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We present a mathematical model that captures the dynamics of the immune system during allergic reactions. Using delay differential equations, we depict the evolution of T cells, APCs, and IL6, considering cell migration between various body compartments. The biological discussions and interpretations within the article revolve around drug desensitization, highlighting one potential application of the model. We conduct stability analysis on certain equilibrium points, demonstrating stability in some cases and only partial stability in others. Numerical simulations validate the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper deals with the time delayed SIR model with saturated treatment function under the effect of pollution. As the pollution rate has hiked in this era and the presence of toxicants are found in various resources, therefore it is important to study their effects on disease dynamic of population. Hence, we consider the populace affected by the pollution. As the effect of treatment is not immediate, rather involves time lag to show its effect, therefore we consider time that is taken by infective to recover as the time delay. Existence of equilibrium points and the boundedness of the system has been obtained. Stability analysis using reproduction number Ro has been done. Global stability of the endemic equilibrium point is established using Lyapunov Lasalles theorem. Considering time delay as the critical parameter, existence of hopf bifurcation has been proved. Also, the direction and stability of Hopf Bifurcation has been obtained. Numerical simulations are done in support of results obtained analytically. It is observed that as pollution increases, infective take more time to recover and we also note that recovered individuals decrease with increase in the pollution. Therefore, it important to reduce pollution and provide timely treatment to the patients. Thus, we prove that, pollution and time delay play a critical role in shaping the dynamics of the system. [ABSTRACT FROM AUTHOR]

• We construct a chain of two interconnected chemostats, assuming competitive exclusion. • The first chemostat cultivates the unadvantaged competitor. The second cultivates two ones. • The output of the first chemostat is the input of the second one. • We carry out a stability analysis by constructing a Lyapunov functional. We study a nonlinear system of differential delay equations describing a model of a chain of two chemostats, where one contains two microbial species in competition for a single limiting nutrient and receives an external input of the less advantaged competitor, which is cultivated in an external chemostat. We obtain sufficient conditions ensuring coexistence of all the species in competition which consist in upper delay bounds. [ABSTRACT FROM AUTHOR]

This article presents an extensive theoretical framework to mathematically defined and information- based routing operators, applied to the continuous-time dynamic traffic assignment problem. Be- cause of the difficulty of the mathematical framework required to provide existence and uniqueness proofs of the solution to the problem in the presence of a routing operator at nodes, the approach is instantiated with a link model, consisting of a system of ordinary delay differential equations and modeling traffic ow macroscopically. The routing operators distributing the incoming ow can en- compass a wide range of information patterns, which can include past knowledge of the network state (statistical, or past deterministic information) up to real time and thus satisfying a nonanticipative character. We show, for a rather broad class of routing operators, the existence and uniqueness of solutions on the full network. This framework can be extended to more advanced traffic ow models such as partial differential equation models. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we introduce a piecewise reproducing kernel method for impulsive delay differential equations with piecewise constant arguments. The method is an improved reproducing kernel method. Compared with the classical reproducing kernel method, the solutions obtained using the present method can give good approximations for a larger time interval. Some numerical examples are used to show the effectiveness and simplicity of the method. [ABSTRACT FROM AUTHOR]

Abstract In the paper, the superconvergence of continuous Galerkin finite element methods (CGFEMs) for linear delay differential equations of neutral type is presented. By the orthogonal analysis method and under the suitable condition, it is proven that the finite element solution is superconvergent at the nodal points and Lobatto points. Numerical experiments further confirm the effectiveness and the superconvergence of the CGFEMs. [ABSTRACT FROM AUTHOR]

In this paper, some new results are obtained for the even order neutral delay difference equation ΔanΔm-1xn+pnxn-kα+qnxn-lβ=0, where m≥2 is an even integer, which ensure that all solutions of the studied equation are oscillatory. Our results extend, include, and correct some of the existing results. Examples are provided to illustrate the importance of the main results. [ABSTRACT FROM AUTHOR]

We propose the generalisation of the p53-Mdm2 protein gene expression model introduced by Monk (2003). We investigate the stability of a unique positive steady state and formulate conditions which guarantee the occurrence of the Hopf bifurcation. We show that oscillatory behaviour can be caused not only by time lag in protein transcription process, but also can be present in the model without time delay. Moreover, we investigate the stability of new born periodic solutions. Theoretical results are illustrated by numerical simulations and interpreted from the biological point of view. [ABSTRACT FROM AUTHOR]

We study the Tricomi problem for equations of mixed type with multiple functional delay and advance. A general solution of the equations is obtained. The problem is uniquely solvable. [ABSTRACT FROM AUTHOR]

