Dynamics of Predator-Prey Interactions, Analyzing the Effects of Time

Delays and Neymark-Saker Bifurcation

Thwiba A. Khalid 1,2,∗

1Department of Mathematics, Faculty of Science, Al-Baha University, Albaha 65525, Saudi Arabia

2Academy of Engineering and Medical Sciences, Department of Mathematics, Khartoum, Sudan

Email: tabdulrhman@bu.edu.sa

Abstract

The study examines the dynamics of a predator-prey model that includes temporal delays, concentrating on

the impact of these delays on system stability and behavior.It delineates criteria for the global stability of the

positive equilibrium using a generalized Lyapunov function and the Razumkin-type theorem, emphasizing the

significance of temporal delays in biological systems. The research highlights the Neymark-Saker (NS) bifur-

cation, examining the impact of fractional configurations on this bifurcation and the system’s overall dynamic

stability. The research utilizes the Lyapunov-Razumihin approach to identify bifurcation points and forecast

the system’s progression in intricate ecological settings. The research examines the presence of periodic solu-

tions and local stability criteria related to the two delays in predator-prey interactions. Numerical simulations

are used to substantiate the theoretical results, specifically for the periodic bifurcation solutions associated

with the Neymark-Saker bifurcation.

Keywords: Fixed Points; Bifurcation; Razumkin Theorem; Predator-Prey Dynamics; Time Delays