unimodal dynamical systems comparison principles spreading speeds and travelling waves abstract reaction diffusion equations with delayed nonlinear reaction terms are used as prototypes to motivate an appropriate abstract formulation of dynamical systems with unimodal nonlinearity for such non monotone dynamical systems we develop a general comparison principle and show how this general comparison principle coupled with some existing results for monotone dynamical systems can be used to establish results on the asymptotic speeds of spread and travelling waves we illustrate our main results by an integral equation which includes a nonlocal delayed reaction diffusion equation and a nonlocal delayed lattice differential system in an unbounded domain with the non monotone nonlinearities including the ricker birth function and the mackey glass hematopoiesis feedback

abstract functional differential equations and reaction diffusion systems functional differential equations and reaction diffusion systems r h martin jr and h l smith abstract several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays suppose q is a bounded domain in rn with aq smooth and a is the laplacian operator on q also let m be a positive integer z a positive number and f fi m a continuous bounded function from 0 xc x q x c z 0 m into rm where c t 0 is the space of continuous functions from z 0 into r the purpose of this paper is to apply abstract results for semilinear functional differential equations in banach spaces to reactiondiffusion systems with time delays having the form a tu x t diau x t i t x ut x t a xeq i 1 m rdd ai x ui x t au i x t 81 x t t a x eaq u x a 0 x x 0 5 t 0 di 0 and ca q 0 o is c and i q x 0 oo r is c2 here an is the outward normal derivative on aq and if di 0 it is assumed that no boundary conditions are specified for this i also tu x t denotes the partial with respect to t whereas ut x denotes the member of c z 0 defined by 0 u x t 0 u x t 0 m our techniques provide basic existence criteria but the main point is that they can also be effectively applied to obtain estimates for solutions especially received by the editors october 7 1987 and in revised form october 7 1988 1980 mathematics subject classification 1985 revision primary 35r10 34k30

a note on hopf bifurcations in a delayed diffusive lotka volterra predator prey system the diffusive lotka volterra predator prey system with two delays is reconsidered here the stability of the coexistence equilibrium and associated hopf bifurcation are investigated by analyzing the characteristic equations and our results complement earlier ones we also obtain that in a special case a hopf bifurcation of spatial inhomogeneous periodic solutions occurs in the system

stability of scalar delay differential equations with dominant delayed terms this paper deals with scalar delay differential equations with dominant delayed terms sufficient conditions are obtained for uniform stability uniformly asymptotic stability and globally asymptotic stability of the equations the criteria extend and improve some existing ones the main results are applied to two physiological models some counterexamples are also given to show the invalidity of some existing results

the dynamics of some discrete population models in this paper we are interested in providing sufficient conditions guaranteeing the fact that all positive solutions of the discrete delay difference equation a ii 1 aa f a e converge as n 00 in e a is a constant real number with 0 0 sufficiently small thus formally equation e comes from x t h x t h yx t d x t r for small h if we set z f x ht then the preceding equation for t n and r h m becomes yh 1 1 h y n hd y n m which is of the form e where a 1 yh and f u hd u u 2 0 clearly h 0 is small if and only if a e 0 1 1069 h

map dynamics versus dynamics of associated delay reaction diffusion equations with a neumann condition in this paper we consider a class of delay reaction diffusion equations drdes with a parameter e 0 a homogeneous neumann boundary condition and non negative initial functions are posed to the equation by letting such an equation is formally reduced to a scalar difference equation or map dynamical system the main concern is the relation of the absolute or delay independent global stability of a steady state of the equation and the dynamics of the nonlinear map in the equation by employing the idea of attracting intervals for solution semiflows of the drdes we prove that the globally stable dynamics of the map indeed ensures the delay independent global stability of a constant steady state of the drdes we also give a counterexample to show that the delay independent global stability of drdes cannot guarantee the globally stable dynamics of the map finally we apply the abstract results to the diffusive delay nicholson blowfly equation and the diffusive mackey glass haematopoiesis equation the resulting criteria for both model equations are amazingly simple and are optimal in some sense although there is no existing result to compare with for the latter

