Double Hopf Bifurcation Research Articles

Bifurcation in an modified model of neutrophil cells with time delay

The hematological stem cells model is introduced with neutrophil dynamics of two department model setting forth. During the cells differentiation and proliferation process, the neutrophils are functioned with negative feedback with delay history, which contains delayed amplification coefficient. In more general view, the new introduction rate is given to replace the familiar Hill function which is helpful to understand the complex dynamics of neutrophils. The double Hopf bifurcation is calculated with the artificial handtools named DDE-Biftool, which is observed as the self-intersection of Hopf lines. The continuation of periodical solutions arising from Hopf points are done and the longer period solutions are manifested with multi-rhythm and bursting oscillation. The near dynamics of double Hopf points is simulated by DDE-Biftool with different route design, the multi-period attractors, quasi-periodical solutions and chaos are observed.

A comprehensive study of spatial spread and multiple time delay in an eco-epidemiological model with infected prey

This paper studies the dynamics of interacting Tilapia fish and Pelican bird population in the Salton Sea. We assume that the diseases spread in Tilapia fish follows the Holling type II response function, and the interaction between Tilapia and Pelican follows the Beddington–DeAngelis response function. The dynamics of diffusive and delayed system are discussed separately. Analytically, all the feasible equilibria and their stability are discussed. The criterion for Turing instability is derived. Based on the normal form theory and center manifold arguments, the existence of stability criterion and the direction of Hopf bifurcation are obtained. Numerical simulation shows the occurrence Hopf bifurcation, double Hopf bifurcation and transcritical bifurcation scenarios. The snap shot shows the spot, spot-strip mix patterns in the whole domain. Further, the stability switching phenomena is observed in the delayed system. Our comprehensive study highlights the effect of different parameters, multiple time delay and extinction in Pelican populations.

Delay-induced nutrient recycling in plankton system: Application to Sundarban mangrove wetland

The paper discusses the nutrient–plankton system with effect of time delay in nutrient recycling and toxin determined function response (TDFR). The designed model system explores the delay-induced system dynamics. We present the local stability analysis of interior equilibrium points in absence as well as in presence of time delay. Further, the direction of Hopf bifurcation is obtained. We perform the numerical computation and observe that time delay in nutrient recycling can generate the periodic solution in a stable nutrient–plankton system. Some other essential parameters, such as input concentration of nutrients and natural removal rate of nutrients, also regulate the dynamical system. The system shows Hopf and double-Hopf bifurcation in the presence of time delay. Our study shows that the delay in the nutrient recycling causes instability transition phenomenon. The delay-induced nutrient recycling and different input concentrations of nutrients can regulate the estuarine system. Finally, the stability switching is observed for delayed system.

Spatiotemporal patterns in a diffusive resource–consumer model with distributed memory and maturation delay

In this paper, we propose a diffusive consumer–resource model in which the consumer is involved with spatiotemporal memory and the resource has maturation delay. Firstly, for the case without maturation delay, the influence of the spatiotemporal distributed memory on the stability of the positive steady state is investigated. It has been shown that memory delay can induce Turing bifurcation for the negative memory-based diffusion coefficient and induce Hopf bifurcation for the positive memory-based diffusion coefficient. Then the joint effect of the memory delay and the maturation delay on the stability of the positive steady state is investigated and it has been shown that the joint effect of the memory delay and the maturation delay can induce more complicated spatiotemporal dynamics via Turing–Hopf bifurcation and double Hopf bifurcation. Finally, we apply our theoretical analysis results to a consumer–resource model with Holling-II type functional response to illustrate the existence of the different spatiotemporal patterns.

Double Hopf bifurcation analysis of time delay coupled active control system

In this paper, the stability analysis of double Hopf bifurcation is carried out based on the time delay coupled active control system by using the related theory of bifurcation analysis. We got the stability switching region of the system with respect to time delay. The time delay and coupling strength are selected as the bifurcation parameters, then the normal form of the coupled time delay active control system is obtained by using the multi-scale method, and the dynamic behavior near the bifurcation point is obtained. So the parameter region of the stable periodic solution and the stable quasi-periodic solution are obtained. The analytical solution of the system is then solved by using the multi-frequency homotopy analysis method (MFHAM) to verify the existence of the unstable periodic solution. Finally, numerical simulation is executed to verify the theoretical results.

