Stability analysis for neutral systems with mixed delays

Relationships between system states contained in the neutral equation are used to address the delay-dependent stability of a neutral system with time-varying state delay. Using linear matrix inequalities, we present a new asymptotic stability criterion, and a new robust stability criterion, for neutral systems with mixed delays. Since the criteria take into account the sizes of the neutral delay, discrete delay and the derivative of discrete delay, they are less conservative than those produced by previous approaches. Numerical examples are presented to demonstrate that these criteria are indeed more effective.     

Sufficient conditions for the asymptotic stability of nonlinear multidelay differential equations with linear parts defined by pairwise permutable matrices

In this paper a generalization of the delayed exponential defined by Khusainov and Shuklin (2003) [1] for autonomous linear delay systems with one delay defined by permutable matrices is given for delay systems with multiple delays and pairwise permutable matrices. Using this multidelay-exponential a solution of a Cauchy initial value problem is represented. By an application of this representation and using Pinto’s integral inequality an asymptotic stability results for some classes of nonlinear multidelay differential equations are proved.     

Permanence and extinction of a three-species ratio-dependent food chain model with delay and prey diffusion

In this paper, we study a diffusive three-species ratio-dependent food chain model, using differential inequality, to obtain sufficient conditions that ensure the permanence of the system and the extinction of predator species. Our results reinforce the main result of Sun Wen, Shihua Chen and Huihai Mei [Positive periodic solution of a more realistic three-species Lotka-Volterra model with delay and density regulation, Chaos, Solitons and Fractals, in press].     

Dynamic transcritical bifurcations in a class of slow-fast predator-prey models

We study the stability loss delay phenomenon in a dynamic transcritical bifurcation in a class of three-dimensional prey and predator systems. The dynamics of the predator is assumed to be slow comparatively to the dynamics of the prey. As an application, a well-known model considered by Clark will be discussed.     

Numerical detection of instability regions for delay models with delay-dependent parameters

In this paper we are interested in gaining local stability insights about the interior equilibria of delay models arising in biomathematics. The models share the property that the corresponding characteristic equations involve delay-dependent coefficients. The presence of such dependence requires the use of suitable criteria which usually makes the analytical work harder so that numerical techniques must be used. Most existing methods for studying stability switching of equilibria fail when applied to such a class of delay models. To this aim, an efficient criterion for stability switches was recently introduced in [E. Beretta, Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal. 33 (2002) 1144–1165] and extended [E. Beretta, Y. Tang, Extension of a geometric stability switch criterion, Funkcial Ekvac 46(3) (2003) 337–361]. We describe how to numerically detect the instability regions of positive equilibria by using such a criterion, considering both discrete and distributed delay models.     

Dynamics of a neutral delay equation for an insect population with long larval and short adult phases

We present a global study on the stability of the equilibria in a nonlinear autonomous neutral delay differential population model formulated by Bocharov and Hadeler. This model may be suitable for describing the intriguing dynamics of an insect population with long larval and short adult phases such as the periodical cicada. We circumvent the usual difficulties associated with the study of the stability of a nonlinear neutral delay differential model by transforming it to an appropriate non-neutral nonautonomous delay differential equation with unbounded delay. In the case that no juveniles give birth, we establish the positivity and boundedness of solutions by ad hoc methods and global stability of the extinction and positive equilibria by the method of iteration. We also show that if the time adjusted instantaneous birth rate at the time of maturation is greater than 1, then the population will grow without bound, regardless of the population death process.     

Periodic solutions for a kind of Liénard equation

Using inequality techniques and coincidence degree theory, new results are provided concerning the existence and uniqueness of T-periodic solutions for a Liénard equations with delay. An illustrative example is provided to demonstrate that the results in this paper hold under weaker conditions than existing results, and are more effective.     

Dynamic behaviors of a delay differential equation model of plankton allelopathy

In this paper, we consider a modified delay differential equation model of the growth of n-species of plankton having competitive and allelopathic effects on each other. We first obtain the sufficient conditions which guarantee the permanence of the system. As a corollary, for periodic case, we obtain a set of delay-dependent condition which ensures the existence of at least one positive periodic solution of the system. After that, by means of a suitable Lyapunov functional, sufficient conditions are derived for the global attractivity of the system. For the two-dimensional case, under some suitable assumptions, we prove that one of the components will be driven to extinction while the other will stabilize at a certain solution of a logistic equation. Examples show the feasibility of the main results.     

Dynamics and stability of impulsive hybrid setvalued integro-differential equations with delay

We study the dynamics and stability theory for impulsive hybrid set integro-differential equations with delay. Sufficient conditions for the stability of the null solution of impulsive hybrid set integro-differential equations with delay are presented.     

Stability in delay difference equations with nonuniform exponential behavior

We study the stability under perturbations for delay difference equations in Banach spaces. Namely, we establish the (nonuniform) stability of linear nonuniform exponential contractions under sufficiently small perturbations. We also obtain a stable manifold theorem for perturbations of linear delay difference equations admitting a nonuniform exponential dichotomy, and show that the stable manifolds are Lipschitz in the perturbation.     

