New Criteria for Analyzing the Permanence, Periodic Solution, and Global Attractiveness of the Competition and Cooperation Model of Two Enterprises with Feedback Controls and Delays
Ahmadjan Muhammadhaji, Yimamu Maimaiti
Abstract
This paper studies a class of the non-autonomous competition and cooperation model of two enterprises involving discrete time delays and feedback controls. The paper proposes new criteria for analyzing the permanence, periodic solution, and global attractiveness of the model. The common mathematical techniques of the Lyapunov method, the continuation theorem, and the comparison principle are used in this paper. By means of the comparison principle and inequality techniques, the concept of permanence is investigated, which refers to the long-term survival of the enterprises within the competitive and cooperative framework. Meanwhile, using the continuation theorem, we establish conditions under which the system exhibits periodic behavior. Additionally, the global attractiveness of the system is derived by constructing multiple Lyapunov functionals. Finally, an example is presented to illustrate the applicability and validity of the proposed criteria in this paper. This example serves as a demonstration that showcases the main results derived from the analysis.