Search

This paper will explore a fractional-order hepatitis B virus (HBV) infection model that takes into account cell-cell and virus-cell transmission and multiple treatment modalities. The desired control strategy is realized by means of a fractional order PD controller. Firstly, we calculated the basic regeneration number and equilibrium point of the HBV model. Afterwards, for the uncontrolled HBV virus system, the adequate conditions for both stability and Hopf bifurcation are systematically investigated via choosing the appropriate time delay as a parameter for bifurcation. Subsequently, under fractional order PD controller, the effect of a proposed controller on system stability and Hopf bifurcation is studied. The desired dynamic characteristics can be obtained afterwards. Finally, numerical simulations show that all three treatments significantly reduce R 0. The onset of oscillations can be delayed by decreasing the order of the fractional order. There are three control pathways for fractional order PD control, and the generation of bifurcation can also be delayed by changing the gain parameter. Using the above methods, the diffusion of HBV virus particles in the body can be effectively controlled. The conclusions drawn in this paper are extremely novel and have potential theoretical value for the future treatment of hepatitis B illness. [ABSTRACT FROM AUTHOR]

The red palm weevil is one of the most dangerous pests affecting date palm trees in the Gulf region and the Middle East. This paper investigates the control of the red palm weevil (RPW) in date palms using several techniques, namely stem injection, sex pheromone traps, and the Sterile Insect Technique. A mathematical model describing the spread of the red palm weevil is presented, and the boundedness of the solution for the RPW system is studied. The conditions for local stability, forward bifurcation, and Hopf bifurcation are also obtained. Additionally, this research highlights the significance of optimal control in managing RPW and simulates the model using the Forward-Backward Sweep numerical method. It is observed that the parameters of the sterile insect technique, the pheromone trap, and the mechanical injection play important roles in controlling the RPW. Mechanical Injection may be more effective in controlling the palm weevil compared to the other two methods. [ABSTRACT FROM AUTHOR]

There are many harmful effects of smoking. It not only engulfs the health and life of smokers, but also pollutes the air, endangers the life and health of others, and brings a heavy burden to public health. To this end, a delayed smoking model including potential smokers, occasional smokers, smokers, temporary quitters, permanent quitters and smokers with some illness, is investigated in the present paper. Firstly, local stability and existence of Hopf bifurcation of the model is conducted. Secondly, global exponential stability is explored. Lastly, we numerically simulate the correctness of the obtained theoretical results in the paper. [ABSTRACT FROM AUTHOR]

To explore transmission dynamics of tuberculosis, a tuberculosis transmission model with vaccination and time delay is developed in current paper. Positivity and boundedness are analyzed. Local stability of tuberculosis-free equilibrium in respect of the time delay due to latent period of tuberculosis is analyzed and we have found threshold value of the time delay for the local stability of tuberculosis-free equilibrium. Then, local stability of tuberculosis-existence equilibrium following exhibition of Hopf bifurcation at the crucial value of the time delay due to latent period of tuberculosis is derived. It is shown that the developed model undergoes a Hopf bifurcation around the tuberculosis-existence when the time delay due to latent period of tuberculosis passes through the threshold value. Direction and stability of the Hopf bifurcation are investigated with the help of the normal form method and center manifold theory. Finally, numerical simulations are carried out in the justification of obtained analytical findings. The results obtained provide significant information for tuberculosis disease controlling. [ABSTRACT FROM AUTHOR]

