Global Bifurcation Analysis Research Articles

Global bifurcation analysis and derivation of an optimal harvesting policy of a prey–predator model with Holling type-IV functional scheme

ABSTRACT This paper establishes a prey–predator model with a Holling type IV functional response and proportional harvesting in prey species and nonlinear harvesting in predator species. We provide a completely local and global bifurcation analysis of the model to characterize some meaningful results that may emerge from the interaction of biological resources. The proposed model exhibits various complex behaviors involving heteroclinic and homoclinic orbits, and we prove analytically that it also undergoes transcritical, Hopf, and Bogdanov–Takens bifurcations in the prey–predator plane. Moreover, the system is unfolded to compute the bifurcation curves, highlighting the critical dynamics by perturbing two bifurcation parameters near the cusp. Furthermore, from the economic point of view, the bionomic equilibrium of the system is studied, and optimal harvesting policy is derived using the Pontryagin’s maximum principle. Numerical simulations are presented to verify our analytical results.

A new model for compressor surge and stall control

This paper compares the bifurcations and closed-loop performances of two compressor models, Moore-Greitzer (MG) and a developed model based on MG (Shahriyari Khaleghi, SK). First, both models are linearized about two equilibrium points (pure surge and fully-developed rotating stall), and the perturbed state-space dynamics and input matrices are obtained. The compressor unstable regions are then identified using an eigenvalue and global bifurcation analysis. Furthermore, optimal LQR controllers are designed, and the performances of closed-loop systems are compared. The LQRs are designed to control the compression system near the peak pressure rise by suppressing surge or stall. Results reveal that if the initial operating point is in the positive slope region of the compressor characteristic and the initial amplitude of the disturbances is small, the LQR controller can stabilize the compressor in both models. However, when the disturbances are intensive, the two models respond differently: although the SK model damps a fair range of disturbances and predicts instability for excessively powerful disturbances, the MG model always damps them, even when extremely intense. Without a controller in the MG model, initial disturbances (even very large) can never grow and are always damped in the compressor’s negative slope region (obviously, the same applies to the controller). However, pending the amplitude of the disturbances (in the absence of a controller), the disturbances in the SK model may be damped or grow. The SK model can successfully control the instabilities if the disturbances are small. Nonetheless, the controller fails to dampen the instabilities for extreme disturbances, which is consistent with reality.

Global Bifurcation Analysis of Higher Codimension in a Ratio Dependent Predator-Prey Model with Prey Refuge and Constant Yield-Harvesting of Both Species

Global Bifurcation Analysis of Higher Codimension in a Ratio Dependent Predator-Prey Model with Prey Refuge and Constant Yield-Harvesting of Both Species

Global Hopf bifurcation for differential equations with multiple threshold‐type state‐dependent delays

Using the ‐equivariant degree, we develop a global Hopf bifurcation theory for system of differential equations with multiple threshold‐type state‐dependent delays whose prototype is the human respiratory system with multiple blood transport time delays. To establish a theoretic framework for modeling practices of periodic breathing, we further investigate the periodic oscillations of carbon dioxide concentrations in the lung, brain, and tissue compartments and conduct a local and global Hopf bifurcation analysis for the model when varying the commensurate scale of the multiple delays in a transformed system. Such a global Hopf bifurcation will indicate the onset and persistence of the periodic oscillations.

Global stability and bifurcation analysis of an infochemical induced three species discrete-time phytoplankton–zooplankton model

This article analyzes a discrete-time three-species model of phytoplankton and zooplankton with infochemical producing, non-infochemical producing, and zooplankton. It is assumed that the grazing of infochemical producing phytoplankton indirectly increases the mortality rate of zooplankton. The fixed points of the system and their respective local stability has been examined. The conditions for Neimark–Sacker bifurcation along with period doubling bifurcation at the positive fixed point are outlined. The sufficient conditions for global asymptotic stability of the positive fixed point are obtained. The application of a hybrid control strategy effectively stabilizes the chaotic behavior of the system. Lastly, our theoretical discussion is solidified through a numerical simulation which enables us to witness the complex behavior of the system.

