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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

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2353-0626
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Global dynamics and parameter identifiability in a predator-prey interaction model

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
/ Suraj S. Meghwani / Swati Tyagi / Syed Abbas
Published Online: 2018-07-20 | DOI: https://doi.org/10.1515/msds-2018-0009

Abstract

This paper discusses a predator-prey model with prey refuge. We investigate the role of prey refuge on the existence and stability of the positive equilibrium. The global asymptotic stability of positive interior equilibrium solution is established using suitable Lyapunov functional, which shows that the prey refuge has no influence on the permanence property of the system. Mathematically, we analyze the effect of increase or decrease of prey reserve on the equilibrium states of prey and predator species. To access the usability of proposed predator-prey model in practical scenarios, we also suggest, the use of Levenberg-Marquardt (LM) method for associated parameter estimation problem. Numerical results demonstrate faithful reconstruction of system dynamics by estimated parameter by LM method. The analytical results found in this paper are illustrated with the help of suitable numerical examples

Keywords: Levenberg-Marquardt; Prey refuge; Parameter estimation; Fluctuation Lemma; Lyapunov functional; Permanence

References

  • [1] J. Smith, Models in ecology, Cambridge University Press, Cambridge, 1974.Google Scholar

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

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

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

  • [5] M. Verma, A.K. Misra, Modeling the Effect of Prey Refuge on a Ratio-Dependent Predator-Prey System with the Allee effect, Bull. Math Biol. (2018), 1-31.Web of ScienceGoogle Scholar

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

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

  • [8] 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

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

  • [10] J.P. Tripathi, S.S. Meghwani, M. Thakur, S. Abbas, A modified Leslie-Gower predator-prey interaction model and parameter identifiability, Commun. Nonlinear Sci. Numer. Simulat. 54 (2018), 331-346.Web of ScienceGoogle Scholar

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

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

  • [13] 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

  • [14] F. Chen, L. Chen, X. Xie, On a Leslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Analysis.: RWA, 10 (2009), 2905-2908.Google Scholar

  • [15] W. Hirsch, H. Hanisch, J. Gabriel, Differential equation models of some parasitic infection: methods for the study of asymptotic behaviour, Commun. Pure Appl. Math. 38 (1985), 733-753.CrossrefGoogle Scholar

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

  • [17] S. Ahmad, M.R.M. Rao, Theory of ordinary differential equations with applications in biology and Engineering, Afiliated East- West Press Private Limited, New Delhi, 1999.Google Scholar

  • [18] S. Abbas, M. Banerjee, N. Hungerbuhler, Existence, uniqueness and stability analysis of allelopathic stimulatory phytoplankton model, J. Math. Anal. Appl. 367 (2010), 249-259.Web of ScienceGoogle Scholar

  • [19] J.P. Tripathi, S. Abbas, M. Thakur, Local and global stability analysis of two prey one predator model with help, Commun. Nonlin. Sci. Simulat. 19 (2014), 3284-3297.Web of ScienceGoogle Scholar

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

  • [21] M.A. Aziz-Alaoui, M.D. 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.CrossrefGoogle Scholar

  • [22] R. Yafia, F. El. Adnani, H.T. Alaoui, Stability of limit cycle in a predator-prey model with modified Leslie-Gower and Hollingtype II schemes with time delay. Appl. Math. Sci, 1(3) (2007), 119-131.Google Scholar

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

  • [24] R.P. Gupta, P. Chandra, Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math. Anal. Appl. 398 (2013), 278-295.Google Scholar

  • [25] M. Ashyraliyev, Y. Fomekong-Nanfack, J.A. Kaandorp, J.G. Blom, Systems biology: parameter estimation for biochemical models, Febs Journal, 276(4) (2009), 886-902.Google Scholar

  • [26] G. Lillacci, M. Khammash, Parameter estimation and model selection in computational biology, PLoS Comput Biol. 6(3) (2010), e1000696.Web of ScienceGoogle Scholar

  • [27] J.M. Walmag, E.J. Delhez, A trust-region method applied to parameter identification of a simple prey-predator model, Appl. Math. Model. 29(3) (2005), 289-307.CrossrefGoogle Scholar

  • [28] L.M. Lawson, Y.H. Spitz, E.E. Hofmann, R.B. Long, A data assimilation technique applied to a predator-prey model, B. Math. Biol. 57(4) (1995), 593-617.CrossrefGoogle Scholar

  • [29] J.J. More, The Levenberg-Marquardt algorithm : implementation and theory, Numerical analysis, Springer Berlin Heidelberg, (1978) 105-116.Google Scholar

  • [30] M.K. Transtrum, J.P. Sethna, Improvements to the Levenberg-Marquardt algorithm for nonlinear least-squares minimization, arXiv preprint arXiv:1201.5885 (2012).Google Scholar

  • [31] J. Pujol, The solution of nonlinear inverse problems and the Levenberg-Marquardt method, Geophysics, 72(4) (2007), W1-W16.Web of ScienceCrossrefGoogle Scholar

  • [32] K. Madsen, H.B. Nielsen, O. Tingleff, Methods for non-linear least squares problems, Informatics and Mathematical Modelling, Technical University of Denmark, DTU, (1999).Google Scholar

  • [33] P. Mendes D. Kell, Non-linear optimization of biochemical pathways: applications to metabolic engineering and parameter estimation, Bioinformatics, 14(10) (1998), 869-883.Google Scholar

  • [34] J.S.R. Jang, E. Mizutani, Levenberg-Marquardt method for ANFIS learning, Fuzzy Information Processing Society, 1996 NAFIPS., Biennial Conference of the North American, IEEE, (1996) 87-91.Google Scholar

  • [35] H.K. Cigizoglu, M. Alp, Generalized regression neural network in modelling river sediment yield, Adv. Eng. Softw. 37(2) (2006), 63-68.Google Scholar

  • [36] H.M. Park, T.Y. Yoon, Solution of the inverse radiation problem using a conjugate gradient method, Int. J. Heat Mass Tran. 43(10) (2000), 1767-1776.Google Scholar

  • [37] M. Thakur, K. Deep, Data Assimilation of a Biological Model Using Genetic Algorithms, In Applications and Innovations in Intelligent Systems XIV, Springer London, (2007), 238-242.Google Scholar

  • [38] J.H. Kim, Z.W. Geem, E.S. Kim, Parameter estimation of the nonlinear Muskingum model using harmony search, 37 (2001), 1131-1138.Web of ScienceGoogle Scholar

  • [39] D. Kusum, M. Thakur, A new mutation operator for real coded genetic algorithms, Appl. Math. Comput. 193 (2007), 211-230Web of ScienceGoogle Scholar

  • [40] M. Schwaab, E.C.Jr. Biscaia, J.L. Monteiro, J.C. Pinto, Nonlinear parameter estimation through particle swarm optimization, Chem. Eng. Sci. 63(6) (2008), 1542-1552.CrossrefWeb of ScienceGoogle Scholar

  • [41] M. Thakur, S.S. Meghwani, H. Jalota, A modified real coded genetic algorithm for constrained optimization, Appl. Math. Comput. 235 (2014), 292-317.Web of ScienceGoogle Scholar

About the article

Received: 2017-02-26

Accepted: 2018-07-01

Published Online: 2018-07-20


Citation Information: Nonautonomous Dynamical Systems, Volume 5, Issue 1, Pages 113–126, ISSN (Online) 2353-0626, DOI: https://doi.org/10.1515/msds-2018-0009.

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© 2018 Syed Abbas, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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