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In this paper, we investigate the mathematical analysis of a mathematical model describing the virotherapy treatment of a cancer with logistic growth and the effect of viral cycle presented by a time delay. The cancer population size is divided into uninfected and infected compartments. Depending on time delay, we prove the positivity and boundedness and the stability of equilibria. We give conditions on which the viral cycle leads to “Jeff’s phenomenon” observed in laboratory and causes oscillations in cancer size via Hopf bifurcation theory. We establish an algorithm that determines the bifurcation elements via center manifold and normal form theories. We give conditions which lead to a supercritical or subcritical bifurcation. We end with numerical simulations illustrating our theoretical results. [ABSTRACT FROM AUTHOR]

Research on nonautonomous delayed differential equations (DDEs) is crucial and very difficult due to nonautonomy and time delay in many fields. The main work of the present paper is to discuss complex dynamics of nonautonomous DDEs, such as Hopf bifurcation, periodic solutions and multistability. We consider three examples of nonautonomous DDEs with time-varying coefficients: a harmonically forced Duffing oscillator with time delayed state feedback and periodic disturbance, generalized van der Pol oscillator with delayed displacement difference feedback and periodic disturbance, and an electro-mechanical system with delayed and periodic disturbance. Firstly, we obtain the amplitude equations of these three examples by the method of multiple scales (MMS), and then analyze the stability of approximate solutions by the Routh–Hurwitz criterion. The obtained amplitude equations are used to construct the bifurcation diagrams, so that we can observe the occurrence of the Hopf bifurcation and judge its type (super- or subcritical) from the bifurcation diagrams. We discover rich dynamic phenomena of the three systems under consideration, such as Hopf bifurcation, quasi-periodic solutions and the coexistence of multiple stable solutions, and then discuss the impact of some parameter changes on the system dynamics. Finally, we validate the correctness of these theoretical conclusions by software WinPP, and the numerical simulations are consistent with our theoretical findings. Therefore, the MMS we use to analyze the dynamics of nonautonomous DDEs is effective, which is of great significance to the research of nonautonomous DDEs in many fields. [ABSTRACT FROM AUTHOR]

This paper discusses the problem of social media addiction that pose a major threat to the human population especially children and teenagers. It is well known that Cognitive Behavioral Therapy (CBT) is an effective treatment to treat the addict individuals and delay in the treatment leads the patient to worst stage even to death. Therefore, it is important to identify the individuals who has addiction symptoms at early stage and to provide proper counseling. We propose and analyze a nonlinear mathematical model for social media addiction problem using case detection strategy to reduce the addiction. The basic reproduction number and equilibria of the model are computed. Further, the deterministic model is extended to delay differential equation model by incorporating transmission delay and treatment delay in the system. The local stability of different equilibria is discussed in detail. Additionally, the model is converted to stochastic model and numerical simulation is carried out to compare the results of both deterministic and stochastic model. Numerical result shows that the introduction of time delays can destabilize the model system and Hopf bifurcation occurs due to periodic oscillations when certain equilibrium point crosses the delay threshold limit. Our results of stochastic model show a smaller number of social media users and addict population when compared with deterministic model. Also, our results reveal that detection and counseling parameters play a vital role in reducing addiction population. Presented results clearly suggest that there is a need to use effective detection strategy and suitable counseling program to reduce the social media addiction level. [ABSTRACT FROM AUTHOR]

In this study, we principally investigate a fractional‐order stage‐structured predator–prey system including distributed time delays and discrete time delays. Taking advantage of transformation of the variable, we obtain an isovalent version of the considered fractional‐order stage‐structured predator–prey system including distributed time delays and discrete time delays. The isovalent version includes fractional‐order and integer‐order equations. Utilizing the stability criterion and bifurcation theory of fractional‐order differential equation, a novel delay‐independent bifurcation condition to ensure the appearance of Hopf bifurcation for the fractional‐order stage‐structured predator–prey system is set up. The impact of time delay on the stability and bifurcation is clearly revealed. Numerical simulation figures are presented to sustain the rationality of the derived key conclusions. [ABSTRACT FROM AUTHOR]

