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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 654619, 13 pages
http://dx.doi.org/10.1155/2013/654619
Research Article

Dynamic Analysis of a Two-Language Competitive Model with Control Strategies

1College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China
2Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea
3Departamento de Análisis Matemático and Instituto de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
4Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received 2 August 2013; Accepted 7 October 2013

Academic Editor: Bashir Ahmad

Copyright © 2013 Lin-Fei Nie 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

The dynamic behavior of a two-language competitive model is analyzed systemically in this paper. By the linearization and the Bendixson-Dulac theorem on dynamical system, some sufficient conditions on the globally asymptotical stability of the trivial equilibria and the existence and the stability of the positive equilibrium of this model are presented. Nextly, in order to protect the endangered language, an optimal control problem relative to this model is explored. We derive some necessary conditions to solve the optimal control problem and present some numerical simulations using a Runge-Kutta fourth-order method. Finally, the languages competitive model is extended to this model assessing the impact of state-dependent pulse control strategy. Using the Poincaré map, differential inequality, and method of qualitative analysis, we prove the existence and stability of positive order-1 periodic solution for this control model. Numerical simulations are carried out to illustrate the main results and the feasibility of state-dependent impulsive control strategy.