Bifurcation Parameter Research Articles

Stability of a delayed prey–predator model with fear, stochastic effects and Beddington–DeAngelis functional response

In view of the importance of predator-dependent functional response and fear of prey induced by powerful predators, we construct a delayed prey–predator model with fear and Beddington–DeAngelis functional response. The existence, uniqueness, and global asymptotic stability of equilibrium points are investigated and some criteria are established. Next, Hopf bifurcation analysis is executed, and the critical values of such bifurcation parameters as fear and delay for the determinate system are obtained. Then we extend it to a random environment and study the boundedness of expectation of solutions and the global asymptotic stability. Finally, the main findings are validated by numerical examples. It is worth noting that the specific influences of fear by predator, time delay, and white noise are explored numerically. Simulation figures intuitively exhibit that fear, delay, and white noise bring serious influences on the stability of the system. Fear from predator leads to a lower equilibrium state of prey and predator, and it can change the system stability from unstable to stable after exceeding a certain critical value. The time delay has a significant impact on the system stability by producing Hopf bifurcations accompanied by limit cycles, and even lead to multiple stabilities. Larger white noise can change the system stability from stable to unstable.

Stability and Hopf Bifurcation Analysis of Fractional Double-Delay Prey–Predator Model Under PDα Control

A fractional-order proportional-derivative controller is designed to address bifurcation issues in a dual-time-delay fractional-order predator–prey model, achieving anti-control of Hopf bifurcation at equilibrium of the system. By selecting different delays as bifurcation parameters, the stability and Hopf bifurcation conditions of the controlled system are derived. The results show that the fractional-order, delays and control parameters play an important role in determining the stability and Hopf bifurcation of the system. By selecting reasonable system parameters (fractional-order, delays, and control parameters), effective system control strategies can be devised. Finally, the key findings of this study are verified through numerical examples. The research in this paper has potential application value for the management of natural ecosystems.

Parametric representation, asymptotic and bifurcation analyses of the electronic plasma oscillations

Parametric representation, asymptotic and bifurcation analyses of the electronic plasma oscillations

Analysis of the Influence of Brood Deaths on Honeybee Population

Many mathematical models using ordinary differential equations (ODEs) have been used to investigate what type of stressors cause honeybee colonies collapse. We propose a simple model of a delayed differential equation system (DDE) to describe the effect of insecticides over brood death rate and its influence over honeybee population dynamics. First, we remember some basic facts for the model with no delay. To analyze our model, we study the equilibria and perform stability and sensitivity analysis of the DDE system. Next, by using the delay time τ as a bifurcation parameter, we find that no Hopf bifurcation could arise as the time lag τ varies within biologically plausible ranges. Numerical simulations with real data are studied for the biological significance of the model.

The Dynamical Behaviors of a Fractional-Order Malware Propagation Model in Information Networks

With the rapid development of communication and information technology, information networks have become increasingly common in our work and daily lives. The aim of this paper is to study the process of introducing a new fractional-order malware propagation model into information networks. Based on the matrix theory of eigenvalues, the conditions of local stability for the above model are discussed. In addition, taking time delay as the bifurcation parameter, Hopf bifurcation is considered. The correctness of our theoretical results is verified through numerical experiments.

Finding critical exponents and parameter space for a family of dissipative two-dimensional mappings.

A family of dissipative two-dimensional nonlinear mappings is considered. The mapping is described by the angle and action variables and parameterized by ε controlling nonlinearity, δ controlling the amount of dissipation, and an exponent γ is a dynamic free parameter that enables a connection with various distinct dynamic systems. The Lyapunov exponents are obtained for different values of the control parameters to characterize the chaotic attractors. We investigated the time evolution for the stationary state at period-doubling bifurcations. The convergence to the stationary state is made using a robust homogeneous and generalized function at the bifurcation parameter, which leads us to obtain a set of universal critical exponents. The parameter space of the mapping is investigated, and tangent, period-doubling, pitchfork, and cusp bifurcations are found, and a street of saddle-area and spring-area structures is observed.