This paper examines the dynamics of a time-delay differential model of the tumour immune system with random noise. The model describes the interactions between healthy tissue cells, tumour cells, and activated immune system cells. We discuss stability and Hopf bifurcation of the deterministic system. We then explore stochastic stability, and the dynamics of the system in view of environmental fluctuations. Criteria for persistence and sustainability are discussed. Using multiple Lyapunov functions, some sufficient criteria for tumour cell persistence and extinction are obtained. Under certain circumstances, stochastic noise can suppress tumour cell growth completely. In contrast to the deterministic model which shows no stable tumour-free state, the white noise can either lead to tumour dormancy or tumour elimination. Some numerical simulations, by using Milstein's scheme, are carried out to show the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we have studied a three-dimensional model to describe the behavior of tumor cells in the absence of the immune response, which is composed of tumor, normal and fat cells (TNF) under time delay effect. The center manifold theory has been used to study the bifurcation behavior. It is proven that TNF model undergoes codimension-1 bifurcation, while Hopf bifurcation does not occur in the non-delay model. For delayed TNF, we take into the late response of fat cells to tumor cells. Consequently, the delay factor is a considerable parameter in TNF model. Hence, we presented a formula of the time delay value for the occurrence of periodic solutions. Additionally, the stability of Hopf bifurcation was analysed. Numerical simulations are provided to illustrate and extend the theoretical results, we provide our study with numerical simulations inclusive families of periodic solutions. [ABSTRACT FROM AUTHOR]

This paper examines economic cycles that do not depend on exogenous economic actions. More precisely, the paper develops a positive model of government behavior in order to define the intertemporal fiscal policies that are optimal for a country and which, determine both the optimal budget level and the optimal level of environmental quality. For this purpose, we establish an optimal control model involving intertemporal subsidy strategies characteristically used by an authoritarian government similar to those found in central Europe. It will be shown that by applying the Hopf bifurcation theorem,a cyclical strategy – that is, waves of regulation, environmental subsidies alternating with deregulation and cuts in social programs – may represent an optimal policy. In this paper, we propose an extremely simple optimal control model which is applied to budget surpluses and environmental subsidies. We investigate the cyclical environmental policies as applied through the bifurcation theorem. A number of propositions are stated during the solution process. The first proposition sets the rules in the model parameters in order to establish the cyclical policies. A second proposition sets the relationship between the discount rate and the opportunity cost of capital, which is used to determine a taxing strategy which becomes the future optimal subsidy policy. [ABSTRACT FROM AUTHOR]

The spread of worm virus has brought great loss to our production and life. In this paper, a new Vulnerable-Exposed-Infectious-Quarantined-Secured (VEIQS) worm propagation model with a saturated incidence and two delays is proposed. The local stability of the worm-existence equilibrium and the occurrence of Hopf bifurcation at the critical values of the two delays are obtained by regarding different combinations of time delays as bifurcation parameters. It shows that the model is ideal stable when the time delay is below the critical value and a Hopf bifurcation occurs when the time delay is above the critical value. In particular, direction and stability of the Hopf bifurcation are determined by using the center manifold theorem. Finally, some numerical simulations are presented in order to verify the analytical results. [ABSTRACT FROM AUTHOR]

Due to the importance of studying the change in glucose which affected by obesity and results in a negative effect on the health of the body, In this paper, glucose metabolism model was developed to study the effect of obesity. The bifurcations and related dynamical behaviors of our model were studied completely. For the non-delay model, it is shown that the model undergoes co-dimension 1 bifurcation such as transcritical, Hopf and saddle bifurcations, and co-dimension 2 bifurcation such as generated hopf, zero-hopf and cusp bifurcations. On the other hand, we consider there is a lag for responding of Insulin sensitivity under obesity effect. Therefore, the time delay τ is deemed as an additional factor in the glucose metabolism model. We set the condition for the existence of the delay parameter value for which the model displays a Hopf bifurcation. Finally, numerical simulations are given to support theoretical analysis and also to show the complex dynamical behaviors including families of periodic curves. The described dynamics of this model can obviously illustrate several clinical observations for obesity effect on glucose level, oscillation and disorder, supporting its potential role in control of blood glucose. [ABSTRACT FROM AUTHOR]

