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Nonautonomous Dynamical Systems

formerly Nonautonomous and Stochastic Dynamical Systems

Editor-in-Chief: Diagana, Toka

Managing Editor: Cánovas, Jose

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Mathematical Citation Quotient (MCQ) 2018: 0.62

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2353-0626
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Dynamics Analysis and Optimality in Selective Harvesting Predator-Prey Model With Modified Leslie-Gower and Holling-Type II

W. Abid / R. Yafia
  • Corresponding author
  • Ibn Tofail Universit, Faculty of Sciences, Department of Mathematics, Campus Universitaire, BP 133, Kénitra, Morocco
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ M. A. Aziz-Alaoui
  • LMAH, FR-CNRS-3335, Université du Havre Normandie, 25 Rue Ph. Lebon, BP:540, 76058 Le Havre Cedex, (Normandie) France
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  • Other articles by this author:
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/ Ahmed Aghriche
Published Online: 2019-03-22 | DOI: https://doi.org/10.1515/msds-2019-0001

Abstract

In this work, we consider the optimal harvesting and stability problems of a prey-predator model with modified Leslie-Gower and Holling-type II functional response. The model is governed by a system of three differential equations which describe the interactions between prey, predator and harvesting effort. Boundedness and existence of solutions for this system are showed. The existence and local stability of the possible steady states are analyzed and the conditions of global stability of the interior equilibrium are established by using the Lyapunov function, we prove also the occurrence of Hopf bifurcation at this point. By using the Pontryagin’s maximal principle, we formulate and we solve the problem of the optimal harvest policy. In the end, some numerical simulations are given to support our theoretical results.

Keywords: Predator-prey model; ordinary differential equations; local and global stability; bifurcation; optimal harvesting policy

References

  • [1] C.W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewal Resources, Wiley, New York, 1976.Google Scholar

  • [2] M. Fan, K.Wang, Optimal harvesting policy for single population with periodic coefficients,Math. Biosci. 152 (1998) 165-177.Google Scholar

  • [3] H. Qiu, J. Lv, K. Wang, The optimal harvesting policy for non-autonomous populations with discount, Appl. Math. Lett. 26 (2013) 244–248.CrossrefWeb of ScienceGoogle Scholar

  • [4] E. Braverman, R.Mamdani, Continuous versus pulse harvesting for population models in constant and variable environment, J. Math. Biol. 57 (2008)413–434.CrossrefWeb of ScienceGoogle Scholar

  • [5] M.A. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling type II shemes, Applied Math. Let., 16, (2003), 1069-1075.Google Scholar

  • [6] M. Daher Okiye, Study and asymptotic analysis of some nonLinear dynamical systems : Application to predator-prey problems, in french, PHD Thesis, Le Havre University, France, 2004.Google Scholar

  • [7] M. A. Aziz-Alaoui, Study of a Leslie-Gower-type tritrophic population model, Chaos, Solitons and Fractals, 14 (8), (2002), 1275-1293.CrossrefGoogle Scholar

  • [8] T.K. Kar, Modelling and analysis of a harvested prey–predator system incorporating a prey refuge, J. Comput. Appl. Math. 185 (2006) 19–33.Google Scholar

  • [9] D. Pal, G.S. Mahaptra, G.P. Samanta, Optimal harvesting of prey–predator system with interval biological parameters: a bioeconomic model, Math.Biosci. 241 (2013) 181–187.Web of ScienceGoogle Scholar

  • [10] J. Hale, Ordinary Differential Equations, Krieger Publ. Co., Malabar, 1980.Google Scholar

  • [11] Kalyan Das, M.N. Srinivas, M.A.S. Srinivas c, N.H. Gazi, d Chaotic dynamics of a three species prey–predator competition model with bionomic harvesting due to delayed environmental noise as external driving force C. R. Biologies 335 (2012) 503–513.Google Scholar