Hemodynamic effects of bifurcation and stenosis geometry on carotid arteries with different degrees of stenosis

Objective.Carotid artery stenosis (CAS) is a key factor in pathological conditions, such as thrombosis, which is closely linked to hemodynamic parameters. Existing research often focuses on analyzing the influence of geometric characteristics at the stenosis site, making it difficult to predict the effects of overall vascular geometry on hemodynamic parameters. The objective of this study is to comprehensively examine the influence of geometric morphology at different degrees of CAS and at bifurcation sites on hemodynamic parameters.Approach.A three-dimensional model is established using computed tomography angiography images, and eight geometric parameters of each patient are measured by MIMICS. Then, computational fluid dynamics is utilized to investigate 60 patients with varying degrees of stenosis (10%-95%). Time and grid tests are conducted to optimize settings, and results are validated through comparison with reference calculations. Subsequently, correlation analysis using SPSS is performed to examine the relationship between the eight geometric parameters and four hemodynamic parameters. In MATLAB, prediction models for the four hemodynamic parameters are developed using back propagation neural networks (BPNN) and multiple linear regression.Main results.The BPNN model significantly outperforms the multiple linear regression model, reducing mean absolute error, mean squared error, and root mean squared error by 91.7%, 93.9%, and 75.5%, respectively, and increasingR2from 19.0% to 88.0%. This greatly improves fitting accuracy and reduces errors. This study elucidates the correlation and patterns of geometric parameters of vascular stenosis and bifurcation in evaluating hemodynamic parameters of CAS.Significance.This study opens up new avenues for improving the diagnosis, treatment, and clinical management strategies of CAS.

Bifurcation and chaos in axially moving ferromagnetic thin plates with imperfections under magnetic field and line load

This paper explores the complex dynamic behavior of axially moving ferromagnetic thin plates with initial geometric imperfection, subject to a magnetic field and a line load. The equation of motion for the imperfect plate is derived using Hamilton's variational principle, based on Kirchhoff's thin plate theory and the magnetoelastic theory of ferromagnetic materials, and discretized via the Galerkin method. When system stiffness becomes negative under specific parameters, leading to structural instability, the Melnikov method is used to predict chaos analytically and to establish the necessary conditions for its onset. Based on these predictions, a chaotic distribution map is generated in the parameter space to visualize chaotic regions. Numerical simulations are performed to produce bifurcation diagrams, maximum Lyapunov exponent plots, and system response curves, providing a dynamic analysis of the system's behavior under varying bifurcation parameters. The results indicate a complex nonlinear coupling between the plate's physical and geometric properties and the external magnetic field conditions. Variations in magnetic field intensity, axial velocity, and frequency ratio result in complex nonlinear dynamics, including bifurcations and transitions between periodic and chaotic states. This study offers critical insights into the dynamic characteristics and evolution of nonlinear systems, with significant implications for controlling and predicting chaotic vibrations in engineering applications.

Coexistence or extinction: Dynamics of multiple lizard species with competition, dispersal and intraguild predation.

Biological invasions significantly impact native ecosystems, altering ecological processes and community behaviors through predation and competition. The introduction of non-native species can lead to either coexistence or extinction within local habitats. Our research develops a lizard population model that integrates aspects of competition, intraguild predation, and the dispersal behavior of intraguild prey. We analyze the model to determine the existence and stability of various ecological equilibria, uncovering the potential for bistability under certain conditions. By employing the dispersal rate as a bifurcation parameter, we reveal complex bifurcation dynamics associated with the positive equilibrium. Additionally, we conduct a two-parameter bifurcation analysis to investigate the combined impact of dispersal and intraguild predation on ecological structures. Our findings indicate that intraguild predation not only influences the movement patterns of brown anoles but also plays a crucial role in sustaining the coexistence of different lizard species in diverse habitats.

Global steady-state bifurcation of a diffusive Leslie–Gower model with both-density-dependent fear effect

This paper mainly focuses on the steady-state bifurcation at the interior positive constant steady state of a diffusive Leslie–Gower model with both-density-dependent fear effect. Taking the growth rate of the predator as the bifurcation parameter and using Crandall–Rabinowitz bifurcation theorem, we discuss the local and the global steady-state bifurcation near the homogeneous steady state, and analyze the stability and the structure of the bifurcated spatially inhomogeneous steady-state solutions. The results show that either too high or too low growth rate or fear intensity is not conducive to the spatial pattern of the species, and a moderate value is beneficial to species survival.