This paper studies both existence and spectral stability properties of bounded spatially periodic traveling wave solutions to a large class of scalar viscous balance laws in one space dimension with a reaction function of monostable or Fisher-KPP type. Under suitable structural assumptions, it is shown that this class of equations underlies two families of periodic waves. The first family consists of small amplitude waves with finite fundamental period which emerge from a Hopf bifurcation around a critical value of the wave speed. The second family pertains to arbitrarily large period waves which arise from a homoclinic bifurcation and tend to a limiting traveling (homoclinic) pulse when their fundamental period tends to infinity. For both families, it is shown that the Floquet (continuous) spectrum of the linearization around the periodic waves intersects the unstable half plane of complex values with positive real part, a property known as spectral instability. For that purpose, in the case of small-amplitude waves it is proved that the spectrum of the linearized operator around the wave can be approximated by that of a constant coefficient operator around the zero solution and determined by a dispersion relation which intersects the unstable complex half plane. In the case of large period waves, we verify that the family satisfies the assumptions of the seminal result by Gardner (Spectral analysis of long wavelength periodic waves and applications , J. Reine Angew. Math. 491 (1997), 149–181) of convergence of periodic spectra in the infinite-period limit to that of the underlying homoclinic wave, which is unstable. A few examples are discussed. [ABSTRACT FROM AUTHOR]

In this paper we will look at the one system of ODE and analyze it. We aim to determine the points of equilibrium; examine their character and establish the existence of a bifurcation for the corresponding parameter value. A detailed analysis of local stability was performed for all values of the given parameter. For a certain value of the parameter, the existence of supercritical Hopf bifurcation of the observed system of differential equations has been proved. Also, the existence of a limit cycle that is always stable has been proved. [ABSTRACT FROM AUTHOR]

In this paper, we will study two independent economies in a country (national, regional and urban), where the dynamics of fluctuations in each economy is described by Keynes's mathematical business cycle model. This is an interaction of two economies which include trade and competition. In the resulting system that consists of two independent economic entities, we show that fluctuations can emerge as two possible types of economic indicators (synchronous and antiphase) when the peaks and downturns of business activities in each of the economies are completely synchronized or on the contrary when the rise of one economy is accompanied by a recession (antiphase cycles). Our aim is to examine the stability question of solutions of the cognate mathematical model. Our analysis of the mathematical model will render methods of the theory of dynamical systems, such as the method of integral manifolds and the Poincare normal forms. This approach will provide a sufficient analysis of the dynamics of solutions of a system of differential equations, which is used as a mathematical model. Asymptotic formulas will be obtained for solutions that depict economic cycles. [ABSTRACT FROM AUTHOR]

In this paper, we proposed and investigated a discretized form of a fractional-order multi-drug antimicrobial resistance model. The existence and the uniqueness of the proposed model is studied. The impacts of varying the parameters of the system are examined. Under certain conditions, the system undergoes some kinds of bifurcations. It is established that the fractional-order parameter affects the stability of the model. The model has an opulent diversity of dynamical behaviors such as Hopf bifurcation, chaotic attractors and attractor crisis. We used some numerical simulations to illustrate our analytical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Hopf bifurcation of a delayed SEIQR (Susceptible-Exposed-Infectious- Quarantined-Recovered) worm propagation model in mobile internet. Firstly, a distinct set of conditions that ensures existence of the Hopf bifurcation are derived by analyzing the roots of the associated characteristic equation and regarding the delay as a bifurcation parameter. Secondly, the normal form theory and the center manifold theorem are employed to determine direction of the Hopf bifurcation and stability of the bifurcating periodic solutions. Finally, some numerical simulations are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Hopf bifurcation of a delayed SEIQR (Susceptible-Exposed-Infectious- Quarantined-Recovered) worm propagation model in mobile internet. Firstly, a distinct set of conditions that ensures existence of the Hopf bifurcation are derived by analyzing the roots of the associated characteristic equation and regarding the delay as a bifurcation parameter. Secondly, the normal form theory and the center manifold theorem are employed to determine direction of the Hopf bifurcation and stability of the bifurcating periodic solutions. Finally, some numerical simulations are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Understanding dynamical behavior of a spatially distributed population is crucial to conservation and management of endangered species. This paper considers predator–prey systems with Beddington–DeAngelis functional response, where the predator moves between source–sink patches asymmetrically and acts as an agent. Our aim is to show how agent-based diffusion affects dynamics of the system and total population abundance of the species. Using dynamical systems theory, we demonstrate stability of positive equilibria in the system, which implies coexistence of the species and change of abundance by diffusion. Moreover, we show Hopf and Bautin bifurcations with multiple limit cycles, which implies multiple oscillations of populations and even extinction of species. Furthermore, this work demonstrates that diffusion in the system may lead to results reversing those without diffusion. The diffusion could change dynamics of the system between coexistence at a steady state and persistence in periodic oscillation, while evolution in asymmetry of diffusion could make the predator reach a total abundance larger than that without diffusion, even reach the maximal abundance. Our results are consistent with experimental observations and are important in studying conservation of biodiversity. • Predator-prey systems with predator's asymmetric diffusion between source–sink patches are considered. • A rigorous analysis on persistence of the system is given. • Subcritical Hopf bifurcation is used to show existence of two limit cycles and bi-stability. • Diffusion in the system could lead to results reversing those if non-diffusing. • Intermediate diffusion to the sink could make the predator reach total abundance higher than if non-diffusing. [ABSTRACT FROM AUTHOR]

