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A diffusive Lotka–Volterra competition system with memory-based diffusion is analyzed. The stability of the boundary steady states depends on memory-based self-diffusion but is independent of memory-based cross-diffusion. However, memory-based cross-diffusion induces instability in the constant coexistence state, increasing the complexity of the spatiotemporal dynamics of the system. We establish the critical conditions under which the constant coexistence state loses its stability through Turing bifurcation, Hopf bifurcation, and double-Hopf bifurcation. The normal form of the double-Hopf bifurcation is also derived. This allows us to prove the existence of an unstable spatially inhomogeneous quasi-periodic solution, and the coexistence of two types of stable spatially inhomogeneous periodic solutions with different wave numbers. In particular, a repulsive memory-based diffusion induces competing species to coexist in spatially segregated steady-state patterns, and an attractive memory-based diffusion can induce competing species to coexist in spatially segregated time-periodic patterns. [ABSTRACT FROM AUTHOR]

This study establishes a novel time-delay fractional SEIHR infectious disease model to investigate the effects of saturated incidence rates and time delays on different populations, including susceptibles, infected individuals, recovered individuals, and latent infected individuals. First, the existence and boundedness of the model's solutions are verified, confirming its well-posedness. Subsequently, the existence of equilibria is analyzed, and the impact of parameter variations on the system is explored by examining the equilibria ϵ 0 and ϵ ∗ , as well as the basic reproduction number R 0 . Additionally, the global dynamics of the equilibria are further analyzed using the Lyapunov method, while Hopf bifurcation theory is applied to explore the conditions under which the system shifts from stability to oscillatory behavior. Numerical simulations further validate the theoretical analysis, showing that time-delay effects significantly influence the system's responsiveness and the rate of disease transmission. Moreover, when the time delay τ crosses the critical threshold τ 0 , the system exhibits periodic oscillations. By predicting periodic fluctuations and incorporating memory effects and persistent influences, we can better control epidemics, emphasizing the importance of time-delay adjustments and enhancing the system's biological realism. [ABSTRACT FROM AUTHOR]

In this paper, we introduce spatial memory into a predator–prey model with herd behavior. Taking memory-based diffusion coefficient and average memory period of predators as control parameters, we obtain the stable conditions of the positive equilibrium of system and prove the existence of Hopf bifurcation. In addition, a double Hopf bifurcation occurs at the intersection of the nonhomogeneous Hopf bifurcation curves, and a spatially nonhomogeneous quasi-periodic pattern can be observed near the double Hopf bifurcation point by numerical simulation. [ABSTRACT FROM AUTHOR]

This study investigates a symmetric fractional-order epidemic model with time delays and non-monotonic incidence rates, considering two viral strains. By confirming the existence, uniqueness, and boundedness of the system's solutions, the research ensures the model's well-posedness, guaranteeing its mathematical soundness and practical relevance. The study calculates and evaluates the equilibrium points and the basic reproduction numbers R 0 1 and R 0 2 to understand the dynamic behavior of the model under different parameter settings. Through the application of the Lyapunov method, the research examines the asymptotic global stability of the system, determining whether it will converge to a particular equilibrium state over time. Furthermore, Hopf bifurcation theory is employed to investigate potential periodic solutions and bifurcation scenarios, highlighting how the system might shift from stability to periodic oscillations under certain conditions. By utilizing the Adams-Bashforth-Moulton numerical simulation method, the theoretical results are validated, reinforcing the conclusions and demonstrating the model's applicability in real-world contexts. It emphasizes the importance of fractional-order models in addressing epidemiological issues related to time delays (τ), individual heterogeneity (m, k), and memory effects (θ), offering greater accuracy compared with traditional integer-order models. In summary, this research provides a theoretical foundation and practical insights, enhancing the understanding and management of epidemic dynamics through fractional-order epidemic models. [ABSTRACT FROM AUTHOR]

In this research, we introduce and analyze a mathematical model for online social networks, incorporating two distinct delays. These delays represent the time it takes for active users within the network to begin disengaging, either with or without contacting non-users of online social platforms. We focus particularly on the user prevailing equilibrium (UPE), denoted as P * , and explore the role of delays as parameters in triggering Hopf bifurcations. In doing so, we find the conditions under which Hopf bifurcations occur, then establish stable regions based on the two delays. Furthermore, we delineate the boundaries of stability regions wherein bifurcations transpire as the delays cross these thresholds. We present numerical simulations to illustrate and validate our theoretical findings. Through this interdisciplinary approach, we aim to deepen our understanding of the dynamics inherent in online social networks. [ABSTRACT FROM AUTHOR]