Bifurcations and pattern formation in a host–parasitoid model with nonlocal effect

In this paper, we analyse Turing instability and bifurcations in a host–parasitoid model with nonlocal effect. For a ordinary differential equation model, we provide some preliminary analysis on Hopf bifurcation. For a reaction–diffusion model with local intraspecific prey competition, we first explore the Turing instability of spatially homogeneous steady states. Next, we show that the model can undergo Hopf bifurcation and Turing–Hopf bifurcation, and find that a pair of spatially nonhomogeneous periodic solutions is stable for a (8,0)-mode Turing–Hopf bifurcation and unstable for a (3,0)-mode Turing–Hopf bifurcation. For a reaction–diffusion model with nonlocal intraspecific prey competition, we study the existence of the Hopf bifurcation, double-Hopf bifurcation, Turing bifurcation, and Turing–Hopf bifurcation successively, and find that a spatially nonhomogeneous quasi-periodic solution is unstable for a (0,1)-mode double-Hopf bifurcation. Our results indicate that the model exhibits complex pattern formations, including transient states, monostability, bistability, and tristability. Finally, numerical simulations are provided to illustrate complex dynamics and verify our theoretical results.

Double Hopf bifurcation induced by spatial memory in a diffusive predator–prey model with Allee effect and maturation delay of predator

In this paper, we delve into double Hopf bifurcation induced by memory-driven directed movement in a spatial predator–prey model with Allee effect and maturation delay of predators. We first adopt a novel technique to handle the associated characteristic equation and thus obtain the crossing curves as well as the double Hopf points. We then calculate explicit formulae of normal form regarding non-resonant double Hopf bifurcation. We thus divide the dynamics of the developed model into several categories near the double Hopf bifurcation points. Our numerical and theoretical results both demonstrate that the model can exhibit various complex phenomena when the parameters are near the double Hopf bifurcation points. For example, the transition from one stable spatially inhomogeneous periodic orbit with mode-5 to another with mode-4 and the coexistence of them can be observed.

Global bifurcation in a general Leslie–Gower type predator[formula omitted]prey system with indirect prey-taxis

The influence of the indirect prey-taxis on the spatiotemporal dynamics of a class predator−prey system with predator functional response is studied. By analyzing the corresponding characteristic equation, the local stability of the positive equilibrium and the related bifurcations of the system are discussed. The critical values of the indirect prey-taxis coefficient leading to the Hopf bifurcation, steady state bifurcation, Turing-Turing bifurcation and double-Hopf bifurcation are derived. Taking the indirect prey-taxis coefficient as the bifurcation parameter, we obtain the global structure of nonconstant steady states bifurcating from the positive equilibrium by an abstract bifurcation theorem. Moreover, the stability condition of bifurcation solutions is carried out. Our results suggest that attractive indirect prey-taxis can destabilize the positive equilibrium and induced the emergence of spatially inhomogeneous periodic solution. And the higher the secretion level of chemoattractant by the prey, the more likely the system is to exhibit spatial patterns.

Nonlinear dynamics near a double Hopf bifurcation for a ship model with time-delay control

Nonlinear dynamics near a double Hopf bifurcation for a ship model with time-delay control

Double-Hopf bifurcation and Pattern Formation of a Gause-Kolmogorov-Type system with indirect prey-taxis and direct predator-taxis

This paper focuses on a general Gause-Kolmogorov-Type predator–prey model with direct predator-taxis and indirect prey-taxis. The global existence and boundedness of solutions to the system are proved. Our approach is applicable to more general situation. Some measures of success for the constant coexistence steady state are rigorously calculated, such as decreasing the indirect prey-taxis sensitivity or the release rate of stimulus, and increasing the predator-taxis sensitivity or the decay rate of stimulus. By choosing the indirect prey-taxis coefficient as the bifurcation parameter, the existence of Hopf and double-Hopf bifurcations is established. We find that both indirect prey-taxis and direct predator-taxis can promote the complexity of the spatiotemporal pattern, e.g. spatially nonhomogeneous periodic patterns with different spatial frequency, spatially nonhomogeneous quasi-periodic patterns. Finally, we apply our theoretical analyses to a Rosenzweig–MacArthur model with indirect prey-taxis and direct predator-taxis, spatially homogeneous periodic patterns.

