Further stability criteria for singular Hopf bifurcation in planar predator-prey systems

Area of Science:

• Mathematical Biology
• Dynamical Systems Theory
• Ecology

Background:

• Singular Hopf bifurcation is crucial in analyzing predator-prey systems.
• Existing research provides Lyapunov coefficients for bifurcations (A≠0), but stability is unclear when A=0.
• Planar predator-prey systems are fundamental ecological models.

Purpose of the Study:

• To investigate the stability of singular Hopf bifurcation in planar predator-prey systems when the first Lyapunov coefficient is zero (A=0).
• To derive and present novel stability criteria for this specific bifurcation case.
• To extend understanding beyond existing results for supercritical (A<0) and subcritical (A>0) bifurcations.

Main Methods:

• Analysis of singular perturbation theory in the context of Hopf bifurcations.
• Derivation of stability criteria for planar predator-prey models.
• Application of methods to Leslie and Gause type predator-prey systems.
• Numerical simulations to validate analytical results.

Main Results:

• New stability criteria for singular Hopf bifurcation at A=0 in planar predator-prey systems are established.
• The analysis specifically addresses the gap in knowledge for the A=0 case.
• Numerical simulations confirm the derived analytical stability criteria.

Conclusions:

• This work provides a comprehensive understanding of singular Hopf bifurcation stability in predator-prey systems, especially for the previously unaddressed A=0 scenario.
• The findings contribute to the theoretical framework of ecological modeling and dynamical systems.
• The validated criteria enhance the predictive power of predator-prey models.