In this paper, an autonomous three-dimensional Helmholtz-type oscillator is designed based on conversion of an autonomous Helmholtz two-dimensional oscillator to a jerk oscillator. For a suitable choice of the parameters, the proposed autonomous Helmholtz jerk oscillator can generate Hopf bifurcation, bistable period-2 limit cycles, two types of one-scroll chaotic attractors and coexistence between period-3 limit cycle and one-scroll chaotic attractors. Using a weak modulation of a parameter of the proposed Helmholtz jerk oscillator, it is possible to destroy the coexisting attractors found and transform the proposed Helmholtz jerk oscillator to period-3 oscillations. Moreover using experiments and OrCAD-PSpice software, circuit implementation of the proposed autonomous Helmholtz jerk oscillator is realized in order to check the one-scroll chaotic attractors and the coexisting attractors found during the numerical simulations. Numerical and experimental/OrCAD-PSpice results have a good qualitative agreement. Finally, by adding two new parameters in the proposed autonomous Helmholtz jerk oscillator, it is possible to control the amplitude of the attractor and the largest Lyapunov exponent. [ABSTRACT FROM AUTHOR]

In this paper, an autonomous Toda jerk oscillator is proposed and analysed. The autonomous Toda jerk oscillator is obtained by converting an autonomous two-dimensional Toda oscillator with an exponential nonlinear term to a jerk oscillator. The existence of Hopf bifurcation is established during the stability analysis of the unique equilibrium point. For a suitable choice of the parameters, the proposed autonomous Toda jerk oscillator can generate antimonotonicity, periodic oscillations, chaotic oscillations and bubbles. By introducing two additional parameters in the proposed autonomous Toda jerk oscillator, it is possible to control partially or totally the amplitude of its signals. In addition, electronic circuit realization of the proposed Toda jerk oscillator is carried out to confirm results found during numerical simulations. The commensurate fractional-order version of the proposed autonomous chaotic Toda jerk oscillator is studied using the stability theorem of fractional-order oscillators and numerical simulations. It is found that periodic oscillations and chaos exist in the fractional-order form of the proposed Toda jerk oscillator with order less than three. Finally, combination synchronization of two fractional-order proposed autonomous chaotic Toda jerk oscillators with another fractional-order proposed autonomous chaotic Toda jerk oscillator is analysed using the nonlinear feedback control method. [ABSTRACT FROM AUTHOR]