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 influenza A (H1N1) dynamics. We begin by presenting the model with a brief discussion, followed by proving the positivity and boundedness of the model solution. We establish sufficient conditions for the global stability of the equilibria (the infection free equilibrium and the infected equilibrium), these conditions are obtained by means of the Lyapunov LaSalle invariance principle of the system. Also we have carried out bifurcation analysis along with an estimated length of delay to preserve the stability behavior. In particular, we show the threshold dynamics in the sense that if (reproduction number) R 0 < 1 the infectious population disappear so the disease dies out, while if R 0 > 1 the infectious population persist. Sensitivity analysis of the influenza A (H1N1) model reveals which parameter values have a major impact on the model dynamics. Numerical simulations with application to H1N1 infection are given to verify the analytical results. [ABSTRACT FROM AUTHOR]

There exists a well-developed stability theory for many classes of functional-differential equations and only a few results on their instability. One of the aims of this paper is to fill this gap. Explicit tests for instability of linear delay differential equations of the second order with damping terms are obtained. [ABSTRACT FROM AUTHOR]

If the Mackey–Glass equation x ˙ ( t ) = r ( t ) [ a x ( h ( t ) ) 1 + x ν ( g ( t ) ) − x ( t ) ] with a > 1 and ν > 0 incorporates not one but two variable delays, some new phenomena arise: there may exist non-oscillatory about the positive equilibrium unstable solutions, the effect of possible absolute stability for certain a and ν disappears. We obtain sufficient conditions for local and global stability of the positive equilibrium and illustrate the stability tests, as well as new effects of two different delays, with examples. [ABSTRACT FROM AUTHOR]

We study the effect of diabetic deficiencies on the production of an oscillatory ultradian regime using a deterministic nonlinear model which incorporates two physiological delays. It is shown that insulin resistance impairs the production of oscillations by dampening the ultradian cycles. Four strategies for restoring healthy regulation are explored. Through the introduction of an instantaneous glucose-dependent insulin response, explicit conditions for the existence of periodic solutions in the linearised model are formulated, significantly reducing the complexity of identifying an oscillatory regime. The model is thus shown to be suitable for representing the effect of diabetes on the oscillatory regulation and for investigating pathways to reinstating a physiological healthy regime. [ABSTRACT FROM AUTHOR]

Briggs et al. (1993) introduced a host–parasitoid model for the dynamics of a system with two parasitoids that attack different juvenile stages of a common host. Their main result was that coexistence of the parasitoids is only possible when there is sufficient variability in the maturation delays of the host juvenile stages. Here, we analyze the phenomenon of coexistence in that model more deeply. We show that with some distribution families for the maturation delays, the coexistence equilibrium is unique, while with other distributions multiple coexistence equilibria can be found. In particular, we find that stable coexistence does not necessarily require mutual invasibility. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of solutions of a class of non-autonomous delay differential equations. These equations have important practical applications and generalize those on which Bernfeld and Haddock conjectured that each solution of the equations tends to a constant. It is shown that every solution of the equations is bounded and tends to a constant as t → + ∞ , which improves and extends some existing ones. [ABSTRACT FROM AUTHOR]

The purpose of this paper is devoted to studying the implicit–explicit (IMEX) one-leg methods for stiff delay differential equations (DDEs) which can be split into the stiff and nonstiff parts. IMEX one-leg methods are composed of implicit one-leg methods for the stiff part and explicit one-leg methods for the nonstiff part. We prove that if the IMEX one-leg methods is consistent of order 2 for the ordinary differential equations, and the implicit one-leg method is A-stable, then the IMEX one-leg methods for stiff DDEs are stable and convergent with order 2. Some numerical examples are given to verify the validity of the obtained theoretical results and the effectiveness of the presented methods. [ABSTRACT FROM PUBLISHER]

We study oscillatory behavior of solutions to a class of even-order neutral differential equations relating oscillation of higher-order equations to that of a pair of associated first-order delay differential equations. As illustrated with two examples in the final part of the paper, our criteria improve a number of related results reported in the literature. [ABSTRACT FROM AUTHOR]

We consider the equation (1) − y ″ ( x ) + q ( x ) y ( x − φ ( x ) ) = f ( x ) , x ∈ R where f ∈ C ( R ) and (2) 0 ≤ φ ∈ C loc ( R ) , 1 ≤ q ∈ C loc ( R ) . Here C loc ( R ) is the set of functions continuous in every point of the number axis. By a solution of (1), we mean any function y , doubly continuously differentiable everywhere in R , which satisfies (1). We show that under certain additional conditions on the functions φ and q to (2), (1) has a unique solution y , satisfying the inequality ‖ y ″ ‖ C ( R ) + ‖ y ′ ‖ C ( R ) + ‖ q y ‖ C ( R ) ≤ c ‖ f ‖ C ( R ) where the constant c ∈ ( 0 , ∞ ) does not depend on the choice of f ∈ C ( R ) . [ABSTRACT FROM AUTHOR]

