Behavior of a competitive system of second-order difference equations

Area of Science:

• Mathematics
• Dynamical Systems
• Difference Equations

Background:

• Rational difference equations are crucial in modeling discrete-time dynamical systems.
• Understanding the qualitative behavior of solutions, such as boundedness and equilibrium, is fundamental.
• Previous research has explored various types of difference equations, but specific systems require dedicated analysis.

Purpose of the Study:

• To investigate the boundedness and persistence of positive solutions for a given system of rational difference equations.
• To determine the existence, uniqueness, and stability of positive equilibrium points.
• To analyze the rate of convergence of positive solutions to the equilibrium.

Main Methods:

• Analytical techniques are employed to study the qualitative properties of the difference equations.
• The analysis includes investigating conditions for boundedness and persistence.
• Existence and uniqueness of positive equilibrium are established through fixed-point theorems or related methods.
• Local and global stability of the equilibrium point are examined using linearization and other dynamical system tools.

Main Results:

• The study establishes conditions under which the positive solutions of the system are bounded and persistent.
• The existence and uniqueness of a positive equilibrium point are proven.
• The local and global behavior of the positive equilibrium point are characterized.
• The rate of convergence of positive solutions to the equilibrium is determined.

Conclusions:

• The theoretical results provide a comprehensive understanding of the dynamics of the studied rational difference equation system.
• The findings contribute to the theory of difference equations and their applications.
• Numerical examples validate the theoretical findings, confirming the established properties of the solutions.