Abstract This paper deals with an age-structured HIV model with antiretrowviral therapy and two infection routes (virus-to-cell and cell-to-cell). The model is first formulated as an abstract non-densely defined Cauchy problem and the existence of the equilibria is obtained under some conditions. Based on the existence of equilibria, the global asymptotical stability of the disease-free equilibrium is studied by applying theory of operator semigroups and spectral analysis. The stability and Hopf bifurcation results with two delays around the endemic equilibrium are also well described under some conditions by the geometric stability switch criteria. Finally some numerical examples are presented to illustrate the obtained results.

Abstract We study the the asymptotic dynamics of elementary cellular automaton 18 through its limit set, generic limit set and $\mu $ -limit set. The dynamics of rule 18 are characterized by persistent local patterns known as kinks. We characterize the configurations of the generic limit set containing at most two kinks. As a corollary, we show that the three limit sets of rule 18 are distinct.

Scale invariance is a hallmark of criticality in complex dynamical systems. While random external inputs or tunable stochastic interactions are typically required to produce critical behavior, it remains unclear whether scale-invariant dynamics can emerge from purely deterministic interactions. Here, we address this question by studying the asymptotic dynamics of the logistic Game of Life (GOL), a deterministic-parameter extension of Conway’s GOL. In this system, we identify three distinct asymptotic phases separated by two fundamentally different critical points. The first critical point, associated with an unusual form of self-organized criticality, separates a sparse-static phase from a sparse-dynamic phase. The second critical point corresponds to a deterministic percolation transition between the sparse-dynamic phase and a third, dense-dynamic phase. In addition, we observe power-law cluster size distributions with unconventional critical exponents not found in standard equilibrium systems. Overall, our work paves the way for studying emergent scale invariance in purely deterministic systems. Criticality and percolation in dynamical systems are widely studied, yet whether they can emerge from purely deterministic interactions and control parameters remains unclear. Here, the authors reveal deterministic critical points associated with percolation and self-organized criticality in the logistic Game of Life, advancing the understanding of emergent scale invariance in deterministic systems.

Abstract A new class of regular, spherically symmetric spacetimes is introduced. These spacetimes possess a cosmological horizon and exhibit a range of asymptotic dynamics, including de Sitter behavior as a special case. The asymptotic behavior is governed by a parameter n in the range 0 ⩽ n ⩽ 1 / 2 , with the spacetime reducing exactly to de Sitter when n = 0. These metrics are solutions to the Einstein equations coupled with nonlinear electrodynamics and satisfy both the weak and dominant energy conditions. Within this framework, regular black hole solutions—encompassing magnetic and electric configurations—are constructed, and their global causal structures are analyzed. It is shown that light propagation remains regular in the magnetic case but exhibits singularities in the electric configurations. Furthermore, the nonlinear correction to the first law of black-hole thermodynamics is derived for these solutions. The analysis reveals a positive heat capacity, indicating their thermodynamic stability.

Bioeconomic systems are inherently governed by fast-slow dynamics, arising from the interplay between rapid market adjustments and slower ecological processes. In this paper, we analyze a four-dimensional fishery model that couples predator-prey dynamics with fishing effort subject to capacity constraints and market-clearing prices. Using geometric singular perturbation theory, we show that the separation of timescales leads to a split critical manifold. The system's operational mode is determined by a single dimensionless bioeconomic parameter, which acts as a structural selector between an Internally-Regulated regime and a Capacity-Saturated regime. Beyond equilibrium stability, we focus on transient behaviors by deriving a closed-form approximation for the transient response time to external shocks. This analytical metric explicitly links recovery duration to the effective net growth budget. Our results demonstrate that while the Capacity-Saturated regime may sustain a stable equilibrium, it incurs significantly larger cumulative ecological deficits and slower recovery rates following perturbations. These findings quantify the trade-off between harvest intensity and system responsiveness, offering a dynamical basis for the vulnerability of high-effort fisheries.

In this paper, we examine a homogeneous diffusive model depicting the interactions among hosts, parasites and hyperparasites. Our primary focus is on the asymptotic dynamics of the solutions in this model. For the corresponding ordinary differential equations, we have investigated both the existence and stability of equilibrium points. Furthermore, we have explored the potential for Hopf bifurcations, delving into their associated characteristics, such as the direction of bifurcation and the stability of the resulting periodic solutions. For the reaction–diffusion equations, our emphasis is on the Turing instability of the periodic solutions that arise from Hopf bifurcations. The results we obtained provide critical insights into the dynamics of the model and help in understanding the complex interactions within this biological system.