The oscillatory behavior of the solutions to a differential equation with several non-monotone delay arguments and non-negative coefficients is studied. A new sufficient oscillation condition, involving lim sup , is obtained. An example illustrating the significance of the result is also given. [ABSTRACT FROM AUTHOR]

This paper studies higher-order nonlinear neutral delay difference equations of the form Using Krasnoselskii's fixed point theorem, we obtain the existence of uncountably many bounded positive solutions to the considered problem. [ABSTRACT FROM AUTHOR]

This paper is concerned with the backward differential formula or BDF methods for a class of nonlinear 2-delay differential algebraic equations. We obtain two sufficient conditions under which the methods are stable and asymptotically stable. At last, examples show that our methods are true. [ABSTRACT FROM AUTHOR]

We investigate the oscillation of a first order delay differential equation with non-negative coefficient and non-monotone arguments. New oscillation criterion of lim sup type is obtained. An example is given to show the applicability and strength of the obtained condition over known ones. [ABSTRACT FROM AUTHOR]

In this work we analyze a nonlinear delay differential equation (DDE) model with negative feedback and constant time delay. The model is constructed from a previously studied biochemical reaction network of gene transcription and protein synthesis. Linear analysis of the associated DDE model gives a critical time delay beyond which a periodic motion is born in a Hopf bifurcation. The method of multiple scales is then used to analyze the nonlinear system to obtain expressions for the amplitude and frequency of oscillation as a function of the system parameters. We use our closed form analytical expressions to study the importance of a well-balanced ratio between synthesis and degradation rates in the existence of periodic solutions. We show that our theoretical results are in agreement with numerical simulations and with experimental evidence found in the biological literature. [ABSTRACT FROM AUTHOR]

We consider a class of systems of differential equations of higher dimension. We show that, if the number of equations is sufficiently large, the last component of the solution can be approximated by solutions to delay differential equations. Bibliography: 14 titles. [ABSTRACT FROM AUTHOR]

This note is concerned with the oscillation of third order nonlinear delay differential equations of the form (r2(t) (r1(t)y'(t))')' + p(t)y'(t) + q(t) f(y(g(t))) = 0 (*) In the papers [A. Tiryaki, M. F. Aktas, Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Anal. Appl. 325 (2007), 54-68] and [M. F. Aktas, A. Tiryaki, A. Zafer, Oscillation criteria for third order nonlinear functional differential equations, Applied Math. Letters 23 (2010), 756-762], the authors established some sufficient conditions which insure that any solution of equation (*) oscillates or converges to zero, provided that the second order equation (r2(t)z'(t))' + (p(t)/r1(t)) z(t) = 0 (**) is nonoscillatory. Here, we shall improve and unify the results given in the above mentioned papers and present some new sufficient conditions which insure that any solution of equation (*) oscillates if equation (**) is nonoscillatory. We also establish results for the oscillation of equation (*) when equation (**) is oscillatory. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical stability of a class of nonlinear neutral delay differential equations. The numerical stability results are obtained for (k,l)-algebraically stable Runge-Kutta methods when they are applied to this type of problem. Numerical examples are given to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

The paper addresses the bifurcations for a delay differential model with parameters which confers a strong Allee effect in Escherichia coli. Stability and local Hopf bifurcations are analyzed when the delayτorσas parameter. It is also found that there is a non-resonant double Hopf bifurcation occur due to the vanishing of the real parts of two pairs of characteristic roots. We transform the original system into a finite dimensional system by the center manifold theory and simplify the system further by the normal form method. Then, we obtain a complete bifurcation diagram of the system. Finally, we provide numerical results to illustrate our conclusions. There are many interesting phenomena, such as attractive quasi-periodic solution and three-dimensional invariant torus. [ABSTRACT FROM AUTHOR]

The nonlinear delay differential equation x ˙ ( t ) = r ( t ) [ g ( t , x t ) − h ( x ( t ) ) ] , t ≥ 0 is considered. Sufficient conditions are established for the uniform permanence of the positive solutions of the equation. In several particular cases, explicit formulas are given for the upper and lower limit of the solutions. In some special cases, we give conditions which imply that all solutions have the same asymptotic behavior, in particular, when they converge to a periodic or constant steady-state. [ABSTRACT FROM AUTHOR]