Flash-back, blow-off, and symmetry breaking of premixed conical flames

Hydrodynamic and thermal-diffusive effects subject premixed flames to intrinsic instabilities that strongly influence their shape and propagation/stabilization characteristics. However, the interaction and coupling of intrinsic flame dynamics with background flow gradients and boundary conditions remain poorly understood. This paper presents a global nonlinear bifurcation analysis of burner-stabilized laminar premixed conical flames with varying reaction rates and reactant diffusivities, respectively parameterized by the Damköhler number Da and the Lewis number Le. Using a dimensionless formulation of the reacting, weakly-compressible Navier–Stokes equations, the dynamics of flash-back, blow-off, and cellular instability are explored in a fully-coupled framework. Our analysis identifies steady conical flame states over a finite range of Da and Le, limited by saddle–node bifurcations corresponding to spontaneous blow-off of the axisymmetric flame below a Le-dependent lower critical Da value and spontaneous boundary-layer flash-back beyond a higher critical Da value. Furthermore, the analysis reveals that the conical flame loses its axisymmetry via circle–pitchfork bifurcations as Le or Da decrease or increase beyond respective critical values. These bifurcations are shown to correspond to stationary three-dimensional global modes describing steady polyhedral or tilted flame structures, each associated with distinct azimuthal periodicities. Statement of SignificanceThis work presents a bifurcation analysis of laminar premixed conical flames within a fully-coupled flame/flow model. By considering variations in reaction rate and reactant diffusivity, the analysis identifies saddle–node bifurcations corresponding to spontaneous flash-back and blow-off of the axisymmetric flame. It also identifies axisymmetry breaking bifurcations associated with transitions to steady three-dimensional polyhedral and tilted flame states. These global instabilities indicate that hydrodynamic and thermal-diffusive effects strongly influence a flame’s steady structure in the azimuthal dimension even when no instabilities appear in the plane of symmetry. As such, future analyses of flame dynamics should rule out symmetry-breaking behaviors before adopting a two-dimensional computational framework.

Dynamic properties of a reaction–diffusion predator–prey model with nonlinear harvesting: A linear and weakly nonlinear analysis

In this work, we have studied the spatiotemporal dynamics of a predator–prey model with selective nonlinear saturated harvesting of both species. The study of the non-diffusive system includes analytical and numerical analysis of local and global bifurcations. Bifurcation analysis of the non-diffusive system unveils that the system may exhibit transcritical, Hopf, saddle–node and homoclinic bifurcation with increased carrying capacity and harvesting efforts. The simulation results further reveal that the paradox of enrichment may disappear in a harvested system for a higher value of carrying capacity. The spatiotemporal study consists of an analysis of different pattern-forming instabilities. A critical ratio of the diffusivities of predator to prey for the occurrence of Turing instability is established. Further, a weakly nonlinear analysis is provided to derive the amplitude equation and predict the form of the pattern. The key highlight of the work is that prey harvesting facilitates spatiotemporal chaos, and predator diffusion controls it. Furthermore, intensive prey harvesting forces spatial segregation, whereas predator harvesting relaxes it. It is also observed that spatial segregation enriches the overall species biomass. A unique observation of this study is the occurrence and nonoccurrence of spatiotemporal chaos in the Hopf-Turing parametric space.

Global stability and bifurcation analysis on the dynamics of three interacting populations in prey-predator system

A mathematical model was proposed and analysed to study the dynamics of a system of a prey, primary predator and a super predator in which the predators show Holing Type II functional response to their respective prey. The study substantially provides mechanism for improving the survivability of honeybee population and to halt the negative impact of Varroa-mite infestation on honeybee population by introducing Pseudoscorpion to serve as a super predator of Varroa-mite. Also, the study significantly shows the impact of virus on honeybee infested with virus-carrying Varroa-mite. The global stability analysis and Hopf-bifurcation of the interior steady state existence of the system are studied and explored. The analytical results matched with the numerical results of the model developed.

Global Stability and Bifurcation Analysis of a Virus Infection Model with Nonlinear Incidence and Multiple Delays

In order to investigate the impact of general nonlinear incidence, cellular infection, and multiple time delays on the dynamical behaviors of a virus infection model, a within-host model describing the virus infection is formulated and studied by taking these factors into account in a single model. Qualitative analysis of the global properties of the equilibria is carried out by utilizing the methods of Lyapunov functionals. The existence and properties of local and global Hopf bifurcations are discussed by regarding immune delay as the bifurcation parameter via the normal form, center manifold theory, and global Hopf bifurcation theorem. This work reveals that the immune delay is mainly responsible for the existence of the Hopf bifurcation and rich dynamics rather than the intracellular delays, and the general nonlinear incidence does not change the global stability of the equilibria. Moreover, ignoring the cell-to-cell infection may underevaluate the infection risk. Numerical simulations are carried out for three kinds of incidence function forms to show the rich dynamics of the model. The bifurcation diagrams and the identification of the stability region show that increasing the immune delay can destabilize the immunity-activated equilibrium and induce a Hopf bifurcation, stability switches, and oscillation solutions. The obtained results are a generalization of some existing models.