dirichlet problem for the diffusive nicholson s blowflies equation abstract the object of this paper is to consider the asymptotic behavior of solutions of the diffusive nicholson s blowflies equation under dirichlet boundary condition a new approach is developed to study the global attractivity of the positive steady state for a reaction diffusion equation with time delay when monotone or quasi monotone conditions fail to apply this approach should be applicable to other dirichlet problems as well even though our analysis is tailored to the diffusive nicholson s blowflies equation

dynamics of a class of delayed reaction diffusion systems with neumann boundary condition abstract this paper considers a class of delayed reaction diffusion systems under the neumann boundary condition which arise in epidemiology and can describe the temporal and spatial evolutionary phenomena for the bacteria population and the human infective population with the help of the iterative properties of interval mapping and dynamical system approaches some positively invariant sets and attractive basins of the considered systems are analyzed detailedly in addition combining the global attractivity of interval mapping we provide some sufficient conditions to ensure local or global attractivity of steady states of the systems finally we apply these theoretical results to some models with different nonlinearity demonstrating force of infection and then obtain some sufficient conditions about force of infection to describe the survival and extinction of bacteria and infective populations

global dynamics of nicholsonʼs blowflies equation revisited onset and termination of nonlinear oscillations abstract 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

spatial temporal patterns of a two age structured population model with spatial non locality abstract the spatial temporal patterns of the solutions to a differential equation with non local delayed feedback that models the population dynamics of a two stage species have been studied in this paper the effects of the maturation time and immature mobility constant on the existence and stability of both spatially homogeneous and inhomogeneous periodic solutions are investigated by studying the hopf bifurcation at the spatially homogeneous steady state and computing center manifold our results show that the bifurcated homogeneous periodic solutions are stable while inhomogeneous periodic solutions will eventually tend to homogeneous periodic solutions after transient oscillations furthermore increasing of the immature mobility constant will weaken the periodic oscillation and shorten the transient oscillation time numerical simulations are given at last to verify our results

stability and existence results for a time delayed nonlocal model of hematopoietic stem cells dynamics abstract in this paper we consider a time delayed nonlocal model describing the dynamics of hematopoietic stem cells hscs which represent the immature cells in the hematopoiesis process by the method of characteristics the nonlocal model is obtained from an age structured reaction diffusion system in bounded domain with dirichlet boundary conditions along this paper we focus on the mathematical analysis of it firstly we give some results on the existence uniqueness positivity and boundedness of solutions next we obtain a threshold value r s and prove that the trivial steady state is globally asymptotically stable when r s 1 when r s 1 we prove the existence and uniqueness of positive stationary solution under the respective additional conditions on the monotonicity and non monotonicity of the integral term finally we prove the uniform weak persistence of the system when r s 1 some numerical simulations are provided to verify the validity of our theoretical results

on the local and global comparison of generalized bajraktarevic means abstract given two continuous functions f g i r such that g is positive and f g is strictly monotone a measurable space t a a measurable family of d variable means m i d t i and a probability measure on the measurable sets a the d variable mean m f g m i d i is defined by m f g m x f g 1 t f m x 1 x d t d t t g m x 1 x d t d t x x 1 x d i d the aim of this paper is to study the local and global comparison problem of these means i e to find conditions for the generating functions f g and h k for the families of means m and n and for the measures such that the comparison inequality m f g m x m h k n x x i d be satisfied

new periodic stability for a class of nicholson s blowflies models with multiple different delays in this research work the existence and global exponential stability of the positive periodic solutions of nicholson s blowflies equation with multiple different time varying delays are investigat

global exponential stability of positive periodic solutions for a delayed nicholsonʼs blowflies model abstract this paper is concerned with a non autonomous delayed nicholson s blowflies model under proper conditions we employ a novel argument to establish a criterion on the global exponential stability of positive periodic solutions this answers an open problem proposed by berezansky et al 2010 2 we also provide numerical simulations to support the theoretical result