This paper is concerned with equivariant normal forms of semilinear functional differential equations (FDEs) in general Banach spaces. The analysis is based on the theory previously developed for autonomous delay differential equations and on the existence of invariant manifolds. We show that in the neighborhood of trivial solutions, variables can be chosen so that the form of the reduced vector field relies not only on the information of the linearized system at the critical point but also on the inherent symmetry. We observe that the normal forms give critical information about dynamical properties, such as generic local branching spatiotemporal patterns of equilibria and periodic solutions. As an important application of equivariant normal forms, we not only establish equivariant Hopf bifurcation theorem for semilinear FDEs in general Banach spaces, but also in a natural way derive criteria for the existence, stability, and bifurcation direction of branches of bifurcating periodic solutions. We employ these general results to obtain the existence of infinite many small-amplitude wave solutions for a delayed Ginzburg-Landau equation on a two-dimensional disk with the homogeneous Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

This paper proposes a coupled-circuit system composed of two Chua's circuits with lossless transmission lines. By applying the Kirchhoff's voltage and current laws, the equations that describe the coupled-circuit system are reduced to two coupled neutral-type differential equations with a time delay. Subsequently, the conditions for global stability are established using the inequality technology, and those for local stability and Hopf bifurcation are obtained by selecting the length of the transmission line as the bifurcation parameter. By using the normal-form theory and central manifold theorem, the formulas for the Hopf bifurcation direction and bifurcation periodic solution are obtained. Finally, the numerical simulations not only verify the theoretical analysis but also show that chaos exists near the Hopf bifurcation point. [ABSTRACT FROM AUTHOR]

This paper studies the behavior of a signal transduction model with signal amplification delay. By choosing the delay as a bifurcation parameter, we show that the delay model exhibits a Hopf bifurcation at the positive equilibrium. Numerical simulations are also given to support the theoretical predictions. [ABSTRACT FROM AUTHOR]

To reduce the global burden of mosquito-borne diseases, e.g. dengue, malaria, the need to develop new control methods is to be highlighted. The sterile insect technique (SIT) and various genetic modification strategies, have a potential to contribute to a reversal of the current alarming disease trends. In our previous work, the ordinary differential equation (ODE) models with different releasing sterile mosquito strategies are investigated. However, in reality, implementing SIT and the releasing processes of sterile mosquitos are very complex. In particular, the delay phenomena always occur. To achieve suppression of wild mosquito populations, in this paper, we reassess the effect of the delayed releasing of sterile mosquitos on the suppression of interactive mosquito populations. We extend the previous ODE models to the delayed releasing models in two different ways of releasing sterile mosquitos, where both constant and exponentially distributed delays are considered, respectively. By applying the theory and methods of delay differential equations, the effect of time delays on the stability of equilibria in the system is rigorously analyzed. Some sustained oscillation phenomena via Hopf bifurcations in the system are observed. Numerical examples demonstrate rich dynamical features of the proposed models. Based on the obtained results, we also suggest some new releasing strategies for sterile mosquito populations. [ABSTRACT FROM AUTHOR]

Predator–prey interactions are among the most complicated interactions between biological species, in which there may be both direct effect (through predation) and indirect effect (e.g., fear effect). In the literature, the indirect effect has been largely missing in predator–prey models, until some recent works. Based on the recent work (Wang et al. in J Math Biol 73:1179–1204, 2016) where a fear effect is considered in an ODE model as a cost, in this paper, we also consider a benefit from the anti-predation response in addition to the cost, as well as a time delay in the transfer of biomass from the prey to the predator after predation. This results in a system of delay differential equations (DDEs). By analyzing this nonlinear DDE system, we obtain some insights on how the anti-predation response level (indirect effect) and the biomass transfer delay jointly affect the population dynamics; particularly we show how the nonlinearity in the predation term mediated by the fear effect affects the long term dynamics of the model system. We also perform some numerical computations and simulations to demonstrate our results. These results seem to suggest a need to revisit existing predator–prey models in the literature by incorporating the indirect effect and biomass transfer delay. [ABSTRACT FROM AUTHOR]

• Stability properties of DEs with a general distributed delay are investigated. • Saddle-node and Hopf bifurcation curves are determined in a general framework. • Stability region of the equilibrium is fully characterized in the parameter plane. • Criticality of Hopf bifurcation is analyzed by the multiple times scales method. • Theoretical results are showcased in the framework of a simple neural model. For a differential equation involving a general distributed time delay, a local stability and bifurcation analysis is performed, relying on fundamental properties of the characteristic function of the random variable whose probability density function is the delay distribution. Based on the root locus method, the bifurcation curves are determined in the considered parameter plane, also providing the number of unstable roots of the analyzed characteristic equation in each of the open connected regions delimited by these curves. This leads to a characterisation of the stability region of the considered equilibrium in the corresponding parameter plane. A Hopf bifurcation analysis is also completed in the general setting, and the criticality is analyzed by employing the method of multiple times scales. In contrast with some previously reported results from the literature, our analysis is accomplished in a general context and only then exemplified for particular types of delay distributions (e.g. Dirac, Gamma, uniform and triangular). The theoretical results are showcased in the framework of a simple neural model. [ABSTRACT FROM AUTHOR]