ABSTRACT This paper is devoted to the study of the asymptotic dynamics in the fractional parabolic‐elliptic Keller‐Segel system on with time‐space dependent logistic source. In [36], among others, we established a theory on the global existence, uniqueness, pointwise and uniform persistence of classical solutions to this fractional Keller‐Segel system. In this paper, we prove the existence, uniqueness, and stability of strictly positive entire solutions of this system. In summary, the results of our paper extend some of the results obtained earlier in [36].

We study the long-time dynamics of solutions to a nonlinear gradient flow associated with Problem (2.7), where trajectories are constrained to evolve on a manifold. Using energy methods and spectral properties of the Dirichlet Laplacian, we first establish global existence and precompactness of trajectories in the natural energy space. By proving a Lojasiewicz-Simon inequality for the corresponding energy functional, we deduce convergence of all global solutions to stationary equilibria. Moreover, we provide sharp convergence rates: exponential in the case of nondegenerate equilibria, and polynomial otherwise. Finally, we demonstrate the existence of a compact global attractor in \(\mathcal{V}\cap\mathcal{M}\) that captures the asymptotic behavior of all bounded trajectories. These results place the problem within the general theory of dissipative gradient systems and give a precise description of its asymptotic dynamics. For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2026/13/abstr.html

Strong information delay as a driver of epidemic waves: Mathematical modeling for drug trends and epidemic bio-preparedness.

Cold-water corals (CWCs) are vital deep-sea ecosystem engineers, yet their growth dynamics remain poorly understood. This study quantifies Desmophyllum dianthus growth in the Southern Adriatic Sea using two unplanned but ideal long-term natural experiments. The first derives from an oceanographic mooring lost at ~ 1200 m depth, allowing a rare four-year in situ assessment via high-resolution ROV footage collected in 2024. The second involves physical specimens from a mooring at ~ 500–600 m in Bari Canyon, recovered after one year. Image-based measurements showed an average linear growth rate of 8.06 ± 0.41 mm yr−1, while physical samples recorded a one-year extension of 6.5 ± 0.6 mm. These in situ growth rates exceed previous records and suggest rapid early growth consistent with asymptotic dynamics. The findings offer crucial benchmarks for natural CWCs growth and support effective conservation and restoration efforts, aligning with goals set by the EU Nature Restoration Law.

Asymptotic dynamics in systems of two coupled quadratic maps

Asymptotic relative dynamics for spacecraft on close hyperbolic trajectories

Abstract In this article, the large-time asymptotic wave dynamics of rogue curves are analytically investigated and numerically confirmed in the Davey–Stewartson (DS) I equation. We show that, when time in bilinear expressions of the rogue curves is large, a certain number of localized lump-shaped waves would arise on the uniform background, exhibiting various wave patterns. We further show that, as time increases, the individual lump-shaped wave asymptotically evolves into a line soliton on the constant background that persist at large time. By performing large-time asymptotic analysis, we reveal that such wave patterns as well as the numbers of lump-shaped waves can be analytically determined by the structure of nonzero roots of the Wronskian-Hermite polynomials. Our asymptotic predictions are compared to true solutions quantitatively and excellent agreement is obtained.

The scale-free nature of gravitational interaction in both Newtonian gravity and the general theory of relativity gives rise to the concept of self-similarity, where solutions are scale invariant. As a result of this property, the governing partial differential equations are greatly simplified and can be transformed into ordinary ones. These solutions function as attractors, characterizing the asymptotic dynamics of more general solutions. There exist situations in which self-similarity is only partially realized, giving rise to kinematic self-similar solutions. Our study provides a systematic classification of kinematic self-similar solutions corresponding to the most general spherically symmetric spacetime in the presence of bulk viscous flows.

Abstract We present an exact analytical solution of the Hu–Paz–Zhang master equation in a precise Markovian limit for a system of two harmonically coupled harmonic oscillators interacting with a common thermal bath of harmonic oscillators. The thermal bath is initially considered to be at arbitrary temperatures and characterized by an Ohmic Lorentz–Drude spectral density. In the examined system, couplings between the two harmonic oscillators and the environment ensure a complete decoupling of the center-of-mass and relative degrees of freedom, resulting in undamped dynamics in the relative coordinate. The exact time evolution is used to analyze the system’s entanglement dynamics, quantified through logarithmic negativity and quantum mutual information, while ensuring the positivity of the density operator to confirm the physical validity of the results. We demonstrate that, under certain parameter regimes and initial conditions, the asymptotic dynamics can give rise to periodic entanglement-disentanglement behavior. Furthermore, numerical simulations reveal that for negative values of the direct coupling between the oscillators, which are sufficiently close to a critical lower bound beyond which the system becomes unstable, the system can maintain entanglement across a broad temperature range and for arbitrarily long durations.

