Bifurcation and chaos analysis for a discrete ecological developmental systems.

Area of Science:

• Ecological modeling
• Dynamical systems theory
• Computational mathematics

Background:

• Ecological systems exhibit complex dynamics.
• Discrete models are crucial for understanding population fluctuations.
• Bifurcation and chaos analysis provide insights into system stability.

Purpose of the Study:

• To investigate the dynamic behaviors of a discrete prey-predator-scavenger (PPS) system.
• To identify bifurcation parameters and analyze stability.
• To explore phenomena such as chaos and period-doubling cascades.

Main Methods:

• Euler discretization method applied to derive the discrete PPS system.
• Bifurcation analysis using the step size (h) as a parameter.
• Numerical simulations to verify theoretical findings.
• Calculation of the maximum Lyapunov exponent.

Main Results:

• The discrete PPS system exhibits flip bifurcation (FB) and Neimark-Sacker bifurcation (NSB).
• Observed chaotic sets, quasi-periodic orbits, and period-doubling cascades (2, 4, 8, 16).
• Maximum Lyapunov exponent confirms the system's rich dynamic characteristics.

Conclusions:

• Discrete ecological models demonstrate complex dynamics beyond continuous models.
• Bifurcation analysis is essential for understanding ecological system stability and transitions.
• The studied PPS system displays a wide range of nonlinear behaviors, including chaos.