domain decomposition method for the global dynamics of delay differential equations with unimodal feedback x t zkmx t c f x tkt with unimodal feedback is studied the existence of the global attractor is shown and bounds of the attractor are given we find attractive invariant intervals and give sufficient conditions that guarantee that all solutions enter the domain where f 0 is negative with respect to a positive equilibrium so the results for delayed monotone feedback can be applied to describe the asymptotic behaviour of solutions in particular the existence of heteroclinic orbits from the trivial equilibrium to a periodic orbit oscillating around the positive equilibrium is established numerical examples using nicholson s blowflies equation and the mackey glass equation are provided to illustrate the main results

dynamics for a type of general reaction diffusion model abstract in this paper we discuss the following reaction diffusion model which is a general form of many population models u t x t u t x u t x f u t x we study the oscillatory behavior of solutions about the positive equilibrium k of system with neumann boundary conditions sufficient and necessary conditions are obtained for global attractivity of the zero solution and acceptable conditions are established for the global attractivity of k these results improve and complement existing results for system without diffusion moreover when these results are applied to the diffusive nicholson s blowflies model and the diffusive model of hematopoiesis some new results are obtained for the latter

on the stability and uniform persistence of a discrete model of nicholson s blowflies abstract consider the delay difference equation n n 1 nn n n pn n k e an n k n 0 1 2 1 which is a discrete analogue of the delay differential equation n t n t pn t e an t t 0 cf 8 we show that when p the zero solution of 1 is uniformly asymptotically stable and every non negative solution of 1 tends to zero as n while for p 1 is uniformly persistent moreover if in addition to p we also have 1 k 1 1 ln p 1 then every positive solution of 1 tends to the positive equilibrium n 1 a ln p

hopf bifurcations in a reaction diffusion population model with delay effect abstract a reaction diffusion population model with a general time delayed growth rate per capita is considered the growth rate per capita can be logistic or weak allee effect type from a careful analysis of the characteristic equation the stability of the positive steady state solution and the existence of forward hopf bifurcation from the positive steady state solution are obtained via the implicit function theorem where the time delay is used as the bifurcation parameter the general results are applied to a food limited population model with diffusion and delay effects as well as a weak allee effect population model

global attractivity spreading speeds and traveling waves of delayed nonlocal reaction diffusion systems abstract the purpose of this work is to study the spatial dynamics of some delayed nonlocal reaction diffusion systems in whole space we first establish a series of comparison theorems to investigate the global attractivity of the equilibria for a delayed nonlocal reaction diffusion system with and without quasi monotonicity then we show that the spreading speed of a general system without quasi monotone conditions is coincident with the minimal wave speed applying a fluctuation method we further provide some sufficient conditions to ensure the upward convergence of the spreading speed and traveling wave solutions finally we point out the effects of the delay and nonlocality on the spreading speed of the non quasi monotone systems

saddle point behavior for monotone semiflows and reaction diffusion models the saddle point behavior is established for monotone semiflows with weak bistability structure and then these results are applied to three reaction diffusion systems modeling man environment man epidemics single loop positive feedback and two species competition respectively

attractivity multistability and bifurcation in delayed hopfieldʼs model with non monotonic feedback abstract for a system of delayed neural networks of hopfield type we deal with the study of global attractivity multistability and bifurcations in general we do not assume monotonicity conditions in the activation functions for some architectures of the network and for some families of activation functions we get optimal results on global attractivity our approach relies on a link between a system of functional differential equations and a finite dimensional discrete dynamical system for it we introduce the notion of strong attractor for a discrete dynamical system which is more restrictive than the usual concept of attractor when the dimension of the system is higher than one our principal result shows that a strong attractor of a discrete map gives a globally attractive equilibrium of a corresponding system of delay differential equations our abstract setting is not limited to applications in systems of neural networks we illustrate its use in an equation with distributed delay motivated by biological models we also obtain some results for neural systems with variable coefficients