Families of Liapunov-Krasovskiį functionals and stability for functional differential equations

This paper discusses the stability of solutions of nonautonomous functional differential equations with infinite delay with respect to a parr of admissible phase spaces of Hale and Kato. A one-parameter family of Liapunov-Krasovskiį functional, together with some additional analysis, is used to prove new sufficient conditions of asymptotic and uniform asymptotic stability for such equations. It is also shown that the so-called Razumikhin condition is unessential when families of Liapunov-Krasovskiį functionals are used. Entrata in Redazione il 25 settembre 1997. Invited address at the Second Marrakesh International Conference on Differential Equations, Marrakesh, Morocco, June 1995.     

A population dynamics model with nonautonomous past

Abstract

In this paper we study a two-phase population model, which distinguishes the population by two different stages

By the standard technique of characteristics, this population equation is transformed as the ordinary differential equation with nonautonomous past

where 1 ≤ p < ∞ and I = [?r, 0] (finite delay) or I = (?∞, 0] (infinite delay), E a Banach space, Φ : W^1,p(I, E) → E a linear delay operator and B a nonlinear operator on E. The main result of this paper is the well-posedness of this delay equation by using the (right) multiplicative perturbation result of Desch and Schappacher in [8].     

Blow-up Results for Some Nonlinear Delay Differential Equations

In this work we study the blow up phenomena for some scalar delay differential equations. In particular, we make connection with the blow up of ordinary differential equations that are related to the delay differential equations. The first author is supported by a Grant from TWAS under contract No: 03-030 RG/MATHS/AF/AC. The second author is supported by a grant from the Lebanese National Council for Scientific Research.     

Stability of dichotomies in difference equations with infinite delay

For delay difference equations with infinite delay we consider the notion of nonuniform exponential dichotomy. This includes the notion of uniform exponential dichotomy as a very special case. Our main aim is to establish a stable manifold theorem under sufficiently small nonlinear perturbations. We also establish the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations. Finally, we characterize the nonuniform exponential dichotomies in terms of strict Lyapunov sequences. In particular, we construct explicitly a strict Lyapunov sequence for each exponential dichotomy.     

Global dynamics of Nicholsonʼs blowflies equation revisited: Onset and termination of nonlinear oscillations

We revisit Nicholson?s blowflies model with natural death rate incorporated into the delay feedback. We consider the delay as a bifurcation parameter and examine the onset and termination of Hopf bifurcations of periodic solutions from a positive equilibrium. We show that the model has only a finite number of Hopf bifurcation values and we describe how branches of Hopf bifurcations are paired so the existence of periodic solutions with specific oscillation frequencies occurs only in bounded delay intervals. The bifurcation analysis and the Matlab package DDE-BIFTOOL developed by Engelborghs et al. guide some numerical simulations to identify ranges of parameters for coexisting multiple attractive periodic solutions.     

Stability and the nonexistence of blowing-up solutions of nonlinear delay systems with linear parts defined by permutable matrices

Sufficient conditions for the exponential stability of the trivial solution of nonlinear differential equations with delay and with linear parts of the form Ax(t)+Bx(t−τ),τ>0, where AB=BA, are proved. A result on the nonexistence of blowing-up solutions is also proved.     

Center manifolds theorem for parameterized delay differential equations with applications to zero singularities

The goal of this paper is to develop a center manifold theory for delay differential equations with parameters. As applications, we use the center manifold theorem to establish fold and Bogdanov-Takens bifurcations. In particular, we obtain the versal unfoldings of delayed predator-prey systems with predator harvesting at the Bogdanov-Takens singularity.     

Maximal regularity of second-order equations with delay

We characterize existence and uniqueness of solutions of an inhomogeneous abstract delay equation in Hölder spaces. The method is based on the theory of operator-valued Fourier multipliers.     

Global attractivity and permanence of a SVEIR epidemic model with pulse vaccination and time delay

In this study, we propose a new SVEIR epidemic disease model with time delay, and analyze the dynamic behavior of the model under pulse vaccination. Pulse vaccination is an effective strategy for the elimination of infectious disease. Using the discrete dynamical system determined by the stroboscopic map, we obtain an ‘infection-free’ periodic solution. We also show that the ‘infection-free’ periodic solution is globally attractive when some parameters of the model under appropriate conditions. The permanence of the model is investigated analytically. Our results indicate that a large vaccination rate or a short pulse of vaccination or a long latent period is a sufficient condition for the extinction of the disease.     

Dissipativity of the delay semigroup

Under mild conditions a delay semigroup can be transformed into a (generalized) contraction semigroup by modifying the inner product on the (Hilbert) state space into an equivalent inner product. Applications to stability of differential equations with delay and stochastic differential equations with delay are given as examples.