In the present paper, the Rosenzweig-MacArthur predator-prey model (RM), which is a bitrophic food chain model, is considered. We develop the model by adding two assumptions. First, we assume that both species are of economic interest, that is can be harvested. Second, we assume that each specie has its own time scale which range from fast for the prey to slow for the predator. We consider that both the death rate and the catch of the predator are very small which leads to a fast-slow dynamical system. That is, the RM model is transformed into a singular perturbed system with a perturbation parameter εε ∈ [0,1]. The existence and stability of equilibria are discussed for εε > 0. The model experiences both transcritical and Hopf bifurcations for εε > 0. The singular perturbation model at ε = 0 is discussed by separating the system into two subsystems; fast and slow and studying them simultaneously. When 0 < εε ≪ 1, the model is discussed using geometric singular perturbation techniques. The solution of the model is approximated on the slow manifold and the numerical simulations give very good results for ε = 0.005. [ABSTRACT FROM AUTHOR]

Oscillatory behavior is a ubiquitous phenomenon in various physical and biological processes. Recently, it has been reported that oscillations of wireless sensor networks infected by virus (WSNIVs) can potentially be deleterious to the security strength of systems and may even cause network congestion and paralysis. Moreover, it has been discovered that latency delays are essential for the function of WSNIVs and can drive instability and periodic oscillations, enhance complexity, and even lead to multistability and chaotic motion. However, the precise roles of such delay during the regulation process are still not completely understood. Here, the primary objective of this paper is to study oscillatory behaviors of WSNIVs with latency delay. In particular, the sufficient conditions for local stability and existence with Hopf bifurcation are obtained. Moreover, we further discuss the properties of Hopf bifurcation by using the normal form and the center manifold theorem. The obtained results show that the latency delay can drive the WSNIVs to be oscillatory even when the network is at a stable state, suggesting that such delay might be a potential hazard to the security of wireless sensor systems. Our findings highlight the importance of considering delays when developing safer and more effective wireless sensor networks. Finally, we test and analyze the above research results through numerical calculation with Matlab and simulation experiments with OPNET, and the conclusions are verified to be correct and effective in experiments. [ABSTRACT FROM AUTHOR]

Highlights • Nonlinear harvesting is introduced into the predator-prey model. • Hopf bifurcation and Turing instability of the model are analyzed. • Weakly nonlinear analysis is employed to identify pattern selection. Abstract In this paper, the effect of diffusion on pattern selection in a predator-prey model with Michaelis–Menten type predator nonlinear harvesting is investigated. From the analysis of stability and bifurcation, the stability of the positive equilibrium and the occurrence conditions for the Hopf bifurcation, as well as the Turing instability, are obtained. Consequently, different patterns will occur in the system. It is necessary and imperative to derive the amplitude equations for understanding pattern selection. To this end, the multiple time scale analysis technique is employed to obtain the amplitude equations at the critical value of the Turing instability. Further, by analyzing the amplitude equations, we could identify different patterns, such as spot-like, stripe-like, hexagon-like and mixed state patterns. Finally, numerical simulations show that the rationality of the theoretical analysis and the obtained results will enrich the dynamical investigation of the predator-prey model. [ABSTRACT FROM AUTHOR]

The current paper deals with the problem of a three-dimensional prey-predator system with infection on prey population. This three-dimensional prey-predator system is split into three groups, namely susceptible prey, infected prey and predator. Next we have introduced gestation time delay term with the equation of predator population. We observe the dynamical behaviour of this system around each of the equilibria and point out the exchange of stability of the model without time delay and with time delay. We calculate the length of time delay to maintain the stability of the delay system. We see that, there is a Hopf bifurcation with non delay and delay system respectively. In the delay system it is investigated that Hopf bifurcation happen when the time delay crosses a certain critical value. Lastly computer simulations have been carried out to illustrate various analytical results. [ABSTRACT FROM AUTHOR]

This paper considers the application of a semi-analytical method to the Schnakenberg model of a reaction-diffusion cell. The semi-analytical method is based on the Galerkin method which approximates the original governing partial differential equations as a system of ordinary differential equations. Steady-state curves, bifurcation diagrams and the region of parameter space in which Hopf bifurcations occur are presented for semi-analytical solutions and the numerical solution. The effect of feedback control, via altering various concentrations in the boundary reservoirs in response to concentrations in the cell centre, is examined. It is shown that increasing the magnitude of feedback leads to destabilization of the system, whereas decreasing this parameter to negative values of large magnitude stabilizes the system. The semi-analytical solutions agree well with numerical solutions of the governing equations. [ABSTRACT FROM AUTHOR]

In this paper, in view of the senescence of plant and the decay of wrack, time delays are introduced into the plant-wrack model. The effects of wrack decay and time delay on the dynamical behaviors of the diffusive plant-wrack model are studied analytically and numerically. When the delay is zero, the wrack decay will induce the change of stability of the unique equilibrium point, further lead to the occurrence of the Hopf bifurcation and the Turing instability. When the delay is present, the conditions for the occurrence of the Hopf bifurcation are established. By comparing the results of the model without and with delay, it is found that the increases of delay may induce no stability switches, a single stability switch or multiple stability switches, when the value of wrack decay can stabilize model with zero delay. When the value of wrack decay can destabilize model with zero delay, numerical simulations show that the small delay may cause homogeneous distributions of vegetation, while the larger delay may cause the emergence of periodic oscillation of vegetation. The obtained results provide a basis for understanding the spatiotemporal evolution of such a plant-wrack model with delay. • Plant-wrack model with delay is proposed. • Effects of wrack decay and delay are analyzed. • Complex dynamics are presented. [ABSTRACT FROM AUTHOR]

This paper analyses the dynamics of infectious disease with a concurrent spread of disease awareness. The model includes local awareness due to contacts with aware individuals, as well as global awareness due to reported cases of infection and awareness campaigns. We investigate the effects of time delay in response of unaware individuals to available information on the epidemic dynamics by establishing conditions for the Hopf bifurcation of the endemic steady state of the model. Analytical results are supported by numerical bifurcation analysis and simulations. [ABSTRACT FROM AUTHOR]

RNA interference (RNAi) is a fundamental cellular process that inhibits gene expression through cleavage and destruction of target mRNA. It is responsible for a number of important intracellular functions, from being the first line of immune defence against pathogens to regulating development and morphogenesis. In this paper we consider a mathematical model of RNAi with particular emphasis on time delays associated with two aspects of primed amplification: binding of siRNA to aberrant RNA, and binding of siRNA to mRNA, both of which result in the expanded production of dsRNA responsible for RNA silencing. Analytical and numerical stability analyses are performed to identify regions of stability of different steady states and to determine conditions on parameters that lead to instability. Our results suggest that while the original model without time delays exhibits a bi-stability due to the presence of a hysteresis loop, under the influence of time delays, one of the two steady states with the high (default) or small (silenced) concentration of mRNA can actually lose its stability via a Hopf bifurcation. This leads to the co-existence of a stable steady state and a stable periodic orbit, which has a profound effect on the dynamics of the system. [ABSTRACT FROM AUTHOR]

Predator foraging facilitation may strongly influence the dynamics of a predator–prey system. This behavioral pattern is well-observed in real life interactions, but less is known about its possible impacts on the predator–prey dynamics. In this paper we analyze a modified Rosenzweig–MacArthur model, where a predator-dependent family of functions describing predator foraging facilitation is introduced into the Holling type II functional response. As the general assumption of foraging facilitation is that higher predator densities give rise to an increased foraging efficiency, we model predator facilitation with an increasing encounter rate function. Using the tools of bifurcation analysis we describe all the nonlinear phenomena that occur in the system provoked by foraging facilitation, these include the fold, Hopf, transcritial, homoclinic and Bogdanov–Takens bifurcation. We show that foraging facilitation can stabilize the coexistence in the predator–prey system for specific rates, but in most of the cases it can have fatal consequences for the predators themselves. [ABSTRACT FROM AUTHOR]

In this paper we explore how the time delay induced Hopf-bifurcation interacts with Turing instability to determine the resulting spatial patterns. For this study, we consider a delayed prey–predator model with Holling type-II functional response and intra-specific competition among the predators. Analytical criteria for the delay induced Hopf-bifurcation and for the delayed spatio-temporal model are provided with numerical example to validate the analytical results. Exhaustive numerical simulation reveals the appearance of three types of stationary patterns, cold spot, labyrinthine, mixture of stripe-spot and two non-stationary patterns, quasi-periodic and spatio-temporal chaotic patterns. The qualitative features of the patterns for the non-delayed and the delayed spatio-temporal model are the same but their occurrence is solely controlled by the temporal parameters, rate of diffusivity and magnitude of the time delay. It is evident that the magnitude of time delay parameter beyond the Hopf-bifurcation threshold mostly produces spatio-temporal chaotic patterns. [ABSTRACT FROM AUTHOR]

In this paper, a nutrient–phytoplankton model described by a couple of reaction-diffusion equations with delay is studied analytically and numerically. The aim of this research is to provide an understanding of the impact of delay on the nutrient–phytoplankton dynamics. Significantly, the delay can not only induce instability of a positive equilibrium, but also promote the formation of patchiness (an irregular pattern) via Hopf bifurcation. However, if the delay does not exist, the positive equilibrium is always globally asymptotically stable when it exists. In addition, the numerical analysis indicates that the input rate and the loss rate of nutrient also play an important role in the growth of phytoplankton, which supports that eutrophic conditions may be a significant reason inducing phytoplankton blooms. Numerical results are consistent with the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, a coupled FitzHugh–Nagumo (FHN) neural system with time delay has been proposed and its stability and Hopf bifurcation are researched. Specifically, the stability of equilibrium point is analyzed by employing the corresponding characteristic equation. Sufficient conditions for existence of Hopf bifurcation are obtained. The results show that the FHN neural system exhibits the parameter regions involved the delay-independence stability and delay dependence stability. Increase of time delay can induce the stability switches between resting state and periodic activity. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper the linear stability of the delayed logistic equation with additive Allee effect is investigated. We also analyze the associated characteristic transcendental equation, to show the occurrence of Hopf bifurcation at the positive equilibrium. To determine the direction of Hopf bifurcation and the stability of bifurcating periodic solution, we use the normal form approach and a center manifold theorem. Finally, a numerical example is given to demonstrate the effectiveness of the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, a ratio dependent delayed predator-prey model with Holling-Tanner type functional response is studied. The local stability of a positive equilibrium and the existence of Hopf bifurcations are established. By using the normal form theory and center manifold theorem, the explicit algorithm determining the stability, direction of the bifurcating periodic solutions are derived. Finally, numerical simulations for justifying the theoretical analysis are also presented. [ABSTRACT FROM AUTHOR]

In this paper, a ratio dependent delayed predator-prey model with Holling-Tanner type functional response is studied. The local stability of a positive equilibrium and the existence of Hopf bifurcations are established. By using the normal form theory and center manifold theorem, the explicit algorithm determining the stability, direction of the bifurcating periodic solutions are derived. Finally, numerical simulations for justifying the theoretical analysis are also presented. [ABSTRACT FROM AUTHOR]

In this paper, a discrete-time food chain characterized by three species is modeled by a system of three nonlinear difference equations. The existence and local stability of the equilibrium points of the discrete dynamical system are analyzed. It is shown that for some parameter values the interior equilibrium point loses its stability through a discrete Hopf bifurcation. Basic properties of the discrete system are analyzed by means of phase portraits, bifurcation diagrams and Lyapunov exponents. We have varied the result through numerical calculation. [ABSTRACT FROM AUTHOR]

• The Schnakenberg Model has two different behaviours of the bifurcation diagram. • We used singularity to calculate the cusp points. • We determined the Hopf region of the model. • Excellent agreement between the semi-analytical model and the numerical solution is observed. This paper discusses how a semi-analytical approach can be used alongside the reversible Schnakenberg model in a reaction-diffusion cell. It is worth noting that semi-analytical techniques are aligned to the Galerkin method, which offers an approximation of the governing partial differential equations through a system of ordinary differential equations. The bifurcation diagrams, steady-state curves and regions of the parameter space where bifurcations take place are part of the semi-analytical solutions and the numerical solution. [ABSTRACT FROM AUTHOR]