on the basins of attraction for a class of delay differential equations with non monotone bistable nonlinearities abstract we consider the delay differential equation dde x t g x t f x t which shares the same equilibria with the corresponding ordinary differential equation ode x t g x t f x t for the bistable case both the dde and ode share three equilibria x 0 0 x 1 x 2 with x 0 and x 2 being stable and x 1 being unstable for the ode we are concerned with stability of these equilibria for the dde and the basins of attraction of x 0 and x 2 when they are asymptotically stable for the dde combining the idea of relating the dynamics of a map to the dynamics of a dde and invariance arguments for the solution semiflow we are able to characterize some subsets of basins of attraction of these equilibria for the dde in addition existence of heteroclinic orbits is also explored the general results are applied to a particular model equation describing the matured population of some species demonstrating the allee effect

reaction diffusion systems with time delays monotonicity invariance comparison and convergence in a recent paper we have obtained differential inequality results developed comparison theorems and established invariance criteria for mild solutions of abstract functional differential equations in a banach space the primary application for these results is to reaction diffusion systems in which the reaction nonlinearity contains time delays in the present paper we apply these results to reaction diffusion systems with time delays of the form u t d u f x u t x x u u n 0 x where is a bounded open set in r n with smooth boundary u r n d diag d 1 d n with d i 0

inequalities for sums of powers be valid for all n e n and x x 0 it is necessary ana s cient thut min a b min c d and max a 6 d max c d 5 in i brenner showed independently that 5 is a sufficient condition for 4 he also obtained a number of more general results there exist other generalizations of theorem a see for instance losonczi 3 pales s but now we show one way that can also be applied to the inequality 1 assume that a b and c d and then rearrange 4 to obtain l q xy h xf p h x xt xi i xf x i cd c it seems to be natural to consider the following more general inequality

global dynamics of a delay differential equation with spatial non locality in an unbounded domain abstract in this paper we study the global dynamics of a class of differential equations with temporal delay and spatial non locality in an unbounded domain adopting the compact open topology we describe the delicate asymptotic properties of the nonlocal delayed effect and establish some a priori estimate for nontrivial solutions which enables us to show the permanence of the equation combining these results with a dynamical systems approach we determine the global dynamics of the equation under appropriate conditions applying the main results to the model with ricker s birth function and mackey glass s hematopoiesis function we obtain threshold results for the global dynamics of these two models we explain why our results on the global attractivity of the positive equilibrium in c 0 under the compact open topology becomes invalid in c 0 with respect to the usual supremum norm and we identify a subset of c 0 in which the positive equilibrium remains attractive with respect to the supremum norm

symmetric functional differential equations and neural networks with memory we establish an analytic local hopf bifurcation theorem and a topological global hopf bifurcation theorem to detect the existence and to describe the spatial temporal pattern the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry we apply these general results to obtain the coexistence of multiple large amplitude wave solutions for the delayed hopfield cohen grossberg model of neural networks with a symmetric circulant connection matrix

fundamental groups of galois closures of generic projections for the galois closure x gal of a generic projection from a surface x it is believed that 1 x gal gives rise to new invariants of x however in all examples this group is surprisingly simple in this article we offer an explanation for this phenomenon we compute a quotient of 1 x gal that depends on 1 x and data from the generic projection only in all known examples this quotient is in fact isomorphic to 1 x gal as a byproduct we simplify the computations of moishezon teicher and others

a reaction diffusion model for a single species with age structure i travelling wavefronts on unbounded domains in this paper we derive the equation for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment we show that if the mature deat

geometric methods of global attraction in systems of delay differential equations abstract in this paper we deduce criteria of global attraction in systems of delay differential equations our methodology is new and consists in dominating the nonlinear terms of the system by a scalar function and then studying some dynamical properties of that function one of the crucial benefits of our approach is that we obtain delay dependent results of global attraction that cover the best delay independent conditions we apply our results in a gene regulatory model and the classical nicholson s blowfly equation with patch structure

