| Abstract: Bifurcation theory serves as a powerful mathematical tool for understanding and analyzing the behavior of dynamic systems. In this talk, I will provide an introductory overview of bifurcation theory, focusing on its fundamental principles and significance in studying the qualitative changes in system behaviour as parameters are varied. I will discuss some high-codimension bifurcations, such as the Hopf bifurcation and the cusp point of codimension 2 bifurcation, which play a pivotal role in shaping the dynamics of nonlinear systems. The second part of the talk explores the application of bifurcation theory in population dynamics. Specifically, I will introduce and investigate a predator-prey model with a Holling type IV functional response, a model renowned for its ability to capture complex ecological interactions. Through bifurcation analysis, I will study the rich and complex dynamics inherent in the system, with insights contributing not only to our theoretical understanding of bifurcation phenomena but also to significant implications for ecological modeling and understanding the behavior of real-world ecosystems. |