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

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.