on a system of q laplace transform of two variables with applications abstract in this paper we introduce the definition of double laplace transform in q calculus by using the functions of several variables certain results on convergence absolute convergence convolution and its properties were discussed later on to illustrate this integral transform several problems were discussed for the effectiveness and performance of the proposed method also this paper surveys recent applications to solve generalized diffusion wave and space time telegraphic equations

propagation and heterogeneous steady states in a delayed nonlocal r d equation without spatial translation invariance abstract we consider a delayed reaction diffusion equation that models the population dynamics of a single species with the mature population living in the 1 d whole space r while the immature population only living in the half space r with homogeneous dirichlet condition for the immatures at the boundary point one of the important features of this model system is that it does have the translational invariance by linking the non translational invariant solution map for this equation to travelling wave maps for another related 1 d spatial homogeneous delay reaction diffusion equation we obtain some traveling like a priori estimates for nontrivial solutions we then establish the existence uniqueness and attractivity of heterogeneous steady states as a result we are able to describe the traveling like asymptotic behaviours of nontrivial solutions in space time region these enable us to develop a new method for exploring the spreading speeds and asymptotic propagation phenomena for a class of non translation invariant delay reaction diffusion equations on r as a corollary we also recover some results on the asymptotic spreading speeds and traveling waves for monostable and spatial homogeneous delay reaction diffusion equations in r

monotone waves for non monotone and non local monostable reaction diffusion equations abstract we propose a new approach for proving existence of monotone wavefronts in non monotone and non local monostable diffusive equations this allows to extend recent results established for the particular case of equations with local delayed reaction in addition we demonstrate the uniqueness modulo translations of obtained monotone wavefront within the class of all monotone wavefronts such a kind of conditional uniqueness was recently established for the non local kpp fisher equation by fang and zhao moreover we show that if delayed reaction is local then each monotone wavefront is unique modulo translations within the class of all non constant traveling waves our approach is based on the construction of suitable fundamental solutions for linear integral differential equations we consider two alternative scenarios in the first one the fundamental solution is negative typically holds for the mackey glass diffusive equations while in the second one the fundamental solution is non negative typically holds for the kpp fisher diffusive equations

stable coexistence and bi stability for competitive systems on ordered banach spaces abstract autonomous and asymptotically autonomous semiflows modeling two species competition are studied which are strongly order preserving on a convex subset of a banach space with competitive order conditions are derived for stable coexistence bi stability and competitive exclusion a complete classification of all possible outcomes is obtained in case that there is at most one positive steady state representing coexistence of both species

the fundamental group of a mathcal cp 2 complement of a branch curve as an extension of a solvable group by a symmetric group the main result in this paper is as follows theorem let s be the branch curve in cp 2 of a generic projection of a veronese surface then 1 cp 2 s is an extension of a solvable group by a symmetric group a group with the property mentioned in the theorem is almost solvable in the sense that it contains a solvable normal subgroup of finite index we pose the following question question for which families of simply connected algebraic surfaces of general type is the fundamental group of the complement of the branch curve of a generic projec tion to cp 2 an extension of a solvable group by a symmetric group

local monotonicity properties of two variable gini means and the comparison theorem revisited abstract in this paper we examine local monotonicity properties of the so called gini means defined by g a b x y x a y a x b y b 1 a b if a b exp x a log x y a log y x a y a if a b with respect to the parameters a b as a consequence of our results we obtain a new proof of the comparison theorem of two variable gini means