The aim of these notes is to provide an overview of the ideas in the recent proof of global well-posedness for the massive Maxwell-Dirac system in the Lorenz gauge in ℝ 1 + 3 , for small and decaying initial data of limiting regularity. The result also includes an in-depth study of the asymptotic dynamics of the global solutions, which can be described as modified scattering. While heuristically we exploit the close connection between the massive Maxwell-Dirac and the wave-Klein-Gordon equations, for the proof of the results we develop a novel approach which applies directly at the level of the Dirac equations. The modified scattering result follows from a precise description of the asymptotic behavior of the solutions inside the light cone, which is derived via the method of testing with wave packets of Ifrim-Tataru.

In purely coherent transport on finite networks, destructive interference can significantly suppress transfer probabilities, which can only reach high values through careful fine-tuning of the evolution time or tailored initial-state preparations. We address this issue by investigating excitation transfer on a ring, modeling it as a locally monitored continuous-time chiral quantum walk. Chirality, introduced through time-reversal symmetry breaking, imparts a directional bias to the coherent dynamics and can lift dark states. Local monitoring, implemented via stroboscopic projective measurements at the target site, provides a practical detection protocol without requiring fine-tuning of the evolution time. By analyzing the interplay between chirality and measurement frequency, we identify optimal conditions for maximizing the asymptotic detection probability. The optimization of this transfer protocol relies on the spectral properties of the Perron-Frobenius operator, which capture the asymptotic nonunitary dynamics, and on the analysis of dark states. Our approach offers a general framework for enhancing quantum transport in monitored systems.

In this paper, the stability and bifurcation analysis for a Leslie–Gower model with simplified Holling IV functional response, strong Allee effect and predator cannibalism are investigated. First, we examine the existence and stability of possible equilibria, as well as the boundedness of the system. Specifically, the asymptotic dynamics near the origin is performed using a blow-up transformation. Furthermore, various bifurcations are explored, including saddle-node bifurcation, supercritical and subcritical Hopf bifurcations, Bautin bifurcation and Bogdanov–Takens bifurcations of codimensions 2 and 3. The system exhibits diverse dynamical phenomena such as the emergence of a semi-stable limit cycle, the coexistence of a homoclinic loop and a limit cycle and the coexistence of two distinct limit cycles. In particular, with a specific set of parameters, two semi-stable limit cycles appear simultaneously, indicating that the system enters a dual-critical state. Finally, numerical simulations are carried out to validate the theoretical results.

This paper investigates the asymptotic behavior of a weighted scheduled traffic process, an extension of the traditional scheduled traffic model where events are subject to random perturbations and carry variable weights. Under the assumption that the perturbations follow a heavy-tailed distribution, we demonstrate that the appropriately rescaled process converges weakly to a fractional Brownian motion. Applications of this framework span diverse fields such as queueing theory, telecommunications, finance, and healthcare, where the model provides insights into workload accumulation, network traffic variability, and transaction flow dynamics.

We obtain an inverse of Furstenberg’s correspondence principle in the setting of countable cancellative, amenable semigroups. Besides being of intrinsic interest on its own, this result allows us to answer a variety of questions concerning sets of recurrence and van der Corput (vdC) sets, which were posed by Bergelson and Lesigne [Colloq. Math. 110 (2008), pp. 1–49], Bergelson and Ferré Moragues [Israel J. Math. 245 (2021), pp. 921–962], Kelly and Lê [Arch. Math. (Basel) 110 (2018), pp. 343–349], and Moreira [ Sets of nice recurrence , I Can’t Believe It’s Not Random, https://joelmoreira.wordpress.com/2013/03/04/323/]. We also prove a spectral characterization of vdC sets and prove some of their basic properties in the context of countable amenable groups. Several results in this article were independently found by Sohail Farhangi and Robin Tucker-Drob, see [ Asymptotic dynamics on amenable groups and van der Corput sets , Preprint, arXiv: 2409.00806 , 2024].