Abstract
In this study, we develop a delayed predator–prey model that incorporates two stage structures for both predator and prey populations, along with two constant maturation delays and fear effect. The model is formulated as a delay differential equation system, and its well-posedness is rigorously demonstrated. We conduct a comprehensive analysis of the system, including stability evaluations and the identification of Hopf bifurcations caused by the delay terms. Numerical simulations are used to validate and extend the theoretical findings, leading to several key insights. Notably, variations in the delay parameters significantly affect the stability of the system. With a single delay, the system undergoes a transition from stability to oscillation and back to stability as the delay increases, showing a characteristic stability-oscillation-stability pattern.
Furthermore, when both delays are taken into account, extrema-based bifurcation diagrams and time series reveal multi-branch structures, periodic windows, and complex irregular oscillatory segments. Delay-embedding phase portraits uncover typical patterns such as period-doubling cascades and the reappearance of periodic windows. Additionally, two types of numerical diagnostics provide further evidence: significant sensitivity to initial histories is observed in the complex segment, while coexisting attractors are detected in the branch-splitting segment, suggesting bistability in the system. These results indicate that stage structure coupled with dual maturation delays can significantly enrich the bifurcation structure and long-term dynamical behavior of the system.