study on monostable and bistable reaction diffusion equations by iteration of travelling wave maps abstract in this paper based on the iterative properties of travelling wave maps we develop a new method to obtain spreading speeds and asymptotic propagation for monostable and bistable reaction diffusion equations precisely for dirichlet problems of monostable reaction diffusion equations on the half line by making links between travelling wave maps and integral operators associated with the dirichlet diffusion kernel the latter is not invariant under translation we obtain some iteration properties of the dirichlet diffusion and some a priori estimates on nontrivial solutions of dirichlet problems under travelling wave transformation we then provide the asymptotic behavior of nontrivial solutions in the space time region for dirichlet problems these enable us to develop a unified method to obtain results on heterogeneous steady states travelling waves spreading speeds and asymptotic spreading behavior for dirichlet problem of monostable reaction diffusion equations on r as well as of monostable bistable reaction diffusion equations on r

new results on pseudo almost periodic solutions of quaternion valued fuzzy cellular neural networks with delays abstract this article investigates quaternion valued fuzzy cellular neural networks with delays with the help of fixed point theory exponential dichotomy of linear equations and related inequalities some new sufficient conditions which guarantee the existence and global exponentially stability of pseudo almost periodic solutions to quaternion valued fuzzy cellular neural networks with delays are established computer simulations are carried out to check the practicability of the obtained main results so far no researchers have investigated this topic the derived results of this article are completely innovative and complement some earlier investigations to some degree

travelling waves for delayed reaction diffusion equations with global response we develop a new approach to obtain the existence of travelling wave solutions for reaction diffusion equations with delayed non local response the approach is based on an abstract formulation of the wave profile as a solution of an operational equation in a certain banach space coupled with an index formula of the associated fredholm operator and some careful estimation of the nonlinear perturbation the general result relates the existence of travelling wave solutions to the existence of heteroclinic connecting orbits of a corresponding functional differential equation and this result is illustrated by an application to a model describing the population growth when the species has two age classes and the diffusion of the individual during the maturation process leads to an interesting non local and delayed response for the matured population

stability and stabilization of quaternion valued neural networks with uncertain time delayed impulses direct quaternion method abstract in this paper the exponential stability of impulsive quaternion valued neural networks iqvnns with time delays is investigated both the differential equation and impulses are affected by time varying delays which may be nonidentical uncertain parameters in impulses are also considered firstly the existence and uniqueness of the equilibrium of the iqvnns are discussed with the help of homeomorphic mapping theorem then by utilizing lyapunov functions and a newly developed inequality several sufficient criteria are established in the form of quaternion valued linear matrix inequalities lmis it should be noted that compared to most of existing results which are obtained by decomposing quaternion valued neural networks qvnns into four real valued neural networks rvnns or two complex valued neural networks cvnns our results are easier to be verified since the quaternion valued lmis exhibit lower dimensions then based on obtained results one impulsive controller is designed to realize the stabilization of delayed qvnns finally two numerical examples are presented to verify the effectiveness and merits of the theoretical analysis

dynamic analysis on a delayed nonlinear density dependent mortality nicholson s blowflies model this paper discusses a class of nonlinear density dependent mortality nicholson s blowflies equation with multiple pairs of time varying delays by utilising differential inequality techniques and

functional equations involving means and gauss compositions of means abstract in this paper we give a review of results on functional equations involving means and their gauss compositions we also formulate some open problems

comparison of means generated by two functions and a measure abstract given two continuous functions f g i r such that g is positive and f g is strictly monotone and a probability measure on the borel subsets of 0 1 the two variable mean m f g i 2 i is defined by m f g x y f g 1 0 1 f t x 1 t y d t 0 1 g t x 1 t y d t x y i the aim of this paper is to study the comparison problem of these means i e to find conditions for the generating functions f g and h k and for the measures such that the comparison inequality m f g x y m h k x y x y i holds

the number and stability of limit cycles for planar piecewise linear systems of node saddle type abstract the objective of this paper is to study the number and stability of limit cycles for planar piecewise linear pwl systems of node saddle type with two linear regions firstly we give a thorough analysis of limit cycles for lienard pwl systems of this type proving one is the maximum number of limit cycles and obtaining necessary and sufficient conditions for the existence and stability of a unique limit cycle these conditions can be easily verified directly according to the parameters in the systems and play an important role in giving birth to two limit cycles for general pwl systems in this step the tool of a bendixon like theorem is successfully employed to derive the existence of a limit cycle secondly making use of the results gained in the first step we obtain parameter regions where the general pwl systems have at least one at least two and no limit cycles respectively in addition for the general pwl systems some sufficient conditions are presented for the existence and stability of a unique one and exactly two limit cycles respectively finally some numerical examples are given to illustrate the results and especially to show the existence and stability of two nested limit cycles

