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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


The Effect of non-Selective Harvesting in Predator-Prey Model with Modified Leslie-Gower and Holling Type II Schemes

Discontinuity, Nonlinearity, and Complexity 7(4) (2018) 413--427 | DOI:10.5890/DNC.2018.12.006

I. El Harraki, R. Yafia, A. Boutoulout, M. A. Aziz-Alaoui

Ecole Nationale Sup┬┤erieure des Mines de Rabat, Morocco

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Abstract

In this paper, we study the effect of harvesting on the qualitative properties of predator-prey model with modified Leslie-Gower and Holling Type II functional responses. The model is given by a system of two ordinary differential equations with non-selective constants harvesting terms. We investigate the impact of harvesting terms on the boundedness of solutions, on the existence of the attraction set, on the stability of different equilibrium points. A Lyapunov function is used to prove the global stability of the interior equilibrium. We also, discussed the policy of optimal harvest and we got the solution for the interior equilibriumby the Pontryaginmaximum criterion. Finally, our theoretical results are illustrated by a numerical simulations.

References

  1. [1]  M.A. Aziz-Alaoui, Study of a Leslie-Gower-type tritrophic population, Chaos Sol. and Fractals 14 (8), 1275- 1293, (2002).
  2. [2]  M.A. Aziz-Alaoui,M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie- Gower and Holling-type IIschemes, Appl. Math. Lett. 16 (2003) 1069-1075.
  3. [3]  Brauer, F., Soudack, A.C.: Stability regions in predator-prey systems with constant-rate prey harvesting. J. Math. Biol. 8, 55-71 (1979).
  4. [4]  L. Chen, F. Chen, L. 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 (1) (2010).
  5. [5]  S. Chen, J. Shi, J.Wei, Global stability and Hopf bifurcation in a delayed diffusive Leslie-Gower predator-prey system, Internat. J. Bifur. Chaos 22(03) (2012). 246-252.
  6. [6]  C.W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources. Wiley, New York (1976).
  7. [7]  P.H. Leslie, J.C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika 47 (1960) 219-231.
  8. [8]  A.J. Lotka, Elements of Physical Biology,Waverly Press, Williams Wilkins Company, Baltimore, MD,USA, 1925.
  9. [9]  E.P. Odum, Fundamentals of Ecology, Saunders, Philadelphia, 1971.
  10. [10]  A. Rojas-Palma, E. Gonzalez-Olivares, Optimal harvesting in a predator-prey model with Allee effect and sigmoid functional response. Appl. Math. Model. 36, 1864-1874 (2012).
  11. [11]  V. Volterra, Variations and fluctuations of the number of individuals in animal species living together, ICES J.Mar. Sci. 3 (1928) 3-51.
  12. [12]  R. Yafia, 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), pp: 2055-2067.
  13. [13]  R. Yafia, F. El Adnani and H. Talibi, Stability of limit cycle in a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay. Applied Mathematical Sciences, Vol. 1, no. 3, (2007), pp: 119-131.
  14. [14]  R. Zhang, J. Sun, H. Yang, Analysis of a prey-predator fishery model with prey researve. Appl.Math. Sci. 1, 2481-2492 (2007).
  15. [15]  Z. Zeng, M. Fan, Study on a non-autonomous predator-prey system with Beddington-DeAngelis functional response, Math. Comput. Modelling 48(11) (2008) 1755-1764.