Delay Τ1 Research Articles

Dynamic analysis of a cytokine-enhanced viral infection model with time delays and CTL immune response

In this paper, a cytokine-enhanced viral infection model with time delays and CTL immune response is proposed and analyzed. Multiple time delays, namely intracellular delay τ1, virus replication delay τ2 and immune response delay τ3 are included. Firstly, two key parameters with strong biological significance, namely the virus reproductive number R0 and the CTL immune reproductive number R1 are derived. And then the stability of equilibria of the model is investigated by constructing suitable Lyapunov functionals and using LaSalle’s invariance principle. We prove that if R0<1, then the infection-free equilibrium E0 is globally asymptotically stable for any τ1,τ2,τ3≥0; if R0>1,R1<1, then the immunity-inactivated equilibrium E1 is globally asymptotically stable for any τ1,τ2,τ3≥0; and if R1>1, then the immunity-activated equilibrium E∗ is globally asymptotically stable for any τ1,τ2≥0 and τ3=0. For τ1,τ2≥0,τ3>0, theoretical analysis and numerical simulations show that with the increase of τ3, the equilibrium E∗ loses its stability and the system generates Hopf bifurcation, which means the system produces periodic oscillation under certain conditions.

Complex dynamical behaviour of predator–prey model with harvesting and multiple delays

Abstract In this work, we investigate a predator–prey model with Crowley–Martin functional response and constant harvesting. The model is extended by incorporating two constant time delays, where the first delay(τ 1) is for density dependent feedback mechanism in the logistic growth of the prey and the second one is for gestation delay(τ 2) of the predator population. The dynamical behaviours such as positivity, boundedness, extinction criteria and existence, stability and bifurcations of the equilibria of the non-delay model are qualitatively discussed. The existence of periodic solutions via Hopf-bifurcation with respect to absence of delay, single delay and both delays are established. Finally, numerical simulations have been carried out to confirm our numerical results.

Dynamical Behaviors of an Environmental Protection Expenses Model in Protected Areas with Two Delays

This paper investigates an environmental protection expenses model, which considers the relations between the visitors to the protected areas V, the quality of the environmental resource E, and the capital stock K. In this model, the total tourism income is used partly to increase the capital stock or as the environmental protection expenses. Two time delays are introduced into the number of visitors, since the visitors need time to respond the changes of the environment, and the environment will take time to respond to the input of money. Stability crossing curves in the plane of delays (τ1,τ2) are used to obtain the stable region of equilibrium. Numerical simulations represent the mutual transformation of the supercritical bifurcation and the subcritical bifurcation. Our model shows that under some parameter conditions, the share of tourism income η is related closely to the delay τ1, while the capital stock and the environmental quality can be maintained persistently if the delay τ1 is not too large.

Hopf bifurcation analysis in a predator-prey model with predator-age structure and predator-prey reaction time delay

A predator-prey model with predator-age structure and predator-prey reaction time delay τ1 is investigated. We consider the predator fertility function as a piecewise function related to predator development time τ2. Considering the complexity of the characteristic equation, we theoretically prove that a non-trivial periodic oscillation phenomena through Hopf bifurcation appears when parameter τ1=τ2:=τ passes through some critical values, by employing the integrated semigroup theory and Hopf bifurcation theory for semilinear equations with non-dense domain. We perform model calibration, data fitting and parameter estimation by utilizing real data, and analyze the rationality and deficiencies of our model. The corresponding theoretical results are certificated by numerical simulations. Additionally, when τ1 and τ2 take different values, we numerically analyze their influence on the model and conclude that the dynamic behavior of the system around the positive equilibrium changes between local asymptotic stability and sustained periodic oscillations.

Stability and Hopf Bifurcation of Three-Species Prey-Predator System with Time Delays and Allee Effect

Allee effect is one of the important factors in ecology, and taking it into account can cause significant impacts in the system dynamics. In this paper, we study the dynamics of a two-prey one-predator system, where the growth of both prey populations is subject to Allee effects, and there is a direct competition between the two-prey species having a common predator. Two discrete time delays τ1 and τ2 are incorporated into the model to represent reaction time of predators. Sufficient conditions for local stability of positive interior equilibrium and existence of Hopf bifurcations in terms of threshold parameters τ1∗ and τ2∗ are obtained. A Lyapunov functional is deducted to investigate the global stability of positive interior equilibrium. Sensitivity analysis to evaluate the uncertainty of the state variables to small changes in the Allee parameters is also investigated. Presence of Allee effect and time delays in the model increases the complexity of the model and enriches the dynamics of the system. Some numerical simulations are provided to illustrate the effectiveness of the theoretical results. The model is highly sensitive to small changes in Allee parameters at the early stages and with low population densities, and this sensitivity decreases with time.

Inverse problems for Sturm – Liouville operator. With potential functions from L2[0,π

This paper deals with non-self-adjoint second-order differential operatorswith two constant delays. We consider four boundary value problems 𝐷𝑖,𝑘,𝑖=0,1,𝑘=1,2−𝑦′′(𝑥)+𝑞1(𝑥)𝑦(𝑥−𝜏1)+(−1)𝑖𝑞2(𝑥)𝑦(𝑥−𝜏2)=𝜆𝑦(𝑥),𝑥∈[0,𝜋]𝑦′(0)−ℎ𝑦(0)=0, 𝑦′(𝜋)+𝐻𝑘𝑦(𝜋)=0,where𝜋3≤𝜏2&lt;𝜋2≤2𝜏2≤𝜏1&lt;𝜋, ℎ,𝐻1,𝐻2∈ 𝑅\{0} and 𝜆 is a spectral parameter. We assumethat 𝑞1,𝑞2are real-valued potential functions from𝐿2[0,𝜋]such that 𝑞1(𝑥)=0,𝑥∈[0,𝜏1)and𝑞2(𝑥)=0,𝑥∈[0,𝜏2).The inverse spectral problem of recovering operators from their spectra hasbeen studied. We provethat delays 𝜏1,𝜏2and parameters ℎ,𝐻1,𝐻2are uniquely determined from thespectra. Then we prove that potentials are uniquely determined by Volterra linear integral equations.

A Fractional‐Order Model for Zika Virus Infection with Multiple Delays

Time delays and fractional order play a vital role in biological systems with memory. In this paper, we propose an epidemic model for Zika virus infection using delay differential equations with fractional order. Multiple time delays are incorporated in the model to consider the latency of the infection in a vector and the latency of the infection in the infected host. We investigate the necessary and sufficient conditions for stability of the steady states and Hopf bifurcation with respect to three time delays τ1, τ2, and τ3. The model undergoes a Hopf bifurcation at the threshold parameters , , and . Some numerical simulations are given to show the effectiveness of obtained results. The numerical simulations confirm that combination of fractional order and time delays in the epidemic model effectively enriches the dynamics and strengthens the stability condition of the model.

Stability and Hopf bifurcation analysis in a delayed three-node circuit involving interlinked positive and negative feedback loops

A system involving interlinked positive and negative feedback loops is a flexible motif that can tune itself in gene regulatory networks. It is well known that time delay is inevitable in gene regulatory networks due to transcription and translation not being physically co-located. In this paper, we systematically consider the effect of time delay on the dynamical behavior of the three-node circuit with three time delays. Based on linear stability analysis and bifurcation theory, sufficient conditions for stability of equilibria and oscillatory behaviors via Hopf bifurcation are derived when choosing positive and negative feedback strengths as well as time delays τ1, τ2, τ3 as the bifurcation parameters, respectively. Moreover, stability and direction of the Hopf bifurcation of time delay are studied by using the normal form method and the center manifold theorem. Finally, several examples are performed to illustrate some analytical results we obtained.

Nonlinear dynamic analysis on maglev train system with flexible guideway and double time-delay feedback control

In this paper, the dynamic behavior of time-delayed feedback control for maglev train system with double discrete time delays is considered with flexible guideway. Considering the maglev guideway as Beroulli-Euler beam, the mathematical model of maglev system with flexible guideway is constructed. The time delay of the two state feedback signals in the maglev system occurs simultaneously, and the values are different. The present treatment method only considers one single feedback delay, which are insufficiency. Thus, the Hopf bifurcation with double time-delay feedback of maglev train running on the flexible guideway is analyzed considering time-delayed position feedback signal τ1 and velocity feedback signal τ2. A novel method is presented to develop the double-parametric Hopf bifurcation diagram in relation to τ1 and τ2. Sufficient numerical simulations are provided to illustrate the complex dynamical behavior of the discrete delays τ1 and τ2 for maglev system and we verify the obtained theoretical analysis. Finally, the field experiments are carried out to validate the effectiveness of the Hopf bifurcation analytical method preliminarily.

Stability and Bifurcation Analysis in a Maglev System with Multiple Delays

This paper considers the time-delayed feedback control for Maglev system with two discrete time delays. We determine constraints on the feedback time delays which ensure the stability of the Maglev system. An algorithm is developed for drawing a two-parametric bifurcation diagram with respect to two delays τ1 and τ2. Direction and stability of periodic solutions are also determined using the normal form method and center manifold theory by Hassard. The complex dynamical behavior of the Maglev system near the domain of stability is confirmed by exhaustive numerical simulation.

Nonlinear delay reaction–diffusion equations with varying transfer coefficients: Exact methods and new solutions

The paper deals with one-dimensional nonlinear delay reaction–diffusion equations with varying transfer coefficients of the form ut=[G(u)ux]x+F(u,ū), where u=u(x,t) and ū=u(x,t−τ), with τ denoting the delay time. Generalized and functional separable solutions for this class of equations have been obtained and presented for the first time; these equations have not been known to have such solutions so far. To construct these solutions and solutions of some other delay PDEs, we developed a few exact methods that rely on using invariant subspaces for corresponding nonlinear differential operators. Many of the results are extendable to more complex nonlinear reaction–diffusion equations with several delay times, τ1,…,τm, and equations with time-varying delay, τ=τ(t). All of the equations considered involve several free parameters (or an arbitrary function) and so their solutions can be suitable for testing approximate analytical and numerical methods for nonlinear delay reaction–diffusion equations. The exact methods described may also be applied to other classes of nonlinear delay PDEs.

Global asymptotic stability of nonlinear neutral differential equation

In this paper, the following nonlinear neutral differential equationx′(t)=-a(t)x(t)+c(t)x′(t-τ1(t))+q(t,x(t),x(t-τ2(t)))is considered. By using fixed point theory, we give some new conditions to ensure that the zero solution is globally asymptotically stable in C1. By weakening the assumptions on the neutral coefficient c and the delay τ1 and by obtaining more manageable criteria, some existing results are improved and generalized.

Stochastic tumor growth system with two different kinds of time delay

Abstract The dynamical properties of a noise-driven tumor cell growth system are investigated when there exist two different kinds of time delays, in the deterministic and fluctuating forces, respectively. Using the approximation probability density approach, the delayed Fokker-Planck equation is obtained. The effects of two different time delays on the stationary probability distribution (SPD), the mean value and the mean passage time (MFPT) are discussed. It is found that the time delay τ1 in the deterministic force can enhance tumor cell number, while the time delay τ2 in the fluctuating force can induce a decrease in tumor cell numbers. On the other hand, while τ1 can hold back the extinction of tumor cells, τ2 can speed up their extinction.

Ultrafast Coherent Control in the Electronic Ground State of a Molecule

AbstractWe report on a coherent control experiment to selectively excite a well localized nuclear probability density in the electronic ground state of isolated, jet cooled potassium dimer molecules. The basic idea of this control scheme is to take advantage of the motion of a vibrational wave packet created in an intermediate excited electronic state, from which a Franck‐Condon window to the desired final state in the ground state emerges. To realize this concept we apply a resonant stimulated Raman process, in which the two involved laser pulses are separated by a time delay τ1 and their wavelengths are independently tunable. The method enables us to excite ground state vibrational wave packets, whereby the vibrational energy of the wave packet and the position of its generation along the internuclear separation coordinate of the ground state potential energy surface can in principle be freely chosen by the experimentalist. The results we present for the chosen model system K2 strongly support the idea that this control concept might be one possible approach to selectively populate a desired chemical reaction channel in the electronic ground state of a molecule.

Participation of Inert Species in Air in Piston-Compression Ignition.

The effect of inert species on low-temperature piston-compression ignition is examined. Pseudo-air is prepared by replacing the inert species in air with pure argon. Lean mixtures are composed of leaded or unleaded gasoline as the fuel and the pseudo-air or real air as the oxidizer. Cool-flame ignition delay τ1 does not differ between the case of using unleaded and that of using leaded gasoline. A difference between the effects of the two oxidizers can be found in the blue flame ignition delay τ2 under considerably lean conditions at an equivalence ratio of 0.7. An anti-knock additive, tetramethyllead, does not seem to be responsible for this difference. At onset of cool flame and blue flame in a pancake chamber a sliced-pineapplelike structure which has a central core bulk and a surrounding ring portion near the cylinder wall appears. Only when air is used as the oxidizer, does a blue flame appear in the ring portion near the cylinder wall earlier than in the central core. Inert species such as nitrogen or argon might participate in the eIementary chemical reactions appearing in preflame periods. Argon is chemically inert, but nitrogen or nitrogen compounds probably participate in the preflame reactions in the case of low-temperature ignition.

Inert Species Participation of Air in Piston Compression Ignition.

The effect of inert species on low-temperature piston-compression ignition is examined. Pseudo-air is prepared, replacing inert species in the air with pure argon. Lean mixtures are composed of leaded or unleaded gasoline as a fuel and the pseudo-air or real air as an oxidizer. Cool-flame ignition delay τ1 shows no differences. A distinction between the two oxidizers can be found in blue flame ignition delay τ2 under considerably lean conditions at equivalence ratios of 0.7. An anti-knock additive, tetramethyl lead, does not seem to be responsible for this distinction. When the pseudo-air is used as an oxidizer, onset of cool and blue flames in a pancake chamber shows a sliced-pineapple-like structure having a central core and a ring portion near the cylinder wall. Only when air is used as an oxidizer, does a blue flame appear in the ring portion in the vicinity of the cylinder wall earlier than in the central core. Inert species, nitrogen or argon, would participate as the third body in elementary chemical reactions appearing in preflame periods. Argon may be quite inert chemically, but nitrogen or nitrogen compounds probably act as a chemical species on the preflame reactions in low-temperature ignitions.

Oscillations of second order linear delay differential equations

Consider the second order linear delay differential equation where the coefficients P1,P2,q1,q2 and the delays τ1,τ2,σ1,σ2 are nonnegative real numbers. By using elementary techniques, we give a rigorous proof of the following result. Theorem. Every solution of Eq. (1) oscillates if and only id the characteristic equation has no real roots

Maximum survival capability of an aircraft in a severe windshear

This paper is concerned with guidance strategies and piloting techniques which ensure near-optimum performance and maximum survival capability in a severe windshear. The take-off problem is considered with reference to flight in a vertical plane. In addition to the horizontal shear, the presence of a downdraft is assumed. First, six particular guidance schemes are considered, namely: constant alpha guidance; maximum alpha guidance; constant velocity guidance; constant absolute path inclination guidance; constant rate of climb guidance; and constant pitch guidance. Among these, it is concluded that the best one is the constant pitch guidance. Next, in an effort to improve over the constant pitch guidance, three additional trajectories are considered: the optimal trajectory, which minimizes the maximum deviation of the absolute path inclination from a reference value, while employing global information on the wind flow field; the gamma guidance trajectory, which is based on the absolute path inclination and which approximates the behavior of the optimal trajectory, while employing local information on the windshear and the downdraft; and the simplified gamma guidance trajectory, which is the limiting case of the gamma guidance trajectory in a severe windshear and which does not require precise information on the windshear and the downdraft. The essence of the simplified gamma guidance trajectory is that it yields a quick transition to horizontal flight. Comparative numerical experiments show that the survival capability of the simplified gamma guidance trajectory is superior to that of the constant pitch trajectory and is close to that of the optimal trajectory. Next, with reference to the simplified gamma guidance trajectory, the effect of the feedback gain coefficient is studied. It is shown that larger values of the gain coefficient improve the survival capability in a severe windshear; however, excessive values of the gain coefficient are undesirable, because they result in larger altitude oscillations and lower average altitude. Finally, with reference to the simplified gamma guidance trajectory, the effect of time delays is studied, more specifically, the time delay τ1 in reacting to windshear onset and the time delay τ2 in reacting to windshear termination. While time delay τ2 has little effect on survival capability, time delay τ1 appears to be critical in the following sense: smaller values of τ1 correspond to better survival capability in a severe windshear, while larger values of τ1 are associated with a worsening of the survival capability in a severe windshear.

An Application of Livengood-Wu Integral Method to Predict Time of Cool Flame Explosion During Piston Compression Process

Livengood and Wu offered the following integration equation to predict the time of knock te, ∫te0 (1/τ)dt=1, along the end gas pressure time and temperature time histories, under the assumptions of zero order reaction and critical concentration of chain carriers, and showed that the agreement between the predicted and actual times of occurrence of autoignition is reasonable. An attempt is made in this paper to check on the utility of the integration procedure for cool flame explosions along the state time history of mixture during the motoring compression period. The pre-cool flame reaction which is primarily a degenerated branching reaction is taken to be similar to the zero order reaction and the critical concentration of active intermediate products is supposed to give rise to cool flame. Empirical coefficient and exponent in the general Arrhenius equation for the cool flame ignition delay τ1 are found by the results obtained using a rapid compression machine developed here. Experimental results verify that this integration procedure is valid in the limited range of motoring compression temperatures where cool flame explosions occur Tc no more than the temperature at the end of the compression stroke of the rapid compression machine TE, but falls over in the range of TE≠Tc, especially TE<Tc. It can be suggested that the reaction mechanism depends on the mixture temperature and seems to change from the low temperature autoignition mechanism to the high temperature one.

Stability and Hopf bifurcation analysis in ecological system with two delays

This paper aims to study the effect of time-delay on a food chain model. Two delays (τ1 and τ2) are considered in the model to describe the time that juveniles of prey and predator take to mature. The stability analysis of the proposed model is carried out. The Hopf bifurcation conditions of the interior equilibrium point are established. Finally, numerical simulations are done to support the analytical findings. In addition, critical value of time delays are determined and it is found that maturation delay always acts as a destabilizing factor.