A dynamical systems model for the total fission rate in Drp1-dependent mitochondrial fission.

Mitochondrial hyperfission in response to cellular insult is associated with reduced energy production and programmed cell death. Thus, there is a critical need to understand the molecular mechanisms coordinating and regulating the complex process of mitochondrial fission. We develop a nonlinear dynamical systems model of dynamin related protein one (Drp1)-dependent mitochondrial fission and use it to identify parameters which can regulate the total fission rate (TFR) as a function of time. The TFR defined from a nondimensionalization of the model undergoes a Hopf bifurcation with bifurcation parameter [Formula: see text] where [Formula: see text] is the total concentration of mitochondrial fission factor (Mff) and k+ and k- are the association and dissociation rate constants between oligomers on the outer mitochondrial membrane. The variable μ can be thought of as the maximum build rate over the disassembling rate of oligomers. Though the nondimensionalization of the system results in four dimensionless parameters, we found the TFR and the cumulative total fission (TF) depend strongly on only one, μ. Interestingly, the cumulative TF does not monotonically increase as μ increases. Instead it increases with μ to a certain point and then begins to decrease as μ continues to increase. This non-monotone dependence on μ suggests interventions targeting k+, k-, or [Formula: see text] may have a non-intuitive impact on the fission mechanism. Thus understanding the impact of regulatory parameters, such as μ, may assist future therapeutic target selection.

Hopf bifurcations for a delayed discrete single population patch model in advective environments

Abstract In this paper, we consider a delayed discrete single population patch model in advective environments. The individuals are subject to both random and directed movements, and there is a net loss of individuals at the downstream end due to the flow into a lake. Choosing time delay as a bifurcation parameter, we show the existence of Hopf bifurcations for the model. In homogeneous non-advective environments, it is well known that the first Hopf bifurcation value is independent of the dispersal rate. In contrast, for homogeneous advective environments, the first Hopf bifurcation value depends on the dispersal rate. Moreover, we show that the first Hopf bifurcation value in advective environments is larger than that in non-advective environments if the dispersal rate is large or small, which suggests that directed movements of the individuals inhibit the occurrence of Hopf bifurcations.

The Effect of Arterial Elongation on Isolated Common Iliac Artery Pathologies.

to investigate the effects of vessel geometry on steno-occlusive and dilatative common iliac artery (CIA) pathologies. this single-center, retrospective study included 100 participants, namely 60 participants with a unilateral, isolated CIA pathology who were divided into three pathology-based groups (a stenosis group, n = 20, an occlusion group, n = 20, and an aneurysm group, n = 20) and 40 participants without a CIA pathology (control group). All participants underwent abdominal and pelvic computed tomography angiography. The aortoiliac region of the participants was reconstructed into three-dimensional models. Elongation parameters (tortuosity index (TI) and absolute average curvature (AAC)) and bifurcation parameters (iliac take-off angle, iliac planarity angle, and bifurcation angle) were determined using an in-house-written piece of software. Demographic data, anthropometric data, cardiovascular risk factor data, and medical history data were obtained from participants' electronic health records. The following statistical methods were used: one-way ANOVA, chi-square test, t-tests, Wilcoxon test, Kruskal-Wallis test, and multivariate linear regression. in the occlusion group, both TI and AAC values were significantly higher on the contralateral side than on the ipsilateral side (both p < 0.001), whereas in the aneurysm group the AAC values were significantly higher on the ipsilateral side than on the contralateral side (p = 0.001). The ipsilateral and contralateral TI and AAC values of the iliac arteries were significantly higher in the aneurysm group than in the other three groups (all p < 0.001). Age significantly affected all of the elongation parameters except for the TI of the infrarenal aorta (all p < 0.010 except the TI of the infrarenal aorta). In addition, the AAC values for the iliac arteries were significantly associated with obesity (ipsilateral iliac artery, p = 0.045; contralateral iliac artery, p = 0.047). Aortic bifurcation parameters did not differ significantly either within each group (ipsilateral versus contralateral side) or between the individual groups. occlusions tend to develop in relatively straight iliac arteries, whereas unilateral, isolated CIA aneurysms are more likely to occur in elongated aortoiliac systems.

