TY - JOUR
T1 - Revisiting Hopf Bifurcation in Single-Species Models with Time Delays
AU - Shu , Hongying
AU - Wei , Junjie
JO - CSIAM Transactions  on Life Sciences
VL - 3
SP - 377
EP - 408
PY - 2025
DA - 2025/10
SN - 1
DO - http://doi.org/10.4208/csiam-ls.SO-2025-0008
UR - https://global-sci.org/intro/article_detail/csiam-ls/24508.html
KW - Hopf bifurcation, delay differential equations, single-species model, geometric criterion, periodic solutions.
AB - <p style="text-align: justify;">Hopf bifurcation in delay differential equations has been a central topic in the study of complex dynamical behaviors in biological and ecological systems. In this review, we revisit Hopf bifurcation phenomena in single-species models that incorporate time delays, emphasizing recent progress in both ordinary and partial differential equation frameworks. We present a comprehensive overview of classic and contemporary models, such as Wright’s equation, Nicholson’s blowflies equation, and diffusive logistic models, highlighting criteria for local and global bifurcations, the geometric and analytical methods used to determine critical values, and the stability of emerging periodic solutions. The review also covers structured models with age, stage, advection, and spatial effects, as well as equations with multiple delays. Through this survey, we aim to consolidate theoretical insights and provide a unified understanding of delay-induced oscillations in population models, laying the groundwork for future developments in delay-driven dynamics.</p>