  • [12] Manju Agarwal and Rachana Pathak, Persistence and optimal harvesting of prey-predator model with Holling Type III functional response, International Journal of Engineering, Science and Technology Vol. 4, No. 3, (2012) pp. 78-96.Google Scholar

  • [13] Xiao D, Ruan S. Bogdanov-Takens bifurcations in predator–prey systems with constant rate harvesting. Fields Inst Commun (1999)21:493–506.Google Scholar

  • [14] Myerscough MR, Gray BF, Hogarth WL, Norbury J. An analysis of an ordinary differential equations model for a two species predator–prey system with harvesting and stocking. J Math Biol (1992) 30:389–411.Google Scholar

  • [15] Brauer F, Soudack AC. Stability regions and transition phenomena for harvested predator–prey systems. J Math Biol (1979) 7:319–37.CrossrefGoogle Scholar

  • [16] M.I.S. Costa, E. Kaszkurewicz, A. Bhaya, L. Hsu, Achieving global convergence to an equilibriumpopulation in predator–prey systems by the use of a discontinuous harvesting policy, Ecol. Model. 128 (2000) 89.Google Scholar

  • [17] Chen, FD: On a nonlinear non-autonomous predator-prey model with diffusion and distributed delay. J. Comput. Appl.Math. 180, 33-49 (2005).Google Scholar

  • [18] J. Rebaza, Dynamics of prey threshold harvesting and refuge, J. Comput. Appl. Math. 236 (2012) 1743.Web of ScienceGoogle Scholar

  • [19] S. Sarwardi, M. Haque, P.K. Mandal, Ratio-dependent predator–prey model of interactin population with delay effect, Nonlinear Dyn. 69 (2012) 817-836.Google Scholar

  • [20] K. Chakraborty, S. Jana, T.K. Kar, Global dynamics and bifurcation in a stage structured prey–predator _shery model with harvesting, Appl. Math.Comput. 218 (2012) 9271–9290.Google Scholar

  • [21] D. L. Ragozin and G. Brown, “Harvest policies and nonmarket valuation in a predatorprey system”, J. Envirn. Econ. Manag. 12 (1985) 155-168.Google Scholar

  • [22] L. S. Pontryagin, V. G. Boltyonskü, R. V. Gamkrelidre and E. F. Mishchenko, The mathematical theory of optimal processes, Wiley, New York, 1962.Google Scholar

  • [23] T. Das, R.N.Mukherjee, K.S. Chaudhuri, Harvesting of a prey–predator fishery in the presence of toxicity, Appl.Math. Model. 33 (2009) 2282.CrossrefWeb of ScienceGoogle Scholar

  • [24] M. Liu, K. Wang, Analysis of a stochastic autonomous mutualism model, J. Math. Anal. Appl. 402 (2013) 392-403.CrossrefGoogle Scholar

  • [25] J.R. Beddington, R.M. May, Harvesting natural populations in a randomly fluctuating environment, Science 197 (1977) 463–465.Google Scholar

  • [26] W. Li, K. Wang, Optimal harvesting policy for general stochastic logistic population model, J. Math. Anal. Appl. 368 (2010) 420–428.Web of ScienceGoogle Scholar

  • [27] A.R. Palma, E.G. Olivares, Optimal harvesting in a predator–prey model with Allee effect and sigmoid functional response, Appl. Math. Model. 5 (2012) 1864.Web of ScienceGoogle Scholar

  • [28] C. Chen, C. Hsui, Fishery policy when considering the future opportunity of harvesting, Math. Biosci. 207 (2007) 138.Web of ScienceGoogle Scholar

  • [29] Dubey, B.P. Chandra, et al. A model for fishery resourse with reserve area. Nonlinear Analysis : Real World Applications., 4(4) (2003). 625-637.Google Scholar

  • [30] K. R Fister and S. Lenhart, optimal harvesting in an age-structured predator-prey model, Appl. Math. Optim. 54 (2006) 1-15.Google Scholar

