Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

BIFURCATIONS AND FEEDBACK CONTROL IN AN EXPLOITED PREY-PREDATOR SYSTEM WITH STAGE STRUCTURE FOR PREY

  • Kar, T.K. (Department of Mathematics, Bengal Engineering and Science University) ;
  • Pahari, U.K. (Bengal Engineering and Science University)
  • Received : 2010.07.24
  • Accepted : 2011.05.26
  • Published : 2011.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

In the present paper we consider a differential-algebraic prey-predator model with stage structure for prey and harvesting of predator species. Stability and instability of the equilibrium points are discussed and it is observed that the model exhibits a singular induced bifurcation when the economic profit is zero. It indicates that the zero economic profit brings impulse, i.e. rapid expansion of the population and the system collapses. For the purpose of stabilizing the system around the positive equilibrium, a state feedback controller is designed. Finally, numerical simulations are given to show the consistency with theoretical analysis.

Keywords

References

  1. W. G. Aiello and H.I. Freedman, A time-delay model of single species growth with stage structure, Math. Biosci. 101 (1990), 139-153. https://doi.org/10.1016/0025-5564(90)90019-U
  2. W.G. Aiello , H.I. Freedman and J. H.Wu, Analysis of a model representing stage-structured population growth with stage- dependent time delay, SIAM J. Appl.Math. 52(1992), 855- 869. https://doi.org/10.1137/0152048
  3. C. W. Clark Mathematical Bioeconomics: The Optimal Management of Renewable re- sources, 2nd Ed., John Wiley and Sons, New York, 1990.
  4. L. Dai, Singular Control System ,Springer, 1989.
  5. H.I. Freedman and J.H. Wu , Persistence and global asymptotic stability of single species dispersal models with stage structure, Quart. Appl. Math. 49(1991), 351-371.
  6. M. Kot, Elements of Mathematical Biology, Cambridge, 2001.
  7. H. Matsuda and P. A. Abrams, Effects of predators-prey interaction and adaptive change on sustainable yield, Can. J. Fish. Aquat. Sci./J. Can. Sci. Halieut. Aquat. 61(2)(2004), 175-184. https://doi.org/10.1139/f03-147
  8. J.D. Murray, Mathematical Biology, 2nd corrected ed., Springer, Heidelberg, 1993.
  9. XY. Song and LS. Chen, Optimal harvesting and stability for a predator-prey system with stage structure, Acta Mathematicae Applicatae Sinica 18(3)(2002), 423-430. https://doi.org/10.1007/s102550200042
  10. V. Venkatasubramanian, H. Schaettler and J. Zaborszky, Local bifurcations and feasibility regions in differential -algebraic systems, IEEE Trans. Auto. Contr. 40 (1995), 1992-2013. https://doi.org/10.1109/9.478226
  11. Y. Zhang and Q.L. Zhang, Chaotic control based on descriptor bioeconomic systems, Contr. Deci. 22 (2007), 445-452.
  12. Y. Zhang ,Q.L. Zhang and L. C. Zhao, Bifurcations and control in singular biological economical model with stage structure, J. Syst. Eng. 22 (2007), 232-238.
  13. X. Zhang and Q. L. Zhang , Bifurcation analysis and control of a class of hybrid biological economic models, Nonlinear Analysis: Hybrid Systems, 3 (4)(2009), 578-587. https://doi.org/10.1016/j.nahs.2009.04.009