Effect of spatial memory on a predator–prey model with herd behavior

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.

Double Hopf bifurcation induced by gestation and memory delays in a spatial model with directed movement

We narrow our focus on double Hopf bifurcations triggered by gestation and memory delays in a spatial model with tactical directed movement in this letter. We generalize the calculation algorithm of normal form concerning the double Hopf bifurcation to a spatial model with different delays in both the reaction and advection terms. We then divide the dynamics around the double Hopf bifurcation points into 6 categories and thus we observe that stable periodic orbits with different spatial modes can coexist.

Bifurcation and optimal control for an infectious disease model with the impact of information

A nonlinear infectious disease model with information-influenced vaccination behavior and contact patterns is proposed in this paper, and the impact of information related to disease prevalence on increasing vaccination coverage and reducing disease incidence during the outbreak is considered. First, we perform the analysis for the existence of equilibria and the stability properties of the proposed model. In particular, the geometric approach is used to obtain the sufficient condition which guarantees the global asymptotic stability of the unique endemic equilibrium [Formula: see text] when the basic reproduction number [Formula: see text]. Second, mathematical derivation combined with numerical simulation shows the existence of the double Hopf bifurcation around [Formula: see text]. Third, based on the numerical results, it is shown that the information coverage and the average information delay may lead to more complex dynamical behaviors. Finally, the optimal control problem is established with information-influenced vaccination and treatment as control variables. The corresponding optimal paths are obtained analytically by using Pontryagin’s maximum principle, and the applicability and validity of virous intervention strategies for the proposed controls are presented by numerical experiments.

Spatiotemporal dynamics of a three-component chemotaxis model for Alopecia Areata

In this paper, we investigate the influence of two indirect chemotaxis terms χ1,χ2 on the spatiotemporal dynamics of a three-component model for Alopecia areata. The stability of the unique positive equilibrium and bifurcations are discussed by choosing the bifurcating parameters χ1,χ2. When χ1=0, χ2 can lead to complex spatial pattern via Turing bifurcation. And fixing χ2 in a certain region, χ1 can lead to spatially nonhomogeneous periodic solutions via Hopf bifurcation. It is shown that there exist a codimension-2 Turing–Turing bifurcation due to the joint effect of χ2 and proliferation rate and Double-Hopf bifurcation due to nonhomogeneous Hopf bifurcations with different modes induced by χ1 and proliferation rate.

Double Hopf bifurcation in nonlocal reaction-diffusion systems with spatial average kernel

<p style='text-indent:20px;'>In this paper, we consider a general reaction-diffusion system with nonlocal effects and Neumann boundary conditions, where a spatial average kernel is chosen to be the nonlocal kernel. By virtue of the center manifold reduction technique and normal form theory, we present a new algorithm for computing normal forms associated with the codimension-two double Hopf bifurcation. The theoretical results are applied to a predator-prey model, and complex dynamic behaviors such as spatially nonhomogeneous periodic oscillations and spatially nonhomogeneous quasi-periodic oscillations could occur.</p>

Dynamics of an HIV infection model with two time delays

In this paper, a mathematical model for HIV-1 infection and immune response is considered, involving two discrete time delays in the intracellular as well as in activation of immune response. Using a recently developed geometric method for studying a class of transcendental equation with two time delays and delay dependent coefficients, we obtain the stability and bifurcation results at the non-trivial equilibrium. In particular, the crossing curves on the two-delays parameter plane can be completely characterized, on which Hopf and double-Hopf bifurcation will take place. In the case of Hopf bifurcation, there exist stability switches, and the direction and stability of delay induced Hopf-bifurcation can be determined using normal form theory and center manifold theorem. These results imply the model will exhibit complex temporal dynamics, such as period oscillations, quasi-periodic solutions, etc. Numerical examples are also carried out to verify these results.

