On the System of High Order Rational Difference Equations.

Area of Science:

• Dynamical Systems
• Nonlinear Analysis
• Mathematical Biology

Background:

• Rational difference equations model various phenomena in applied mathematics.
• Understanding the long-term behavior of solutions is crucial for applications.
• Previous studies often focused on single equations or simpler systems.

Purpose of the Study:

• To investigate the boundedness of positive solutions for a specific system of two rational difference equations.
• To determine the conditions for the persistence of these solutions.
• To analyze the global asymptotic behavior of the system's positive solutions.

Main Methods:

• Utilized techniques from the theory of difference equations.
• Employed stability analysis and invariant principle methods.
• Established bounds and convergence properties through iterative analysis.

Main Results:

• Proved that all positive solutions of the system are bounded.
• Established criteria for the persistence of the system, ensuring solutions remain positive.
• Demonstrated that all positive solutions converge to a unique equilibrium point.

Conclusions:

• The system of rational difference equations exhibits stable and persistent positive solutions.
• The findings contribute to the understanding of complex dynamics in discrete mathematical models.
• Results have implications for modeling systems where positive feedback and resource limitation interact.