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.

During chronic viral infection, sustained antigen stimulation leads to exhaustion of virus-specific CD8⁺ T cells, characterized by elevated expression of inhibitory receptors and progressive functional impairment, including loss of cytokine production, reduced cytotoxicity, and diminished proliferative capacity. In this paper, to investigate how T cell exhaustion influences viral persistence, we developed a within-host mathematical model integrating viral infection dynamics with adaptive immune responses. The model demonstrates three non-trivial equilibria: infectionfree equilibrium (S₁ ), uncontrolled-infection state (S₂), and immune-controlled equilibrium (S₃). Through dynamical systems analysis, we established the local stability of all states (S₁-S₃) and prove global stability for both S₁ (complete viral clearance) and S₂ (chronic infection). Notably, the system exhibits Hopf bifurcations at S₂ and S₃, with distinct critical thresholds governing oscillatory dynamics. Numerical simulations reveal that successful immune-mediated control of viral load and infected cell levels requires maintenance of low CD8⁺ T cell exhaustion rates.

In this paper, we propose an age-structured epidemic model with strain mutation and age-based vaccination. We define the reproduction numbers of both the original and mutant strains ($R_{0}^{1}$ and $R_{0}^{2}$). If the reproduction number R₀ < 1, the disease-free steady state is locally asymptotically stable. If the reproduction number $R_{0}^{2}$ >1, there exists a dominant steady state of the mutant strain. Conditions for local stability of this dominant steady state are also obtained. If both reproduction numbers $R_{0}^{1}$ and $R_{0}^{2}$ are greater than 1, a coexistence steady state may occur. Finally, the uniform persistence of the disease described by our age structured model is strictly proved when the reproduction number $R_{0}^{1}$ > 1. By using the data of the COVID-19 epidemic in Wuhan and the theoretical results obtained in this paper, some numerical calculations are carried out to prove the effect of the age-based vaccination strategy.

Dispersal strategies that lead to the ideal free distribution (IFD) were shown to be evolutionarily stable in various ecological models. In this paper, we investigate this phenomenon in time-periodic environments where N species - identical except for dispersal strategies - compete. We extend the notions of IFD and joint IFD, previously established in spatially continuous models, to time-periodic and spatially discrete models and derive sufficient and necessary conditions for IFD to be feasible. Under these conditions, we demonstrate two competitive advantages of ideal free dispersal: if there exists a subset of species that can achieve a joint IFD, then the persisting collection of species must converge to a joint IFD for large time; if a unique subcollection of species achieves a joint IFD, then that group will dominate and competitively exclude all the other species. Furthermore, we show that ideal free dispersal strategies are the only evolutionarily stable strategies. Our results generalize previous work by construction of Lyapunov functions in multi-species, time-periodic setting.

A mathematical model is proposed that describes the adaptive spatial movement of prey towards higher population density to reduce predation risk. The model admits the increased nonlinearity and the global existence of solutions of the system is established in Sobolev space through analytical estimates. The conditions for the Turing instability from a coexistence steady state are obtained, and sharp conditions for the asymptotical stability of the positive equilibrium in a large region are established with the help of a Lyapunov function. Numerical simulations are presented to support the theoretical results and demonstrate the versatility of spatial models.

Personal diverse interests and many group interactions naturally create partially overlapping groups within a population. Those who belong to multiple groups play a crucial role in spreading of infectious diseases across the whole population. We develop an algorithm to decompose the microscopic overlap structure of groups with representing a population of partially overlapping groups as a hypergraph of partially overlapping hyperedges, and characterize it using a newly defined overlap matrix. We formulate a specific multi-group SIR epidemic model, and address a one-time preventive vaccine allocation problem aimed at effectively reducing the basic reproduction number. By leveraging perturbation theory, we derive a principled ranking index to measure the vaccination priority of different groups, and establish a ranking vaccination strategy, which usually outperforms random vaccination strategies as verified by a series of numerical examples. These results offer a theoretical foundation for public health decision-making to develop effective vaccination allocation plans.

This paper investigates a threshold control discrete population model with Allee effects, characterized by density-dependent growth functions separated at a critical population threshold. The model captures diverse ecological scenarios through simple switching mechanisms while maintaining biological realism. We overcome analytical challenges in piecewise systems by developing a complete classification of five distinct dynamics, revealing how critical transitions emerge at specific parameter boundaries. One key theoretical contribution identifies the precise conditions generating persistent oscillations, a counterintuitive result demonstrating how discontinuous switching can sustain periodic behavior despite monotonic growth functions. These findings provide actionable conservation strategies, including extinction prevention protocols and sustainable harvesting policies. The framework offers both theoretical advances in piecewise dynamical systems and practical tools for ecological management, with potential applications in species conservation and ecosystem restoration.