Nonlinear dynamics and safety aspects of pressure relief valves

Direct spring operated pressure relief valves are regular elements of high-pressure hydraulic systems; however, valve vibrations may still occur endangering the valve and the protected system. The damping of these vibrations is a crucial issue, whereas the external viscous damping at the valve cannot be increased boundlessly based on safety aspects. The vibrations emerge via Hopf bifurcations of either super- or subcritical type, and the nonlinear characteristics have an important role in the valve dynamics. Thorough nonlinear analysis is needed to guarantee safe operation. The linear stability analysis is extended by local and global bifurcation analysis to reveal the stable domains bounded by subcritical Hopf points and the possible bistabilities that exclude the safe operation. Afterwards, the effect of a downstream pipe is analysed, which can also carry safety risks based on the literature; however, the time delay originated in the wave propagation in the pipe may also have stabilising effects. The nonlinear analysis of the presented system compares analytical, semi-analytical and numerical results and assigns the safe parameter domains of the valve with respect to a wide range of relevant parameter variations.

Bifurcation Control for a Ship Maneuvering Model with Nonsmooth Nonlinearities

We consider a widely used form of models for ship maneuvering, whose nonlinearities entail continuous but nonsmooth second-order modulus terms. For such models bifurcations of straight motion are not amenable to standard center manifold reduction and normal forms. Based on a recently developed analytical approach, we nevertheless determine the character of local bifurcations when stabilizing the straight motion course with standard proportional control. For a specific model class we perform a detailed analysis of the linearization to determine the location of these bifurcations in the control parameter space and its dependence on selected design parameters. By computing the analytically derived characteristic parameters, we find that “safe” supercritical Andronov–Hopf bifurcations are typical. Through numerical continuation we provide a more global bifurcation analysis, which identifies the arrangement and relative location of stable and unstable equilibria and periodic orbits.

Global stability, periodicity, and bifurcation analysis of a difference equation

This research aims to discuss the existence, global stability, periodicity, and bifurcation analysis of a modified version of the ecological model proposed by Tilman and Wedlin [Nature 353, 653–655 (1991)].

Global Hopf bifurcation of a cholera model with media coverage.

We propose a model for cholera under the impact of delayed mass media, including human-to-human and environment-to-human transmission routes. First, we establish the extinction and uniform persistence of the disease with respect to the basic reproduction number. Then, we conduct a local and global Hopf bifurcation analysis by treating the delay as a bifurcation parameter. Finally, we carry out numerical simulations to demonstrate theoretical results. The impact of the media with the time delay is found to not influence the threshold dynamics of the model, but is a factor that induces periodic oscillations of the disease.

Global bifurcation analysis in a predator–prey system with simplified Holling IV functional response and antipredator behavior

In this paper, we study a predator–prey system with the simplified Holling IV functional response and antipredator behavior such that the adult prey can attack vulnerable predators. The model has been investigated by Tang and Xiao, and the existence and stability of all possible equilibria are determined. In addition, they performed a bifurcation analysis and showed that the system undergoes a Codimension 2 Bogdanov–Takens bifurcation. In this paper, for the same model, we further show that the cusp‐type Bogdanov–Takens bifurcation can be of Codimension 3, which acts as an organizing center for the whole bifurcation set. In addition, we propose the existence of Hopf bifurcation of Codimension 2 and the coexistence of stable limit cycle and unstable limit cycle. In particular, we show that the antipredator behavior has great effect on the dynamics of the model, it may cause the predator population to extinct while the prey population will increase up to the carrying capacity. Numerical simulations including bifurcation diagrams and phase portraits are performed to illustrate and confirm the theoretical results. These results may enrich the dynamics of predator–prey systems.

Global stability and bifurcation analysis in a discrete-time two prey-one predator model with help

ABSTRACT This article examines a discrete-time predator-prey model where prey teams cooperate. Positivity and boundedness of the model solution are investigated. Positive fixed points are examined. Using an iteration scheme and the comparison principle of difference equations, we determined the sufficient condition for global stability of the positive fixed point. It is shown that the sufficient criterion for Neimark-Sacker bifurcation and flip bifurcation can be established. It is observed that the system behaves in a chaotic manner when a specific set of system parameters is selected, which are controlled by a hybrid control method. Examples are provided to illustrate our conclusions.

