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Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 727531, 9 pages
http://dx.doi.org/10.1155/2015/727531
Research Article

Discrete Optimal Control Method Based on the Optimal Strategy of Fishing

1Mathematics School and Institute, Jilin University, Changchun, Jilin 130012, China
2College of Mathematics, Beihua University, Jilin City, Jilin 132013, China

Received 24 September 2014; Revised 11 December 2014; Accepted 11 December 2014

Academic Editor: Manuel De la Sen

Copyright © 2015 Lichun Zhang and Qingdao Huang. 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

Consideration was given to the discrete optimal control method for the optimal fishing strategy. Our method is new and efficient for discrete optimal control problem, which is different from the other optimal methods such as the traditional variational method, the Pontryagin principle of maximum, and the dynamic programming. The basic construction of the model is the traditional logistic function relating to the growth of fry. The discrete optimal control method for optimal fishing strategy was used to construct the optimal rate of each fishing strategy; the main focus of our work is on the rigorous mathematical analysis of the optimal control problem. The analysis allows one to obtain the optimal initial investment amount of the fry and the optimal size of the total catch. Furthermore, when the initial investment amount of the fry is below or above the optimal value and the intrinsic growth rate of fish is too small, we derive that fishing operations should not be started in the last few years to make the overall fishing amount optimal. At last, several typical examples are given to illustrate the obtained results.