Turing Instability and Dynamic Bifurcation for the One‐Dimensional Gray–Scott Model

ABSTRACTWe study the dynamic bifurcation of the one‐dimensional Gray–Scott model by taking the diffusion coefficient of the reactor as a bifurcation parameter. We define a parameter space of for which the Turing instability may happen. Then, we show that it really occurs below the critical number and obtain rigorous formula for the bifurcated stable patterns. When the critical eigenvalue is simple, the bifurcation leads to a continuous (resp. jump) transition for if is negative (resp. positive). We prove that when lies near the Bogdanov–Takens point . When the critical eigenvalue is double, we have a supercritical bifurcation that produces an ‐attractor . We prove that consists of four asymptotically stable static solutions, four saddle static solutions, and orbits connecting them. We also provide numerical results that illustrate the main theorems.

A stochastic mechanism causing long or short transients near a bifurcation point

When a bifurcation parameter in a deterministic model becomes a stochastic process, additional new stochastic processes may arise near a critical parameter value. By examining the sample paths of a population model with a strong Allee effect, in which the bifurcation parameter is defined as an Ornstein–Uhlenbeck (OU) process, we find that transient times before extinction may be extended or shortened according to the excursions of the OU process. We find the distributions of transient times as they depend on process parameters.

Study of functional properties of different bifurcation types of the splenic vasculature

Background. The study using numerical modeling of functional properties (conductive, distributive, pillar) of digital models of 4 types of bifurcations of the intraorgan arterial vasculature is a valuable tool to find its morphometric reference and subsequently the criterion of normality.Aim: To establish the functional properties of different types of splenic arterial bifurcations through their numerical modeling based on morphometry results.Material and Methods. Modelling was carried out on the basis of previously obtained morphometric characteristics of different types of splenic arterial bifurcations: type 1, the diameter of the parent (proximal) segment (D) is not equal to the diameters of the larger (dmax) and smaller (dmin) subsidiary branches (distal segments) D ≠ dmax ≠ dmin; type 2, D = dmax, D ≠ dmin; type 3, D ≠ dmax, dmin = dmax; type 4, D = dmax = dmin. The ANSYS Student computer software was used to calculate the values of splenic arterial bifurcation indices characterizing the conductive and support functions, and the Vasculograph computer software was used to calculate the distribution function.Results. It was found that the value of the bifurcation parameter of splenic arterial bifurcations of different types characterizing: 1) conductive function decreases in the order of type 1 complete asymmetry, type 2 lateral asymmetry, type 4 complete symmetry and type 3 unilateral symmetry; 2) the distributive function decreases in the direction of type 1 complete asymmetry, type 2 lateral asymmetry, type 3 unilateral symmetry, and type 4 complete symmetry 3) the pilar function decreases in the direction of type 1 complete asymmetry, type 2 lateral asymmetry, type 3 unilateral symmetry, and type 4 complete symmetry.Conclusion. The obtained results indicate that different types of splenic arterial bifurcations are oriented to fulfil heterogeneous functions. This should be taken into account when seeking a reference and subsequently a morphometric criterion of splenic vasculature norm, which can be used for radial diagnostics.

Data-driven bifurcation analysis using parameter-dependent trajectories

Identification of bifurcation diagrams in nonlinear systems is of great importance for resilient design and stability analysis of dynamical systems. Data-driven identification of bifurcation diagrams has a significant advantage for large dimensional systems where analysis of the equations is not possible, and for experimental systems where accurate system equations are not available. In this work, a novel forecasting approach to predict bifurcation diagrams in nonlinear systems is presented using system trajectories before instabilities occur. Unlike previous techniques, the proposed method considers a varying bifurcation parameter during the system response to perturbations. Combined with an asymptotic analysis provided by the method of multiple scales eliminates the requirement of using multiple measurements and allows the novel technique to predict the bifurcation diagram using a single system recovery. The proposed approach allows stability analyses of nonlinear systems with limited access to experimental or surrogate data. The novel technique is demonstrated through its application to a Hopf bifurcation, highlighting its inherent advantages. Subsequently, the method is employed in the analysis of an aeroelastic system that shows both supercritical and subcritical Hopf bifurcations. The findings reveal great accuracy, achieved with a reduced number of measurements, while enhancing versatility.