We study the asymptotic integration problem for second-order nonlinear delay differential equations of the form ( p ( t ) x ′ ( t ) ) ′ + q ( t ) x ( t ) = f ( t , x ( g ( t ) ) ) . It is shown that if u and v are principal and nonprincipal solutions of equation ( p ( t ) x ′ ) ′ + q ( t ) x = 0 , then there are solutions x 1 ( t ) and x 2 ( t ) of the above nonlinear equation such that x 1 ( t ) = a u ( t ) + o ( u ( t ) ) , t → ∞ and x 2 ( t ) = b v ( t ) + o ( v ( t ) ) , t → ∞ . [ABSTRACT FROM AUTHOR]

We present several oscillation criteria for a second-order nonlinear delay differential equation with a nonpositive neutral coefficient. Two examples are given to illustrate the main results. [ABSTRACT FROM AUTHOR]

The existence of the nontrivial periodic solutions for the nonautonomous first order delay differential equation $x'(t)=-[f(t, x(t-1))+f(t, x(t-2))+\cdots+f(t, x(t-(2N-1)))]$ is investigated, where $f\in C({\mathbf{R}}\times{\mathbf{R}}, {\mathbf{R}})$ is 2 N-periodic in t and odd in x, N is a positive integer. We prove several new existence results by some recent critical point theorems. [ABSTRACT FROM AUTHOR]

The paper deals with nonlinear delay reaction–diffusion equations of the form u t = a u x x + F ( u , u ̄ ) , where u = u ( x , t ) and u ̄ = u ( x , t − τ ) , with τ denoting the delay time. We present a number of traveling-wave solutions of the form u = w ( z ) , z = k x + λ t , that can be represented in terms of elementary functions. We consider equations with quadratic, power-law, exponential and logarithmic nonlinearities as well as more complex equations with the kinetic function dependent on one to four arbitrary functions of a single argument. All of the solutions obtained involve free parameters and so may be suitable for solving certain model problems as well as testing numerical and approximate analytical methods for delay reaction–diffusion equations and more complex nonlinear delay PDEs. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study oscillation of a forced second-order neutral differential equation. By using the generalized Riccati substitution and integral technique, a new sufficient condition is obtained which insures that all solutions to the studied equation are oscillatory. An illustrative example is included. [ABSTRACT FROM AUTHOR]

In this paper, by using the fixed point method, we prove two new results on the Hyers-Ulam-Rassias and the Hyers-Ulam stability for the first order delay differential equation of the form y'(t) = F(t, y(t), y(t -t )). Our results improve some related results in the literature. [ABSTRACT FROM AUTHOR]

The existence of uncountably many positive solutions and convergence of Mann iterative schemes for a fourth order neutral delay difference equation are proved. Seven examples are included. [ABSTRACT FROM AUTHOR]

In this paper, the method of fractional finite difference presents and used for solving a number of famous fractional order version of scientific models. The proposed method besides being simple is so exact which is sensible in the solved problems. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to give sufficient conditions for the uniform stability of the zero solution of a certain fourth-order vector delay differential equation of the following form: X(4)+F(X˙,X¨)X⃛+Φ(X¨)+G(X˙(t-r))+H(X(t-r))=0. By constructing a Lyapunov functional, we obtained the result of stability. [ABSTRACT FROM AUTHOR]

In this paper, we investigate existence of oscillatory solutions for a forced second order nonlinear delay differential equations [ r ( t ) Φ ( x ′ ( t ) ) ] ′ + ∑ i = 1 m f i ( t , x ( g i ( t ) ) ) = q ( t ) , where f i ∈ C ( [ t 0 , ∞ ) × R , R ) , g i ( t ) ≤ t , lim t → ∞ g i ( t ) = ∞ , i = 1 , 2 , ⋯ , m , Φ ∈ C 1 ( R , R ) , Φ ( u ) is an increasing function for all u ∈ R , Φ − 1 ( u ) satisfies the local L i p i s c h i t z condition. A new sufficient condition for global existence of oscillatory solution is obtained by the Schauder–Tychonoff theorem. When Φ ( u ) = u α with α ≥ 1 being the ratio of two positive odd integers has also been studied. We give examples to illustrate the applicability of our results. [ABSTRACT FROM AUTHOR]

The existence, multiplicity, and properties of positive solutions for a third order nonlinear neutral delay difference equation are discussed. Six examples are given to illustrate the results presented in this paper. [ABSTRACT FROM AUTHOR]

In this paper we use variational methods to investigate the existence of periodic solutions for some second order delay differential systems with impulsive effects. Some new methods and results are obtained. [ABSTRACT FROM AUTHOR]

By using the standard Liapunov-Krasovskii functional approach, in this paper, new stability, boundedness and ultimately boundedness criteria are established for a class of vector functional differential equations of third order with retarded argument. [ABSTRACT FROM AUTHOR]