HOPF BIFURCATION AND CHAOS OF COMBINATIONAL IMMUNE ANTI-TUMOR MODEL WITH DOUBLE DELAYS

In order to investigate the relations between tumor species growth and T cell activation assisted by dendritic cells, we establish a combinational immune anti-tumor model with double delays. Taking the activation rate of T cell and two time delays of tumor species growth and dendritic cell activation as parameters, we investigate the dynamical properties of the double delayed model, including the stability switches and the Hopf bifurcations of tumor-escape equilibrium and tumor-present equilibrium. With Hopf bifurcation curves, the center manifold theory and the normal form method, we find bi-stability states, the coexistence of two periodic solutions with different stabilities, two double Hopf bifurcation points, and use numerical simulations to show rich dynamic behaviors around the double Hopf bifurcation points, including the phase portraits and the corresponding Poincaré maps of chaotic attractors, as well as the progress transmission of unstable-oscillation-stable-oscillation. The theoretical and numerical results reveal the new methods of controlling tumor cells.

Dynamic Analysis of a New Financial System with Diffusion Effect and Two Delays

The dynamic behavior of a new financial system with diffusion effect and two delays is studied. The local stability of equilibrium is studied by analyzing the corresponding characteristic equation and developing the crossing curve method. Hopf bifurcation curves are obtained by three procedures and the critical value of delay of Hopf bifurcation is obtained. By using the central manifold theorem and the normal form theory of partial differential equations, the normal form of Hopf bifurcation with two delays is derived, and the bifurcation direction and the stability of periodic solution are determined. The numerical simulations not only verify these theoretical results, but also show that double Hopf bifurcation may occur and the two delays play a key role in chaos suppression. This study has important theoretical value of innovation and practical value in the macro-economic system.

Thermal convection in rotating spherical geometry: A numerical overview of the transitions from periodic axisymmetric to temporally complex three-dimensional flows.

The aim of this work is to elucidate the type of transitions that take place when the periodic axisymmetric flows, which can set up at the onset of thermal convection in rotating fluid spheres, lose stability and to study the behavior of the new stable velocity fields until the flows become temporally chaotic. The computations for Prandtl numbers Pr=0.715, 0.1, and 0.01 show that when it decreases, the range of stability of these flows becomes narrower because the kinetic energy of the axisymmetric periodic solutions increases very fast, favoring their instability. From the stability analysis and direct three-dimensional simulations it is found that the transition to stable quasiperiodic flows through Neimark-Sacker bifurcations is supercritical when Pr≥0.01. For Pr=0.1 two branches of stable periodic flows emerging from the conduction state have been found due to the proximity to a double Hopf bifurcation. However, only the branches bifurcating from the azimuthal rotating waves are stable at large Rayleigh numbers. Far from this bifurcation the stable flows keep the influence of the axisymmetric dynamics up to large Rayleigh numbers. For small Pr they behave as repeated transients of mixed dynamics, controlled by the azimuthal wave numbers m=0,m=1, and m=2.

Stability and bifurcation analysis in a diffusive predator–prey model with delay and spatial average

In this paper, we study the effect of spatial average and time delay on the dynamics of a diffusive predator–prey model under the Neumann boundary condition. Compared to the model without spatial average, the delay‐induced Hopf bifurcation at the first critical value of delay is nonhomogeneous due to the joint effects of spatial average and delay, and spatiotemporal patterns arise. Also, we show that the spatially homogeneous and nonhomogeneous periodic patterns may switch for different bifurcation parameter values. Moreover, a double Hopf bifurcation occurs at the intersection point of the homogeneous and nonhomogeneous Hopf bifurcation curves, and spatially nonhomogeneous quasi‐periodic patterns can be observed via numerical simulations near the double Hopf bifurcation point. The normal form algorithm for the spatially nonhomogeneous/homogeneous Hopf bifurcation is derived for a general reaction‐diffusion system with spatial average and delay.