In this paper, we study a fractional-order model with time-delay to describe the dynamics of Ebola virus infection with cytotoxic T-lymphocyte (CTL) response in vivo. The time-delay is introduced in the CTL response term to represent time required to stimulate the immune system. Based on fractional Laplace transform, some conditions on stability and Hopf bifurcation are derived for the model. The analysis shows that the fractional-order with time-delay can effectively enrich the dynamics and strengthen the stability condition of fractional-order infection model. Finally, the derived theoretical results are justified by some numerical simulations. [ABSTRACT FROM AUTHOR]

We study stability and Hopf bifurcation analysis of a model that refers to the competition between the immune system and an aggressive host such as a tumor. The model which describes this competition is governed by a reaction-diffusion system including time delay under the Neumann boundary conditions, and is based on Kuznetsov-Taylor's model. Choosing the delay parameter as a bifurcation parameter, we first show that Hopf bifurcation occurs. Second, we determine two properties of the periodic solution, namely its direction and stability, by applying the normal form theory and the center manifold reduction for partial functional differential equations. Furthermore, we discuss the effects of diffusion on the dynamics by analyzing a model with constant coefficients and perform some numerical simulations to support the analytical results. The results show that diffusion has an important effects on the dynamics of a mathematical model. [ABSTRACT FROM AUTHOR]

We consider the system of delay differential equations (DDE) representing the models containing two cells with time-delayed connections. We investigate global, local stability and the bifurcations of the trivial solution under some generic conditions on the Taylor coefficients of the DDE. Regarding eigenvalues of the connection matrix as bifurcation parameters, we obtain codimension one bifurcations (including pitchfork, transcritical and Hopf bifurcation) and Takens-Bogdanov bifurcation as a codimension two bifurcation. For application purposes, this is important since one can now identify the possible asymptotic dynamics of the DDE near the bifurcation points by computing quantities which depend explicitly on the Taylor coefficients of the original DDE. Finally, we show that the analytical results agree with numerical simulations. [ABSTRACT FROM AUTHOR]

In this work, we develop a mathematical model to study the effect of drug on the development of cancer including the quiescent compartment. The model is governed by a system of delay differential equations where the delay represents the time that the cancer cell take to proliferate. Our analytical study of the stability shows that by considering the time delay as a parameter of bifurcation, it is possible to have stability switch and oscillations through a Hopf bifurcation. Moreover, by introducing the drug intervention term, the critical delay value increases. This indicates that the system can tolerate a longer delay before oscillations start. In the end, we present some numerical simulations illustrating our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we give a detailed Hopf bifurcation analysis of a recurrent neural network system involving both discrete and distributed delays. Choosing the sum of the discrete delay terms as a bifurcation parameter the existence of Hopf bifurcation is demonstrated. In particular, the formulae determining the direction of the bifurcations and the stability of the bifurcating periodic solutions are studied by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included. [ABSTRACT FROM AUTHOR]

For the model of periodic chronic myelogenous leukemia considered by Pujo-Menjouet, Mackey et al., model consisting of two delay differen- tial equations, the equation for the density of so-called "resting cells" was studied from numerical and qualitative point of view in several works. In this paper we focus on the equation for the density of proliferating cells and study it from a qualitative point of view. [ABSTRACT FROM AUTHOR]

Abstract: It is acknowledged that coral reefs are globally threatened. P.J. Mumby et al. [10] constructed a mathematical model with ordinary differential equations to investigate the dynamics of coral reefs. In this paper, we first provide a detailed global analysis of the coral reef ODE model in [10]. Next we incorporate the inherent time delay to obtain a mathematical model with delay differential equations. We consider the grazing intensity and the time delay as focused parameters and perform local stability analysis for the coral reef DDE model. If the time delay is sufficiently small, the stability results remain the same. However, if the time delay is large enough, macroalgae only state and coral only state are both unstable, while they are both stable in the ODE model. Meanwhile, if the grazing intensity and the time delay are endowed some suitable values, the DDE model possesses a nontrivial periodic solution, whereas the ODE model has no nontrivial periodic solutions for any grazing rate. We study the existence and property of the Hopf bifurcation points and the corresponding stability switching directions. [Copyright &y& Elsevier]

