/10.1515/msds-2018-0011.

Export Citation

© by Jai Prakash Tripathi, et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Comments (0)

Please log in or register to comment.
Log in