  • [31] Zhang X, chen L, Neumann UA (2000) The stage structured predator prey model and optimal harvesting policy. Math Biosci 168 : 201-210.Google Scholar

  • [32] Hsu, S.B. and Hwang T.W., Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55(3), (1995) 763-789.CrossrefGoogle Scholar

  • [33] Hsu, S.B. and Hwang T.W., Uniqueness of limit cycles for a predator-prey system of Holling and Leslie type. Can. Appl.Math. Q., 6(2), 91-117, (1998).Google Scholar

  • [34] Daher, O. M. and Aziz-Alaoui, M. On the dynamics of a predator-prey model with the Holling-Tanner functional Editor V. Capasso, Proc. ESMTB conf, (2002) 270-278.Google Scholar

  • [35] R. Ya_a, F. El Adnani and H. Talibi Limit cycle and numerical similations for small and large delays in a predator-prey model with modified Leslie-Gower and Holling-type II scheme, Nonlinear Analysis: Real World Applications Vol.9, (2008) 2055-2067.Google Scholar

  • [36] R. Ya_a, F. El Adnani and H. Talibi, Stability of limit cycle in a predator-prey model with modified Leslie-Gower and Hollingtype II schemes with time delay. Applied Mathematical Sciences, Vol. 1, no. 3, (2007) pp 119 - 131.Google Scholar

  • [37] E. Beretta, Y. Kuang, Global analyses in some delayed ratio-depended predator–prey systems, Nonlinear Anal. Theory Methods Appl. 32 (3) (1998) 381-408.Google Scholar

  • [38] A.F. Nindjin, M.A. Aziz-Alaoui and M. Cadivel, Analysis of a Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes with Time Delay, Nonlinear Anal. Real World Appl., 7(5), (2006) 1104-1118. Theoretical Biology 245 (2007) 220–229.Google Scholar

  • [39] Sze-Bi Hsu and Tzy-Wei Hwang, Hopf bifurcation for a predator-prey system of Holling and Leslie type, Taywanese journal of Mathematics Vol. 3, No. 1, pp. 35-53, March 1999.Google Scholar

  • [40] S. Chakraborty, S. Pal, N. Bairagi, Predator–prey interaction with harvesting: mathematical study with biological ramifications, Applied Mathematical Modelling 36 (2012) 4044–4059.Google Scholar

  • [41] C.W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, second ed., John Wiley and Sons, New York, 1990.Google Scholar

  • [42] B. Roy and S. K. Roy, Analysis of prey-predator three species models with vertebral and invertebral predators, International Journal Dynamics and Control 3 (3), (2015) 306-312.Google Scholar

  • [43] S. K. Roy and B. Roy, Analysis of prey-predator three species _shery model with harvesting including prey refuge and migration, International Journal of Bifurcation and Chaos 26 (02), (2016) 1650022.Google Scholar

  • [44] B. Roy, S. K. Roy and D. B. Gurung, Holling-Tanner model with Beddington-DeAngelis functional response and time delay introducing harvesting, Mathematics and Computers in Simulation 142 (2017) 1-14.Web of ScienceGoogle Scholar

  • [45] B. Roy, S. K. Roy and M. H. A. Biswas, Effects on prey-predator with different functional response, International Journal of Biomathematics 10 (08), (2017) 1750113.Google Scholar

  • [46] Birkoff G. and Rota G.C., Ordinary Differential Equations. Ginn; (1982).Google Scholar

About the article

Received: 2018-04-02

Accepted: 2019-02-25

Published Online: 2019-03-22

Published in Print: 2019-03-01


Citation Information: Nonautonomous Dynamical Systems, Volume 6, Issue 1, Pages 1–17, ISSN (Online) 2353-0626, DOI: https://doi.org/10.1515/msds-2019-0001.

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© 2019 W. Abid et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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