Abstract: Recently, a large number of mathematical models that are described by delay differential equations (DDEs) have appeared in the life sciences. In this paper, we present a delay differential model to describe the interactions between the effector and tumour cells. The existence of the possible steady states and their local stability and change of stability via Hopf bifurcation are theoretically and numerically investigated. Parameter estimation problem for given real observations, using least squares approach, is studied. The global stability and sensitivity analysis are also numerically proved for the model. The stability and periodicity of the solutions may depend on the time-lag parameter. The model is qualitatively consistent with the experimental observations of immune-induced tumour dormancy. The model also predicts dormancy as a transient period of growth which necessarily results in either tumour elimination or tumour escape. [Copyright &y& Elsevier]

The aim of this paper is to give a detailed analysis of Hopf bifurcation of a ratio-dependent predator–prey system involving two discrete delays. A delay parameter is chosen as the bifurcation parameter for the analysis. Stability of the bifurcating periodic solutions is determined by using the centre manifold theorem and the normal form theory introduced by Hassard et al. Some of the bifurcation properties including the direction, stability and period are given. Finally, our theoretical results are supported by some numerical simulations. [ABSTRACT FROM AUTHOR]

A mathematical model, coupling the dynamics of short-term stem-like cells and mature leukocytes in leukemia with that of the immune system, is investigated. The model is described by a system of seven delay differential equations with seven delays. Three equilibrium points E0, E1, E2 are highlighted. The stability and the existence of the Hopf bifurcation for the equilibrium points are investigated. In the analysis of the model, the rate of asymmetric division and the rate of symmetric division are very important. [ABSTRACT FROM AUTHOR]

Abstract: This paper continues the work contained in two previous papers of the authors, devoted to the qualitative study of the dynamical system generated by a delay differential equation that models leukemia. The problem depends on five parameters and has two equilibria. As already known, at the non-zero equilibrium solution, for certain values of the parameters, Hopf bifurcation occurs. Our aim here is to investigate the Bautin bifurcation for the considered model. By using a second order approximation of the center manifold, we compute the first Lyapunov coefficient for Hopf bifurcation points. We explore the space of parameters in a zone of biological interest and find, by direct computation, points where the first Lyapunov coefficient equals zero. These are candidates for Bautin type bifurcation. For these points we compute the second Lyapunov coefficient, that determines the type of Bautin bifurcation. The computation of the second Lyapunov coefficient requires a fourth order approximation of the center manifold, that we determine. The points with null first Lyapunov coefficient obtained are given in tables and are also plotted on surfaces of Hopf bifurcation obtained by fixing two parameters. Next we vary two parameters around a point in the space of parameters with and numerically explore the behavior of the solution. The results confirm the Bautin type-bifurcation. [Copyright &y& Elsevier]

Abstract: In this paper, we have considered a system of delay differential equations. The system without delayed arises in the Lengyel–Epstein model. Its dynamics are studied in terms of local analysis and Hopf bifurcation analysis. Linear stability is investigated and existence of Hopf bifurcation is demonstrated via analyzing the associated characteristic equation. For the Hopf bifurcation analysis, the delay parameter is chosen as a bifurcation parameter. The stability of the bifurcating periodic solutions is determined by using the center manifold theorem and the normal form theory introduced by Hassard et al. (1981) [7]. Furthermore, the direction of the bifurcation, the stability and the period of periodic solutions are given. Finally, the theoretical results are supported by some numerical simulations. [Copyright &y& Elsevier]

Abstract: A reaction–diffusion model with logistic type growth, nonlocal delay effect and Dirichlet boundary condition is considered, and combined effect of the time delay and nonlocal spatial dispersal provides a more realistic way of modeling the complex spatiotemporal behavior. The stability of the positive spatially nonhomogeneous positive equilibrium and associated Hopf bifurcation are investigated for the case of near equilibrium bifurcation point and the case of spatially homogeneous dispersal kernel. [Copyright &y& Elsevier]

