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Mathematical Problems in Engineering
Volume 2012, Article ID 803270, 17 pages
http://dx.doi.org/10.1155/2012/803270
Research Article

Dynamics and Optimal Taxation Control in a Bioeconomic Model with Stage Structure and Gestation Delay

1State Key Laboratory of Integrated Automation of Process Industry, Institute of Systems Science, Northeastern University, Shenyang 110004, China
2School of Science, Shenyang University of Technology, Shenyang 110870, China
3School of Science, Dalian Jiaotong University, Dalian 116028, China

Received 17 February 2012; Accepted 1 May 2012

Academic Editor: Moez Feki

Copyright © 2012 Yi Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A prey-predator model with gestation delay, stage structure for predator, and selective harvesting effort on mature predator is proposed, where taxation is considered as a control instrument to protect the population resource in prey-predator biosystem from overexploitation. It shows that interior equilibrium is locally asymptotically stable when the gestation delay is zero, and there is no periodic orbit within the interior of the first quadrant of state space around the interior equilibrium. An optimal harvesting policy can be obtained by virtue of Pontryagin's Maximum Principle without considering gestation delay; on the other hand, the interior equilibrium of model system loses as gestation delay increases through critical certain threshold, a phenomenon of Hopf bifurcation occurs, and a stable limit cycle corresponding to the periodic solution of model system is also observed. Finally, numerical simulations are carried out to show consistency with theoretical analysis.