Local and global Hopf bifurcation analysis of an age-infection HIV dynamics model with cell-to-cell transmission

In this study, we formulate an age-infection human immunodeficiency viruses (HIV) model incorporating cell-to-cell transmission and virus-to-cell infection. Our main goal is to investigate the local and global Hopf bifurcations of the model. We apply integrated semi-group, Hopf bifurcation theory and topological degree theorem. The theoretical analysis indicates that local and global bifurcation can occur from the positive steady state with respect to bifurcation parameters. These bifurcations are crucial for clinical treatment and experimental of HIV infections within the host. Because of the periodic oscillation, higher (or lower) virus load detected at a moment does not indicate the same high (or lower) load as time goes to infinity. At the same time, the appearance of periodic solution caused by time delay τ also explain the fluctuation of viral load in plasma. Finally, we presented some numerical simulations to validate our theoretical results.

Global dynamics and bifurcation analysis of an insect-borne plant disease model with two transmission routes

Huanglongbing (HLB) is a plant disease mainly spread by the insect-borne citrus psyllid. It is the most destructive citrus pathosystem worldwide. To understand the impact of sexual transmission on HLB dynamics, we propose a host–vector–HLB compartment model incorporating two transmission routes. The basic reproduction number [Formula: see text] is derived. Various interventions of the disease are assessed. We also investigate the effect of different incidence functions to simulate sexual transmission. For the case of sublinear incidence functions, the disease-free equilibrium is globally asymptotically stable (GAS) provided [Formula: see text]. For mass action incidence of sexual transmission, the endemic equilibrium is GAS provided [Formula: see text]. However, under nonlinear incidence, it is proved that the model may exhibit backward bifurcation. Theoretical and numerical studies reveal that (i) different forces of infection between heterosexual psyllids in the model may have a distinct impact on disease dynamics; (ii) sensitivity analysis shows that for [Formula: see text], the transmission rate between host and vector is more sensitive parameter than that between heterosexual psyllids; (iii) if the sexual transmission is ignored, the disease burden is likely to be underestimated in comparison with realistic scenarios; (iv) in the absence of chemical insecticides, the combined use of yellow sticky traps and injection of nutrient solutions can be more effective in suppressing the spread of HLB. These findings provide valuable insights for public policymakers to determine the long-term viability of implemented HLB management strategies and highlight the urgency of finding sustainable HLB solutions.

Limit cycles of multi-parameter polynomial dynamical systems

This is an overview of our recent works on the global bifurcation analysis of multi-parameter polynomial dynamical systems. In particular, using our bifurcation-geometric approach, we study the global dynamics and solve the problem on the maximum number and the distribution of limit cycles in a polynomial Euler–Lagrange–Liénard-type mechanical system. We also consider a rational endocrine system by carrying out the global bifurcation analysis of a reduced planar quartic Topp system, which models the dynamics of diabetes. By analyzing global bifurcations and applying the Wintner–Perko termination principle, we prove that such a system can have at most two limit cycles.

Global dynamics and bifurcation analysis of a fractional‐order SEIR epidemic model with saturation incidence rate

The present paper studies a fractional‐order SEIR epidemic model for the transmission dynamics of infectious diseases such as HIV and HBV that spreads in the host population. The total host population is considered bounded, and Holling type‐II saturation incidence rate is involved as the infection term. Using the proposed SEIR epidemic model, the threshold quantity, namely, basic reproduction number , is obtained that determines the status of the disease, whether it dies out or persists in the whole population. The model's analysis shows that two equilibria exist, namely, disease‐free equilibrium (DFE) and endemic equilibrium (EE). The global stability of the equilibria is determined using a Lyapunov functional approach. The disease status can be verified based on obtained threshold quantity . If , then DFE is globally stable, leading to eradicating the population's disease. If , a unique EE exists, and that is globally stable under certain conditions in the feasible region. The Caputo type fractional derivative is taken as the fractional operator. The bifurcation and sensitivity analyses are also performed for the proposed model that determines the relative importance of the parameters in disease transmission. The numerical solution of the model is obtained by the generalized Adams–Bashforth–Moulton method. Finally, numerical simulations are performed to illustrate and verify the analytical results. From the obtained results, it is concluded that the order of the fractional derivative plays a significant role in the dynamic process. Also, from the sensitivity analysis results, it is seen that the parameter δ ( infection rate) is positively correlated to , while the parameter μ (inhibition rate) is negatively correlated to , which means these parameters are more sensitive than others in disease dynamics.

Global stability and bifurcation analysis of a discrete time SIR epidemic model

Global stability and bifurcation analysis of a discrete time SIR epidemic model