Abstract: In the machining process, unstable self-excited vibrations known as regenerative chatter can occur, causing excessive tool wear or failure, and a poor surface finish on the machined workpiece, hence the relevant measures must be taken to predict and avoid this phenomenon of instability. In this paper, we propose a weakly nonlinear model with square and cubic terms in both structural stiffness and regenerative terms, to represent self-excited vibrations in machining. It is proved that Hopf bifurcation exists when bifurcation parameter equals a critical value, a formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are given by using the normal form method and center manifold theorem. Numerical simulations show excellent agreement with the theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper, a delayed differential equation model that describes infection of thymus with HIV-1 is considered. We first investigate the existence and stability of the equilibria and then we study the effect of the time delay on the stability of the infected equilibrium. Criteria are given to ensure that the infected equilibrium is asymptotically stable for all delay. Finally, by using a delay as a bifurcation parameter, the existence of Hopf bifurcation is investigated. Numerical simulations are presented to illustrate the analytical results. [Copyright &y& Elsevier]

Abstract: In this paper, we study the influence of time delays on the dynamics of the classical Gompertz model. We consider the models with one discrete delay introduced in two different ways and the model with two delays which generalise those with one delay. We study asymptotic behaviour and bifurcations with respect to the ratio of delays . Our results show that in such model with two delays there is only one stability switch and for a threshold value of bifurcation parameter, Hopf bifurcation (HB) occurs. However, the type of HB, and therefore its stability (i.e. stability of periodic orbits arising due to it), strongly depends on the magnitude of . The function describing stability of HB is periodic with respect to . Within one period of length 4 five changes of HB stability are observed. We also introduce the second model with two delays which has a better biological interpretation than the first one. In that model several stability switches can occur, depending on the model parameters. We illustrate analytical results on the example of tumour growth model with parameters estimated on the basis of experimental data. [Copyright &y& Elsevier]

Abstract: In this paper, effects of the synaptic delay of signal transmissions on the pattern formation of nonlinear waves in a bidirectional ring of neural oscillators is studied. Firstly, the linear stability of the model is investigated by analyzing the associated characteristic transcendental equation. Meanwhile, using the symmetric bifurcation theory of delay differential equations coupled with the representation theory of Lie groups, we discuss the spontaneous bifurcation of multiple branches of periodic solutions and their spatio-temporal patterns. Finally, Hopf bifurcation directions and corresponding stabilities of bifurcating periodic orbits are derived by using the normal form approach and the center manifold theory. These theoretical results are significant to complement experimental and numerical observations made in living neuronal systems and artificial neural networks, in order to better understand the mechanisms underlying the system’s dynamics. [Copyright &y& Elsevier]

In this paper, a model of predator-prey with disease in food chain is investigated - where, prey is infected by bacteria and then the infected prey in turn infects predator, but the disease does not spread among predators. The law for disease development and biodiversity conservation are the focus. Stability and persistence are deduced in terms of system parameters. Next, time required delay is incorporated into the model. Stability and bifurcation analysis of the delay differential equation model are carried out. Furthermore, stability and direction of the bifurcating periodic solutions are performed by the normal form theory and the center manifold argument. Finally, numerical simulations are included for illustrating the theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a differential–algebraic biological economic system with time delay where the model with Holling type II functional response incorporates a constant prey refuge and prey harvesting. By considering time delay as bifurcation parameter, we analyze the stability and the Hopf bifurcation of the differential–algebraic biological economic system based on the new normal form approach of the differential–algebraic system and the normal form approach and the center manifold theory. Finally, numerical simulations illustrate the effectiveness of our results. [Copyright &y& Elsevier]

A five-dimensional delay differential system is proposed to describe oscillatory behaviour in microbial continuous culture. Stability and Hopf bifurcation are discussed, including the changing regularity of the bifurcation value. An explicit algorithm for determining the direction of Hopf bifurcation and the stability of periodic solution is derived, using the theory of normal form and centre manifold. Finally, some numerical simulations are performed to illustrate the analytical results found. [ABSTRACT FROM AUTHOR]

This paper studied the stability and Hopf bifurcation of a type of protein synthesis system with time delay and negative feedback. Firstly, it is proved theoretically that the time delay, nonlinearity in the protein production and the cooperativity in the negative feedback are key factors to generate circadian oscillation; Taking time delay as a parameter, we obtained the critical value of the time delay that Hopf bifurcation generates. Secondly, based on the center manifold and normal form theorem, we derived the formulas for determining the stability of bifurcating periodic solutions and the supercritical or subcritical Hopf bifurcation. Finally, the matlab program is used to simulate the results. [ABSTRACT FROM AUTHOR]