Spatial memory is key in animal movement modeling, but it has been challenging to explicitly model learning to describe memory acquisition. In this paper, we study novel cognitive consumer-resource models with different consumer learning mechanisms and investigate their dynamics. These models consist of two PDEs in composition with one ODE such that the spectrum of the corresponding linearized operator at a constant steady state is unclear. We describe the spectra of the linearized operators and analyze the eigenvalue problems to determine the stability of the constant steady states. We then perform bifurcation analysis by taking the perceptual diffusion rate as the bifurcation parameter. It is found that steady-state and Hopf bifurcations can both occur in these systems, and the bifurcation points are given so that the stability region can be determined. Moreover, rich spatial and spatiotemporal patterns can be generated in such systems via different types of bifurcation. Our effort establishes a new approach to tackling a hybrid model of PDE-ODE composition and provides a deeper understanding of cognitive movement-driven consumer-resource dynamics. [ABSTRACT FROM AUTHOR]

In this paper we present a general approach to rigorously validate Hopf bifurcations as well as saddle-node bifurcations of periodic orbits in systems of ODEs. By a combination of analytic estimates and computer-assisted calculations, we follow solution curves of cycles through folds, checking along the way that a single nondegenerate saddle-node bifurcation occurs. Similarly, we rigorously continue solution curves of cycles starting from their onset at a Hopf bifurcation. We use a blowup analysis to regularize the continuation problem near the Hopf bifurcation point. This extends the applicability of validated continuation methods to the mathematically rigorous computational study of bifurcation problems. [ABSTRACT FROM AUTHOR]

This paper is concerned with the spatial patterns of a predator–prey system with Beddington–DeAngelis functional response, in which the parameter k measuring the mutual interference between predators can play an essential role. By using the bifurcation theory and implicit function theorem we first consider the positive steady state solution bifurcating from the semitrivial steady state solution set of the system and prove that the positive steady state solution is constant. Then we show that nonconstant positive steady state solution may bifurcate from the constant positive steady state solution when k is neither small nor large. Finally, we show that spatially nonhomogeneous periodic orbits may also bifurcate from the constant positive steady state solution as k is not large. [ABSTRACT FROM AUTHOR]

We provide a very brief review of the description of colored invariants for the Hopf link in terms of characters, which need to be taken at a peculiar deformation of the topological locus, depending on one of the two representations associated with the two components of the link. Most important, we extend the description of this locus to conjugate and, generically, to composite representations and also define the “adjoint” Schur functions emerging in the dual description. [ABSTRACT FROM AUTHOR]

The stabilization height of a fiber surface in the 3-sphere is the minimal number of Hopf plumbing operations needed to attain a stable fiber surface from the initial surface. We show that families of fiber surfaces related by iterated Stallings twists have unbounded stabilization height. [ABSTRACT FROM AUTHOR]

This paper analyzes the existence of Hopf bifurcation and establishes the conditions under which the equilibrium path converges toward periodic solutions in a one-sector optimal growth model with delay. We establish the limits and the possibilities of nonlinear dynamics (i.e., cycles) vis-à-vis delays. In particular, we formulate a new method to further comprehend the root distribution of the characteristic equation of a standard optimal growth model with delayed investment structure. We show that nonmonotonic dynamics (limit cycles, persistent oscillations) occurs when the delayed investment causes permanent adjustment failures among the economic variables in the economy. [ABSTRACT FROM AUTHOR]

Neural field equations describe the activity of neural populations at a mesoscopic level. Although the early derivation of these equations introduced space dependent delays coming from the finite speed of signal propagation along axons, there have been few studies concerning their role in shaping the (nonlinear) dynamics of neural activity. This is mainly due to the lack of analytical tractable models. On the other hand, constant delays have been introduced to model the synaptic transmission and the spike initiation dynamics. By incorporating the two kinds of delays into the neural field equations, we are able to find the Hopf bifurcation curves analytically, which produces many Hopf--Hopf interactions. We use normal theory to study two different types of connectivity that reveal a surprisingly rich dynamical portrait. In particular, the shape of the connectivity strongly influences the spatiotemporal dynamics. [ABSTRACT FROM AUTHOR]

We study networks of delay-coupled oscillators with the aim to extend time-delayed feedback control to networks. We show that unstable periodic orbits of a network can be stabilized by a noninvasive, delayed coupling. We state criteria for stabilizing the orbits by delay-coupling in networks and apply these to the case where the local dynamics is close to a subcritical Hopf bifurcation, which is representative of systems with torsion-free unstable periodic orbits. Using the multiple scale method and the master stability function approach, the network system is reduced to the normal form, and the characteristic equations for Floquet exponents are derived in an analytical form, which reveals the coupling parameters for successful stabilization. Finally, we illustrate the results by numerical simulations of the Lorenz system close to a subcritical Hopf bifurcation. The unstable periodic orbits in this system have no torsion, and hence cannot be stabilized by the conventional time delayed-feedback technique. [ABSTRACT FROM AUTHOR]