A new chemical networked system: spatial-temporal evolution and control

Abstract This paper constructs a scale-free chemical network based on the Gierer-Meinhardt (GM) system and investigates its Turing instability. We establish a fractional-order single-node GM system with delay and design a fractional-order proportional derivative (PD) control strategy for the issue of bifurcation control. Using delay as bifurcation parameter, the existence of Hopf bifurcation is proven, and control over bifurcation threshold points is achieved through a fractional-order PD control strategy. For the scale-free chemical network based on the GM system, we obtain the condition of how the Turing instability occurs. We derive how the number of edges for the new nodes changes the stability of the network-organized system and investigate the relationship between degrees of nodes and eigenvalues of the network matrix. We give the instability condition caused by diffusion in the network-organized system. Finally, the numerical simulations verify analytical results.

Hemodynamic characteristics of pulsatile blood flow through bifurcated stenosed carotid artery

PurposeCarotid artery is often associated with plaque deposition because of its shape and associated flow features. The shape of stenosed bifurcation is characterised by bifurcation angle (ß), planarity angle (α) and severity of stenosis (b). In the present work, three-dimensional numerical computations have been performed to analyse the effect of these geometrical parameters of carotid bifurcation on the characteristics of flow.Design/methodology/approachGoverning equations of this study were solved using ANSYS Fluent 20.1 and the blood flow was considered as laminar, pulsatile and non-Newtonian. Instantaneous flow behaviour has been illustrated using vorticity, velocity and helicity contours, whereas the time-averaged wall shear stress (τw¯) and oscillatory shear index (OSI) quantify the time-averaged behaviour.FindingsThe recirculation zone and secondary flow are ascertained to be stronger for higher bifurcation angle as compared to the lower bifurcation angle. Strength of the secondary flow is found to reduce with increase in α from 0° to 10°, whereas it grows as α varies from 10° to 20°. For higher bifurcation angles, τw¯ is lower than 2 Pa and OSI is greater than 0.2 on the outer walls. Similar observations were made for τw¯ and OSI distribution on bottom wall in non-planar cases, which predicted atherogenic locations.Originality/valueThe values for ß were taken as 30°, 45°, 60° and 75°, whereas for α, range of 0°–20° was chosen. The stenosis was considered on the outer wall of internal carotid artery and its severity was considered within the range of 0%–60%.

Mathematical insights into the influence of delay and external recruitment on coral-macroalgae system

As anthropogenic pressures on coral reef ecosystems continue to increase due to global warming and mass coral bleaching events, there is growing interest in developing conservation strategies to restore degraded coral reefs. One such approach is the transplantation of coral larvae onto degraded reefs. In this paper, we use a delayed coral-macroalgae model to explore the effects of external recruitment of coral. By analyzing the local and global stability of macroalgae-free equilibrium and coexistence equilibrium in the system, we find that sustained external recruitment of coral will favor coral competition with macroalgae. In addition, we choose delay as a bifurcation parameter and demonstrate that Hopf bifurcation may occur at a critical delay near the coexistence equilibrium. Interestingly, the delay may cause the globally asymptotically stable equilibrium in the ODE system to become unstable, resulting in the appearance of periodic solutions. Furthermore, we analyze population dynamics using optimal control theory and determine the effect of minimum external recruitment on the population dynamics. In the numerical simulation section, parameters of coral-macroalgae dynamics are estimated by using a 12-year (2005–2017) benthic cover dataset of coral reefs in the Gulf of Mannar, southeastern India. The theoretical results are validated and supported by numerical simulations.