Abstract: A delayed Lotka–Volterra two-species predator–prey system with discrete hunting delay and distributed maturation delay for the predator population described by an integral with a strong delay kernel is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. It is found that under suitable conditions on the parameters the positive equilibrium is asymptotically stable when the hunting delay is less than a certain critical value and unstable when the hunting delay is greater than this critical value. Meanwhile, according to the Hopf bifurcation theorem for functional differential equations (FDEs), we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the hunting delay crosses through a sequence of critical values. In particular, by applying the normal form theory and the center manifold reduction for FDEs, an explicit algorithm determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions occurring through Hopf bifurcations is given. Finally, to verify our theoretical predictions, some numerical simulations are also included at the end of this paper. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, Hopf bifurcation for a delayed SIS epidemic model with stage structure and nonlinear incidence rate is investigated. Through theoretical analysis, we show the positive equilibrium stability and the conditions that Hopf bifurcation occurs. Applying the normal form theory and the center manifold argument, we derive the explicit formulas determining the properties of the bifurcating periodic solutions. In addition, we also study the effect of the inhibition effect on the properties of the bifurcating periodic solutions. To illustrate our theoretical analysis, some numerical simulations are also included. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a neural network model consisting of two coupled oscillators with delayed feedback and excitatory-to-excitatory connection. We study how the strength of the connections between the oscillators affects the dynamics of the neural network. We give a full classification of all equilibria in the parameter space and obtain its linear stability by analyzing the characteristic equation of the linearized system. We also investigate the spatio-temporal patterns of bifurcated periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. Moreover, the stability and bifurcation direction of the bifurcated periodic solutions are obtained by employing center manifold reduction and normal form theory. Some numerical simulations are provided to illustrate the theoretical results. [Copyright &y& Elsevier]

Abstract: By introducing discrete time delay into the model for producing 1,3-propanediol by microbial continuous fermentation, we consider the stability and Hopf bifurcation of the delay differential system. Through numerical simulations, we get the rule of branch value changing with parameter and draw the pictures of periodic solutions and phase diagrams with specified parameters. The effect of time delay suggests that the system can qualitatively describe oscillatory phenomena occurring in the experiment. [Copyright &y& Elsevier]

In this paper we analyze a simple mathematical model which describes the interaction between proteins P53 and Mdm2. For the stationary state we discuss the local stability and the existence of a Hopf bifurcation. We study the direction and stability of the bifurcating periodic solutions by choosing the delay as a bifurcation parameter. Finally, we will offer some numerical simulations and present our conclusions. [ABSTRACT FROM AUTHOR]

Abstract: The dynamics of a physiological control systems described by a first-order nonlinear delay differential equations are investigated. we proved that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived, using the theory of normal form and center manifold. Global existence of periodic solutions are established using a global Hopf bifurcation result due to Wu [Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799–4838]. [Copyright &y& Elsevier]

Abstract: The term leukopoiesis describes processes leading to the production and regulation of white blood cells. It is based on stem cells differentiation and may exhibit abnormalities resulting in severe diseases, such as cyclical neutropenia and leukemias. We consider a nonlinear system of two equations, describing the evolution of a stem cell population and the resulting white blood cell population. Two delays appear in this model to describe the cell cycle duration of the stem cell population and the time required to produce white blood cells. We establish sufficient conditions for the asymptotic stability of the unique nontrivial positive steady state of the model by analysing roots of a second degree exponential polynomial characteristic equation with delay-dependent coefficients. We also prove the existence of a Hopf bifurcation which leads to periodic solutions. Numerical simulations of the model with parameter values reported in the literature demonstrate that periodic oscillations (with short and long periods) agree with observations of cyclical neutropenia in patients. [Copyright &y& Elsevier]

Abstract: We study a mathematical model describing the dynamics of a pluripotent stem cell population involved in the blood production process in the bone marrow. This model is a differential equation with a time delay. The delay describes the cell cycle duration and is uniformly distributed on an interval. We obtain stability conditions independent of the delay and show that the distributed delay can destabilize the entire system. In particular, it is shown that a Hopf bifurcation can occur. [Copyright &y& Elsevier]

In this paper, a two-neuron network with different time delays is investigated. By analyzing the associated characteristic equation, we obtain the conditions for delay-dependent and delay-independent asymptotic stability, respectively. Furthermore, we find that if the delay is used as a bifurcation parameter, Hopf bifurcation would occur. The direction and stability of the bifurcating periodic solutions are determined by using the Nyquist criterion and the graphical Hopf bifurcation theorem. Some examples are included to illustrate our results. [ABSTRACT FROM AUTHOR]