Oscillatory dynamics in a discrete predator-prey model with distributed delays.

Area of Science:

• Mathematical Biology
• Dynamical Systems
• Ecology

Background:

• Predator-prey models are fundamental in ecology for understanding population dynamics.
• Incorporating time delays, such as distributed delays, is crucial for realistic ecological modeling.
• Analyzing the stability of periodic solutions is key to predicting long-term population behavior.

Purpose of the Study:

• To investigate the existence and global asymptotic stability of positive periodic solutions in a predator-prey system with distributed delay.
• To establish rigorous mathematical conditions guaranteeing the stability of ecological interactions over time.

Main Methods:

• Application of the coincidence degree theory to establish the existence of solutions.
• Utilization of Lyapunov function methods to prove global asymptotic stability.
• Numerical simulations to validate the theoretical results.

Main Results:

• Sufficient conditions derived for the existence of at least one positive periodic solution.
• Proof of the global asymptotic stability of the positive periodic solution under specific conditions.
• Simulation results demonstrate the accuracy of the established theoretical framework.

Conclusions:

• The study successfully establishes conditions for the stability of a predator-prey model with distributed delays.
• The findings contribute to a deeper mathematical understanding of ecological population dynamics.
• The employed mathematical techniques provide a robust framework for analyzing similar complex systems.