[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Discontinuity, Nonlinearity and Complexity
ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
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

Dynamics of a Stage-Structured-Prey and Predator Model with Linear Harvesting of Mature Prey and Predator

Discontinuity, Nonlinearity, and Complexity 10(1) (2021) 61--75 | DOI:10.5890/DNC.2021.03.005

Md. Manarul Haque, Sahabuddin Sarwardi

Department of Mathematics $&$ Statistics, Aliah University, IIA/27, New Town, Kolkata - 700 160, India

Download Full Text PDF

 

Abstract

In this paper we have studied the dynamical behaviors of stage-structured-prey and predator model with linear harvesting of both mature prey and predator. Optimal harvesting policy, positivity, boundedness and Hopf-bifurcation have been studied. Finally, some graphical and numerical simulations are given in order to validate our analytical and theoretical findings.

Acknowledgments

Authors are thankful to the Department of Mathematics $\&$ Statistics, Aliah University for providing opportunities to perform the present work. The corresponding author Dr. S. Sarwardi is thankful to his Ph.D. supervisor Prof. Prashanta Kumar Mandal, Department of Mathematics, Visva-Bharati (a Central University) for his generous help and continuous encouragement while preparing

References

  1. [1]  % Format for Journal Reference %Author, Article title, Journal, Volume, page numbers (year) % Format for books %
  2. [2]  Author, Book title, page numbers. Publisher, place (year)
  3. [3]  Chen, L. and Chen, J. (1993), Nonlinear biological dynamics system, Science, Beijing, (in Chinese).
  4. [4]  Chen, L. (1988), Mathematical models and methods in ecology, Science, Beijing, (in Chinese).
  5. [5]  Hofbauer, J. and Sigmund, K. (1998), Evolutionary games and population dynamics, Cambridge University, Cambridge.
  6. [6]  Freedman, H.I. (1980), Deterministic mathematical models in population ecology, Marcel Dekker, New York.
  7. [7]  Murray, J.D. (1993), Mathematical biology, 2nd Ed. Springer, Heidelberg.
  8. [8]  Takeuchi, Y. (1996), Global dynamical properties of Lotka-Volterra system, World Scientific, Singapore.
  9. [9]  May, R.M. (1975), Stability and complexity in model ecosystems, 2nd Ed. Princeton University, Princeton, NJ.
  10. [10]  May, R.M. (1981), Theoritical ecology, principles and applications, 2nd Ed. Blackwell, Oxford.
  11. [11]  Manarul, M.M. and Sarwardi, S. (2016), Effect of toxicity on a harvested fishery model, Model. Earth Syst. Environ., {\bf 122}(2).
  12. [12]  Clark, C.W. (1985), Bieconomic modelling and fisheries management, Wiley, New York.
  13. [13]  Clark, C.W. (1990), Mathematical Bioeconomics: The optimal management of renewable resources, Wiley series, New York.
  14. [14]  Kot, M. (2001), Elements of mathematical ecology, Cambridge University Press, Cambridge.
  15. [15]  Ge, Z. and Yan, J. (2011), Hopf bifurcation of predator-prey system with stage structure and harvesting, Non. Anal. Theo. Meth. Appl., {\bf 74}(2), 652-660.
  16. [16]  Chen, L., Chen, F., and Chen, L. (2010), Qualitative analysis of a predator-prey model with Holling type-II functional response incorporating a constant prey refuge, Non. Anal. Real World Appl., {\bf 11}(1), 246-252.
  17. [17]  Gan, Q., Xu, R., and Yang, P. (2009), Bifurcation and chaos in a ratio-dependent predator-prey system with time delay, Chaos Sol. Fract., {\bf 39}(4), 1883-1895.
  18. [18]  Zhang, X. and Zhang, Q.L. (2009), Bifurcation analysis and control of a class of hybrid biological economic model, Nonl. Anal. Hybrid Syst., {\bf 3}(4), 578-587.
  19. [19]  Zhang, X., Chen, L., and Neumann, A.U. (2000), The stage-structured predator-prey model and optimal harvesting policy, Math. Biosci., {\bf 168}, 201-210.
  20. [20]  Cooke, K.L., Elderkin, R.H., and Huang, W. (2006), Predator-prey interactions with delays due to juvenile maturation, SIAM J. Appl. Math., {\bf 66}, 1050-1079.
  21. [21]  Cushing, J.M. (1998), An introduction to structured population dynamics, SIAM J. App. Math., {\bf 71}, Philadelphia.
  22. [22]  Chen, F. (2006), Permanence of periodic Holling type predator-prey system with stage structure for prey, Appl. Math. Comput., {\bf 182}, 1849-1860.
