Nonlinear Dynamics

, Volume 70, Issue 3, pp 1805–1829 | Cite as

Effort dynamics of a delay-induced prey–predator system with reserve

Original Paper


This paper describes a prey–predator fishery system with prey dispersal in a two-patch environment, one of which is a free fishing zone and the other a protected zone. The proposed system reflects the dynamic interaction between the net economic revenue and the fishing effort used to harvest the population in presence of a suitable tax. Local as well as global stability of the system is analyzed. The optimal taxation policy is formulated and solved with the help of Pontryagin’s maximal principle. The objective of the paper is to achieve the sustainability of the fishery, keeping the ecological balance, and maximize the monetary social benefit. The dynamical behavior of the delay system is further analyzed through incorporating discrete type gestational delay of predators, and the existence of Hopf bifurcation phenomenon is checked at the interior equilibrium point. Moreover, we use normal form method and center manifold theorem to examine the nature of the Hopf bifurcation. Theoretical results are verified with the help of numerical examples and graphical illustrations.


Prey–predator fishery Gestational delay Dynamic reaction Hopf bifurcation Optimal taxation 



We are very grateful to the anonymous reviewers for their careful reading, constructive comments and helpful suggestions, which have helped us to improve the presentation of this work significantly. First author gratefully acknowledges Director, INCOIS for his encouragement and unconditional help. This is INCOIS contribution number 121. Research of Soovoojeet Jana is financially supported by University Grants Commission, Government of India (F. 11-2/2002 (SA-1) dated 19 August, 2011).


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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Kunal Chakraborty
    • 1
  • Soovoojeet Jana
    • 2
  • T. K. Kar
    • 2
  1. 1.Information Services and Ocean Sciences GroupIndian National Centre for Ocean Information Services, Hyderabad “Ocean Valley”HyderabadIndia
  2. 2.Department of MathematicsBengal Engineering and Science UniversityHowrahIndia

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