dynamics of the diffusive nicholson s blowflies equation with distributed delay in this paper we study the diffusive nicholson s blowflies equation generalizing previous works we model the generation delay by using an integral convolution over all past times and results are obtained for general delay kernels as far as possible the linearized stability of the non zero uniform steady state is studied in detail mainly by using the principle of the argument global stability both of this state and of the zero state are studied by using energy methods and by employing a comparison principle for delay equations finally we consider what bifurcations are possible from the non zero uniform state in the case when it is unstable

global stability of a class of delay differential systems this paper is concerned with a class of systems of delay differential equations which are defined on the nonnegative function space under proper conditions we employ a novel proof to establish several criteria of the global stability of positive equilibrium moreover we give two examples to illustrate our main results

global attractivity for a delay logistic equation with instantaneous terms abstract in this paper some global attractivity conditions are obtained for delay logistic differential equation with instantaneous terms x t r t 1 x t cx t f t x t t 0 where c 0 these conditions extend and improve all existing relative results in literature

global attractivity of the diffusive nicholson blowflies equation with neumann boundary condition a non monotone case abstract in this paper we establish the global attractivity of the positive steady state of the diffusive nicholson s equation with homogeneous neumann boundary value under a condition that makes the equation a non monotone dynamical system to achieve this we develop a novel method combining a dynamical systems argument with maximum principle and some subtle inequalities

stability and hopf bifurcation in a diffusive logistic population model with nonlocal delay effect 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

a new algorithm for calculating adomian polynomials for nonlinear operators in this paper a reliable technique for calculating adomian polynomials for nonlinear operators will be developed the new algorithm offers a promising approach for calculating adomian polynomials for all forms of nonlinearity the algorithm will be illustrated by studying suitable forms of nonlinearity a nonlinear evolution model will be investigated

almost periodic solutions for an impulsive delay nicholson s blowflies model by means of the contraction mapping principle and gronwall bellman s inequality we prove the existence and exponential stability of positive almost periodic solution for an impulsive delay nicholson s blowflies model the main results are illustrated by an example

existence multiplicity shape and attractivity of heterogeneous steady states for bistable reaction diffusion equations in the plane abstract we consider a class of bistable reaction diffusion equations in the plane first we introduce a partition of the plane into infinitely many sectors and consider dirichlet problems in these sectors by establish some a priori estimates for nontrivial solutions to these sub systems we obtain the existence and attractivity of a heterogeneous steady state of the dirichlet problem in each of the sectors and prove the existence of a maximum positive steady state and describe the asymptotic behaviours of positive steady states at the infinities we also estimate limit sets at the vicinities of the boundaries of the sectors near origin and at infinities further assuming the sub linearity for the reaction term we obtain the uniqueness and attractivity of a heterogeneous steady state by applying the dynamical and sliding methods these results help us describe the multiplicity shape and attractivity of the heterogeneous steady states for the equation

traveling waves for the diffusive nicholson s blowflies equation we consider traveling wave front solutions for the diffusive nicholson s blowflies equation on the real line the existence of such solutions is proved using the technique developed by j wu and x zou j dyn differ equations 13 3 2001 some numerical simulation using the iteration formula of wu and zou 7 is also provided

hopf bifurcation analysis in a delayed nicholson blowflies equation abstract the dynamics of a nicholson s blowflies equation with a finite delay are investigated we prove 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 of wu trans amer math soc 350 1998 4799 and a bendixson criterion for higher dimensional ordinary differential equations due to li and muldowney j differential equations 106 1994 27