  23. [23]  Chen, F.D. and Youa, M.S. (2008), Permanence, extinction and periodic solution of the predator-prey system with Beddington-DeAngelis functional response and stage structure for prey, Non. Anal. Real World Appl., {\bf 9}, 207-221.
  24. [24]  Yang, W.S., Li, X.P., and Bai, Z.J., (2008), Permanence of periodic Holling type-IV predator-prey system with stage structure for prey, Math. Comput. Model., {\bf 48}, 677-684.
  25. [25]  Du, Y.H., Pang, P.Y.H., and Wang, M.X. (2008), Qualitative analysis of a prey-predator model with stage structure for predator, SIAM J. Appl. Math., {\bf 69}(2), 596-620.
  26. [26]  Liu, S.Q. and Beretta, E. (2006), A stage-structured predator-prey model of Beddington-DeAngelis type, SIAM J. Appl. Math., {\bf 66}(4), 1101-1129.
  27. [27]  Wang, M.X. (2008), Stability and Hobf bifurcation for a prey-predator model with prey-stage structure and diffusion, Math. Biosci., {\bf 212}, 149-160.
  28. [28]  Georgescu, P. and Hsieh, Y.H. (2006), Global dynamics of predator-prey model with stage structure for the predator, SIAM J. Appl. Math., {\bf 67}(5), 1379-1395.
  29. [29]  Wang, Z.G. and Wu, J.H. (2008), Qualitative analysis for a ratio-dependent predator-prey model with stage structure and diffusion, Nonlinear Anal. Real World Appl., {\bf 9}, 2270-2287.
  30. [30]  Zhang, R., Guo, L., and Fu, S.M. (2009), Global behavior for a diffusive predator-prey model with stage structure and nonlinear density restriction-I: the case in $\mathbf R^{n}$, Bound. Val. Probl., 1-26.
  31. [31]  Cao, H.H. and Fu, S.M. (2010), Global existence and convergence of solutions to a cross-diffusion cubic predator-prey system with stage structure for the prey, Bound. Val. Probl., 1-24.
  32. [32] Xu, S.H. (2014), Dynamics of a general prey-predator model with prey-stage structure and diffusive effects, Com. And Math. with Appl., {\bf 68}(3), 405-423.
  33. [33]  Kar, T.K. and Pahari, U.K. (2007), Modellling nad analysis of a prey predator system with stage-structure and harvesting, Nonl. Anal. Real World Appl., {\bf 8}, 601-609.
  34. [34]  Magnusson, K.G. (1999), Destabilizing effect of cannibalism on a structured predator prey system, Math. Biosci., {\bf 155}, 61-75.
  35. [35]  Wang, W., Mulone, G., Salemi, V., and Salone, V. (2001), Permanence and stability of stage-structured predator prey model, J. Math. Anal. Appl., {\bf 262}, 499-528.
  36. [36]  Xu, R., Chaplain, M.A.J., and Davidson, F.A. (2004), Global stability of a Lotka-Voltera type predator prey model with stage structure and time delay, Appl. Math. Comput., {\bf 159}, 863-880.
  37. [37]  Gao, S., Chen, L., and Teng, Z. (2008), Hopf bifurcation and global stability for a delayed predator-prey system with stage structure for predator, Appl. Math. Comput., {\bf 202}, 721-729.
  38. [38]  Xu, R. and Ma, Z. (2008), Stability and hopf bifurcation in a ratio-dependent predator prey system with stage structure, Chaos Sol. Fract., {\bf 38}, 669-684.
  39. [39]  Chakraborty, K., Jana, S., and Kar, T.K. (2012), Global dynamics and bifurcation in a stage structured prey-predator fishery model with harvesting, Appl. Math. Comput., {\bf 218}(18), 9271-9290.
  40. [40]  Birkhoff, G. and Rota, G.C. (1982), Ordinary Differential Equations. Ginn Boston.
  41. [41]  Hale, J. (1989), Ordinary differential equation. Klieger Publishing Company, Malabar.
  42. [42]  Pontryagin, L.S., Boltyanskii, V.S., Gamkrelidze, R.V., and Mishchencko, E.F. (1962), The mathematical theory of optimal processes. Wiley, New York.
  43. [43]  Guckenheimer, J. and Holmes, P. (1990), Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Springer-Verlag.
  44. [44]  Wiggns, S. (2003), Introduction to applied nonlinear dynamical systems and chaos, 2nd Ed. Springer.
  45. [45]  Kuznetsov, Y. (2004), Elements of applied bifurcation theory, 3rd Ed. Springer.
  46. [46]  Liu, W.M. (1994), Criterion of Hopf bifurcations without using eigenvalues, J. Math. Anal. Appl., {\bf 182}, 250-256.
  47. [47]  Sotomayor, J. (1973), Generic bifurcations of dynamical systems, In: Peixoto, M.M. (eds.) Dynamical Systems, Academic Press, New York, pp. 549--560.
  48. [48]  Rudin, W. (1976), Principles of Mathematical Analysis, McGraw-Hill, {\bf3}.

Copyright © 2011-2023 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates