Hopf Bifurcation Of Periodic Solutions Research Articles

Stability and Hopf bifurcation for a quaternion-valued three-neuron neural network with leakage delay and communication delay

This paper proposes the stability and Hopf bifurcation for a quaternion-valued neural network(QVNN) with leakage delay and communication delay. Due to the higher order of the characteristic equation, the order of characteristic equation is reduced via the matrix block theory. Simultaneously, taking the leakage delay and communication delay as bifurcation parameter respectively, a number of conditions ensuring the stability of Hopf bifurcation periodic solutions are established. It lavishly illustrates that the stability of QVNN can be demolished more when considering the leakage delay as bifurcation parameter. Finally one simulations example is given to validate the theoretical results.

Global Hopf bifurcation of a two-delay epidemic model with media coverage and asymptomatic infection

The influence of media coverage on disease transmission has attracted more and more attention, but the impact mechanism of media coverage with asymptomatic infection needs to be studied urgently. Therefore, a SIsIaS−M model with two delays and asymptomatic infection is considered. The two kinds of delays are respectively caused by the lag time of media reports about the number of patients and the lag time of audience feedback after media coverage. By using the geometric approach based on the third compound matrix, the existence of a threshold for the average duration of media coverage ensures that the endemic equilibrium in delay-free differential system is globally asymptotically stable. In the delay system the uniqueness, positivity and boundedness of solutions are investigated, and the system is persistent when R0>1. Through stability analysis, we find that the disease can be eliminated when the proportion of symptomatic patients in total patients is above a threshold. By choosing the sum of two delays as a bifurcation parameter, we analyze the stability switches of the endemic equilibrium, and discuss the onset and termination of Hopf bifurcations of periodic solutions. The properties of local Hopf bifurcation are also obtained. Furthermore, the global Hopf bifurcations are unbounded and a sufficient condition for the system to have multiple periodic solutions is obtained. Three parameters related to media coverage are found in numerical simulations to have thresholds that guarantee the stability of the endemic equilibrium. In numerical simulations, if media coverage is highly responsive to the number of patients or it has a strong influence on individual behavioural changes, this can lead to periodic oscillations. Conversely, although this makes the disease stable, the number of patients who reach steady state is higher. This is the same result as for the dissipation rate of media coverage, and again there is a threshold that makes the endemic equilibrium stable.

Complex dynamical study of a delayed prey–predator model with fear in prey and square root harvesting of both species

In the current study, the dynamics of predator-prey systems under the influence of fear effect on the reproduction of prey population and harvesting on both species has been proposed. Assessing the dynamics of the system with the combined influence of fear and harvesting for various values of n is our central objective. We present comprehensive mathematical findings that cover fundamental dynamical features, the presence of positive equilibria, and the stability of all equilibria. Hopf-bifurcating periodic solutions have been demonstrated to emerge around the positive equilibrium point, and the direction of the Hopf-bifurcating limit cycle is determined using the first Lyapunov coefficient. Furthermore, in two-parameter space, we have seen that the system experiences the Bogdanov-Takens bifurcation. Moreover, we have included predator gestation delay and noticed some chaotic dynamics in the system. In addition, we run through numerical simulations to numerically validate our mathematical findings. The article is concluded with a conclusion at the end.

A Delayed Dynamical Model Describing the Effect of Tumor-Induced Vascular Endothelial Growth Factor on Angiogenesis: Backward Bifurcations and Global Dynamics

Dynamic models have been successfully used for the prediction of long-term evolutionary behavior of the growth of tumor cells. In this paper, based on previous research, time delay in the process of the growth of tumor cells is further considered and a class of dynamic models with two-time delays and general functional response function is proposed. This model describes the effect of tumor-induced vascular endothelial growth factor (VEGF) on angiogenesis. First, the model is reduced to a two-dimensional one by means of the quasi-steady-state approximation method. This model has forward and backward bifurcations of the equilibria. Then, local stability of all equilibria, global stability of the tumor cell-free equilibrium and the vascular endothelial cell-free equilibrium, and the persistence of this model are studied in detail. It is found that time delay of tumor-induced vascular endothelial cells proliferation does not change local stability of the equilibria, while time delay associated with the growth of tumor cells can lead to the existence of Hopf bifurcation periodic solutions.

Dynamical analysis for the impact of asymptomatic infective and infection delay on disease transmission

The influence of asymptomatic patients on disease transmission has attracted more and more attention, but the mechanism of some factors affecting disease transmission needs to be studied urgently. Considering the self-healing rate of asymptomatic patients, the cure rate of symptomatic patients, the transformation rate from asymptomatic to symptomatic and the infection delay, a type of infectious disease dynamics model SIsIaS with asymptomatic infection and infection delay is established in this paper. It is found that both the infection delay and the difference size between the cure rate and the self-healing rate not only affect the minimum value of the total number of patients in the persistent state of the disease, but also lead to disease extinction to be controlled by the proportion of symptomatic patients in patients. Moreover, the infection delay can lead to local Hopf bifurcation of periodic solutions. By using the normal form and center manifold theory the direction of Hopf bifurcations and the stability of bifurcated periodic solutions are discussed. At last, sensitivity analysis shows that the infection delay can change the correlation of the proportion of symptomatic patients in patients and the transformation rate to the total number of patients.

Dynamics of a Predator–Prey Model with the Additive Predation in Prey

In this paper, we consider a predator–prey model, in which the prey’s growth is affected by the additive predation of its potential predators. Due to the additive predation term in prey, the model may exhibit the cases of the strong Allee effect, weak Allee effect and no Allee effect. In each case, the dynamics of global features of the model are investigated. Compared to the well-known Lotka–Volterra type model, the model proposed in this paper exhibits much richer and more complex dynamic behaviors, such as the Allee effect, the sensitivity to the initial conditions caused by the strong Allee effect, the oscillatory behavior and the Hopf and heteroclinic bifurcations. Furthermore, the stability and Hopf bifurcation of the model with the density dependent feedback time delay in prey are investigated. By the normal form method and center manifold theory, the explicit formulas are presented to determine the direction of Hopf bifurcation and the stability and period of Hopf-bifurcating periodic solutions. Theoretical analysis and numerical simulation indicate that the delay may destabilize the model, and cause the Hopf bifurcation not only at the interior equilibrium but also at a boundary equilibrium.

Global Hopf bifurcation and dynamics of a stage-structured model with delays for tick population

In this paper, we study a three-stage tick population model with three development delays. Using the total delay τ as the bifurcation parameter, we conduct local and global Hopf bifurcation analysis. Especially, we examine the onset and termination of Hopf bifurcations of periodic solutions from the unique positive equilibrium. We locate all of the stability switches for the equilibrium and demonstrate that the global Hopf bifurcation branches are bounded. This result implies the system undergoes oscillatory behavior only with bounded delays. The key step in the proof is the exclusion of 2τ-periodic solutions due to the negative Lozinskiĭ measure of the corresponding compound matrix. Numerical simulations are provided to show the complicated patterns of the tick population when the development delays are altered by the temperature and other environmental factors.

Dynamics in a ratio-dependent eco-epidemiological predator-prey model having cross species disease transmission

We have proposed and analyzed an eco-epidemiological predator-prey interaction model having disease in both the population with ratio-dependent functional response. The total population has been classified into susceptible prey, infected prey, susceptible predator, and infected predator. The infection propagation is considered directly proportional to the number of individuals come in to contact with one infected individual. The predation ability of infected predator is neglected during the disease. The infection is considered to be weak infection in case of predator. Predator can recover themselves due to their natural immune system or application of external curative stimulants. Though such weak infections are not generally considered for study but these cannot be completely ignored. The positivity of the solutions, the existence of various biologically feasible equilibrium points, their stability are investigated. The numerical analysis is carried out with hypothetical set of parameters to substantiate the analytical findings that our model exhibits. The oscillatory coexistence of the species which is very common in nature is observed for disease free as well as coexisting system. The stability nature of the Hopf-bifurcating periodic solutions of the disease free as well as coexisting equilibrium are determined by computing the Lyapunov coefficients. Further, the system undergoes the Bogdanov-Takens bifurcation in two-parameter space around the disease free as well as coexisting equilibrium. It is also observed that the system will be disease free through proper predational strategies.

Complex dynamics in a delay differential equation with two delays in tick growth with diapause

We consider a delay differential equation for tick population with diapause, derived from an age-structured population model, with two time lags due to normal and diapause mediated development. We derive threshold conditions for the global asymptotic stability of biologically important equilibria, and give a general geometric criterion for the appearance of Hopf bifurcations in the delay differential system with delay-dependent parameters. By choosing the normal development time delay as a bifurcation parameter, we analyze the stability switches of the positive equilibrium, and examine the onset and termination of Hopf bifurcations of periodic solutions from the positive equilibrium. Under some technical conditions, we show that global Hopf branches are bounded and connected by a pair of Hopf bifurcation values. This allows us to show that diapause can lead to the occurrence of multiple stability switches, coexistence of two stable limit cycles, among other rich dynamical behaviours.

Periodic forcing on degenerate Hopf bifurcation

<p style='text-indent:20px;'>This paper is devoted to the effect of periodic forcing on a system exhibiting a degenerate Hopf bifurcation. Two methods are employed to investigate bifurcations of periodic solution for the periodically forced system. It is obtained by averaging method that the system undergoes fold bifurcation, transcritical bifurcation, and even degenerate Hopf bifurcation of periodic solution. On the other hand, it is also shown by Poincaré map that the system will undergo fold bifurcation, transcritical bifurcation, Neimark-Sacker bifurcation and flip bifurcation. Finally, we make a comparison between these two methods.

Hopf Bifurcation Analysis and Stability for a Ratio-Dependent Predator–Prey Diffusive System with Time Delay

In this paper, we are focused on a new ratio-dependent predator–prey system that introduced the diffusive and time delay effect simultaneously. By analyzing the characteristic equations and the distribution of eigenvalues, we examine the stability and boundary of positive equilibrium states, and the existence of spatially homogeneous and spatially inhomogeneous bifurcating periodic solutions, respectively. Further, we prove that when [Formula: see text], the system has Hopf bifurcation at the positive equilibrium state. By using the center manifold reduction, we simplify the system so that we can convert an infinite-dimensional system into a low-dimensional finite-dimensional system. By using the normal form theory, we obtain explicit expressions for the direction, stability and period of Hopf bifurcation periodic solutions. Finally, we have illustrated the main results in this thesis by numerical examples, our work may provide some useful measures to save time or cost and to control the ecosystem.

Global dynamics of an immunosuppressive infection model with stage structure.

In this paper, we propose an immunosuppressive infection model incorporating natural mortality of immune cells during the time lag needed for the expansion of immune cells. Starting from a stage structure model for the immune cells with various ages, we use the method of characteristic lines to derive a delay differential equation for the population of mature immune cells. Then, we use Lyapunov functional techniques to obtain the global dynamics of the model system. Specifically, we show that the virus dominant equilibrium is globally asymptotically stable when the delay is large. Next, we conduct local and global Hopf bifurcation analysis for the proposed model via Hopf bifurcation theory of delay differential equations. We choose the delay as a bifurcation parameter and examine the onset and termination of Hopf bifurcations of periodic solutions from the immune control equilibrium. We also prove that the model has only a finite number of Hopf bifurcation values, and the periodic solutions with specific oscillation frequencies occur only in bounded delay intervals. Under some technical conditions, we show that two global Hopf branches bifurcated from different Hopf bifurcation values may connect to each other and thus be bounded. However, unlike the global Hopf bifurcation results in the existing literature, the Hopf branches for our model system are not necessarily bounded, though the delay components are always bounded. Numerical simulation suggests that bounded and unbounded Hopf branches may co-exist in the bifurcation diagram. Moreover, we observe a new interesting phenomenon that a global Hopf branch may have uniformly bounded periodic solutions, bounded delays, but unbounded periods.

Role of Fear in a Predator–Prey Model with Beddington–DeAngelis Functional Response

Abstract In the present article, we investigate the impact of fear effect in a predator–prey model, where predator–prey interaction follows Beddington–DeAngelis functional response. We consider that due to fear of predator the birth rate of prey population reduces. Mathematical properties, such as persistence, equilibria analysis, local and global stability analysis, and bifurcation analysis, have been investigated. We observe that an increase in the cost of fear destabilizes the system and produces periodic solutions via supercritical Hopf bifurcation. However, with further increase in the strength of fear, system undergoes another Hopf bifurcation and becomes stable. The stability of the Hopf-bifurcating periodic solutions is obtained by computing the first Lyapunov coefficient. Our results suggest that fear of predation risk can have both stabilizing and destabilizing effects.

Supercritical and subcritical Hopf-bifurcations in a two-delayed prey–predator system with density-dependent mortality of predator and strong Allee effect in prey

One of the possible ways to visualize the effect of intra- and inter-species synergistic and antagonistic interactions in a natural ecosystem is the detailed analysis of the underlying prey-predator model, and the subsequent analytical findings may provide a definite justification towards the species coexistence, which often maintains biodiversity in nature. Here, our central motivation is to understand the combined effect of the Allee threshold and intra-species competition on the evolution of interacting species, which often experience delays in evolution due to its complex ecological and physiological processes. Therefore, in the present paper, we extensively analyze the stability of a two-delayed prey-predator system in the presence of strong Allee effects in prey and intra-species competition in predator. In addition, we capture the reality of time difference between the lifespan of the prey and predator through proper nondimensionalization. The two delays in the proposed retarded system correspond to the intra-specific competition-induced feedback time lag to the prey and predator gestation period. In the absence of intra-predator competition, the present dynamics unveils supercritical Hopf-bifurcation around the interior point of coexistence which is in-line with the existing literature. It is noteworthy to mention that the proposed Allee system exhibits subcritical Hopf-bifurcation in the presence of intra-species competition in predator. We confirm the occurrence of both supercritical and subcritical Hopf-bifurcations via calculating the direction and stability of Hopf-bifurcating periodic solutions using the normal form method and the center manifold theory. Moreover, the suggested delayed schema presents supercritical Hopf-bifurcation at the boundary steady-state, where the population density of prey exists at its maximum carrying capacity. We recognize the bistability between extinction and coexistence, and the proposed model also exhibits the ‘chaotic concurrence between prey and predator,’ which happens through the period-doubling bifurcation. The existence of chaos is validated using the estimated power spectrum and the spectrum of Lyapunov exponents. The primary finding of this paper is that Allee threshold induces the capability to the density-dependent death rate of predator towards changing the stability of ‘oscillatory coexistence between prey and predator.’

Dynamical Analysis of a &amp;lt;i&amp;gt;Schistosomiasis japonicum&amp;lt;/i&amp;gt; Model with Time Delay

In this paper, a Schistosomiasis japonicum model incorporating time delay is proposed which represents the developmental time from cercaria penetration through skins of human hosts to egg laying. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equations, the local stability of the equilibria is investigated. And it proves that Hopf bifurcations occur when the time delay passes through a sequence of critical value. Furthermore, the explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions are derived by using techniques from the normal form theory and Center Manifold Theorem. Some numerical simulations which support our theoretical analysis are also conducted.

Stability of Hopf-bifurcating limit cycles in a diffusion-driven prey-predator system with Allee effect and time delay.

In this paper, we investigate the effect of the gestation delay on the spatiotemporal pattern formation in a prey-predator system with monotonic functional response with saturation and intraspecific competition among the predator population in presence of the additive Allee effect in prey growth. In this regard, we present rigorous analytical results for determining the delay-induced Hopf bifurcation threshold and the associated properties of the Hopf-bifurcating periodic solutions and verify them with the help of numerical simulations. We derive analytically the delay-induced Hopf bifurcation threshold by employing the linear stability analysis about the unique spatially uniform coexistence steady state. Also, we provide the expressions for determining the direction and stability of the Hopfbifurcating periodic solutions by using the normal form theory and center manifold reduction. The numerical simulation results reveal that the Hopf bifurcation can potentially lead to spatially homogeneous periodic in time distribution of the populations which will eventually settles to chaotic in space and time distribution for sufficiently large value of the time delay. Further, our numerical investigations reveal that the time delay can change one stationary pattern to another through the loss of monotonicity property of the spatially averaged densities.

Graphical Hopf Bifurcation of a Filippov HTLV-1 Model With Delay in Cytotoxic T Cells Response

In this paper, we present a discontinuous cytotoxic T cells (CTLs) response for HTLV-1. Moreover, a delay parameter for the activation of CTLs is considered. In fact, a system of differential equation with discontinuous right-hand side with delay is defined for HTLV-1. For analyzing the dynamical behavior of the system, graphical Hopf bifurcation is used. In general, Hopf bifurcation theory will help to obtain the periodic solutions of a system as parameter varies. Therefore, by applying the frequency domain approach and analyzing the associated characteristic equation, the existence of Hopf bifurcation by using delay immune response as a bifurcation parameter is determined. The stability of Hopf bifurcation periodic solutions is obtained by the Nyquist criterion and the graphical Hopf bifurcation theorem. At the end, numerical simulations demonstrated our results for the system of HTLV-1.

Dynamics of a delayed phytoplankton-zooplankton system with Crowley-Martin functional response

In this paper, a delayed phytoplankton-zooplankton system with Crowley-Martin functional response is investigated analytically. We study the permanence and analyze the stability of the both boundary and positive equilibrium points for the system with delay as well as the system without delay. The global asymptotic stability is discussed by constructing a suitable Lyapunov functional. Numerical analysis indicates that the delay does not change the stability of the positive equilibrium point. Furthermore, we also show that due to the increase of the delay there occurs a Hopf bifurcation of periodic solutions. It is found that population fluctuations will not appear under the condition of certain parameters. In addition, we determine the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions by applying a normal form method and center manifold theory. Finally, some numerical simulations are carried out to support our theoretical analysis results.

Stability, bifurcations and synchronization in a delayed neural network model of [formula omitted]-identical neurons

This paper deals with dynamic behaviors of Hopfield type neural network model of n(≥3) identical neurons with two time-delayed connections coupled in a star configuration. Delay dependent as well as independent local stability conditions about trivial equilibrium is found. Considering synaptic weight and time delay as parameters Hopf-bifurcation, steady-state bifurcation and equivariant steady state bifurcation criteria are given. The criterion for the global stability of the system is presented by constructing a suitable Lyapunov functional. Also conditions for absolute synchronization about the trivial equilibrium are also shown. Using normal form method and the center manifold theory the direction of the Hopf-bifurcation, stability and the properties of Hopf-bifurcating periodic solutions are determined. Numerical simulations are presented to verify the analytical results. The effect of synaptic weight and delay on different types of oscillations, e.g. in-phase, phase-locking, standing wave and oscillation death, has been shown numerically.

Bifurcation Analysis in a Three-Neuron Artificial Neural Network Model with Distributed Delays

In this paper, a three-neuron artificial neural network model with distributed delays is considered. Its dynamics is investigated in term of the linear stability analysis and Hopf bifurcation analysis. By regarding the sum of two delays as a bifurcation parameter and analyzing the associated characteristic equation, we find that Hopf bifurcation occurs when the bifurcation parameter passes through some certain values. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions are derived by using the normal form method and center manifold theory. Finally, computer simulations are given to support the theoretical predictions.