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

on the geometry of wave solutions of a delayed reaction diffusion equation abstract the aim of this paper is to study the existence and the geometry of positive bounded wave solutions to a non local delayed reaction diffusion equation of the monostable type

numerical values of goldberg s coefficients in the series for log e x e y calcul des coefficients de goldberg dans la serie infinie de log e x e y pour x et y non communitatifs avec des mots de longueur 20

a geometric approach in the study of traveling waves for some classes of non monotone reaction diffusion systems abstract in this paper we further extend a recently developed method to investigate the existence of traveling waves solutions and their minimum wave speed for non monotone reaction diffusion systems our approach consists of two steps first we develop a geometrical shooting argument with the aid of the theorem of homotopy invariance on the fundamental group to obtain the positive semi traveling wave solutions for a large class of reaction diffusion systems including the models of predator prey interaction for both predator independent dependent functional responses the models of combustion belousov zhabotinskii reaction si type of disease transmission and the model of biological flow reactor in chemostat next we apply the results obtained from the first step to some models such as the beddinton deangelis model and the model of biolocal flow reactor to show the convergence of these semi traveling wave solutions to an interior equilibrium point by the construction of a lyapunov type function or the convergence of semi traveling waves to another boundary equilibrium point by the further analysis of the asymptotical behavior of semi traveling wave solutions

strongly order preserving semiflows generated by functional differential equations the aim of this paper is to apply some recently developed improvements in the theory of monotone dynamical systems due to the authors 12 13 j to certain systems of functional differential equations fde s which do not enjoy the quasimonotone property considered by one of us in 9 the present work extends our previous results in this direction for scalar equations in 111 to systems in the fundamental paper s m w hirsch establishes that most orbits of a strongly monotone scmiflow on a strongly ordered space x tend to the set e of equilibria employing ideas of hirsch and of h matano 7 the authors have extended the theory to obtain some improvements in the results of hirsch and matano under matano s weaker assumption that the semiflow is strongly order preserving recently in 13 we gave sufficient conditions for most orbits of a strongly order preserving semiflow to converge to an equilibrium these conditions require additional smoothness of the semiflow and a certain strong monotonicity condition for the linearized flow determined by the variational equation about each equilibrium p policik s has also given sufficient conditions for most orbits to converge for semilinear parabolic evolution equations possessing certain strong monotonicity properties in this paper wc apply the convergence results developed in 131 to nonquasimonotone fde s in this introduction wc briefly review the application of monotone dynamical systems theory to fde s enjoying the quasimonotone property this research appeared in 9 133 then we state one of the main results of this paper which applies when the quasimonotone property fails but when another property m holds some simple examples are given which

spreading speeds and travelling waves for non monotone time delayed lattice equations a non monotone time delayed lattice system with global interaction is considered the spreading speed including the upward convergence is established by comparison arguments and a fluctuation met

monotone traveling wavefronts of the kpp fisher delayed equation abstract in the early 2000 s gourley 2000 wu et al 2001 ashwin et al 2002 initiated the study of the positive wavefronts in the delayed kolmogorov petrovskii piskunov fisher equation u t t x u t x u t x 1 u t h x u 0 x r m since then this model has become one of the most popular objects in the studies of traveling waves for the monostable delayed reaction diffusion equations in this paper we give a complete solution to the problem of existence and uniqueness of monotone waves in eq we show that each monotone traveling wave can be found via an iteration procedure the proposed approach is based on the use of special monotone integral operators which are different from the usual wu zou operator and appropriate upper and lower solutions associated to them the analysis of the asymptotic expansions of the eventual traveling fronts at infinity is another key ingredient of our approach

exact and analytic numerical solution of coupled diffusion problems in a semi infinite medium in this paper we consider coupled semi infinite diffusion problems of the form ut x t a2uxx x t x 0 t 0 subject to u 0 t b and u x 0 0 where a is a matrix in cr r and u x t and b are vectors in cr given an admissible error and a bounded domain d x0 x1 t0 x t 0 0 an analytic numerical solution is constructed so that the error with respect to the exact solution is uniformly upper bounded by in d x0 x1 t0

entire solutions in bistable reaction diffusion equations with nonlocal delayed nonlinearity this paper is concerned with entire solutions for bistable reaction diffusion equations with nonlocal delay in one dimensional spatial domain here the entire solutions are defined in the whole space and for all time t r assuming that the equation has an increasing traveling wave solution with nonzero wave speed and using the comparison argument we prove the existence of entire solutions which behave as two traveling wave solutions coming from both ends of the x axis and annihilating at a finite time furthermore we show that such an entire solution is unique up to space time translations and is liapunov stable a key idea is to characterize the asymptotic behavior of the solutions as t in terms of appropriate subsolutions and supersolutions in order to illustrate our main results two models of reaction diffusion equations with nonlocal delay arising from mathematical biology are considered

on the uniqueness of semi wavefronts for non local delayed reaction diffusion equations abstract we establish the uniqueness of semi wavefront solution for a non local delayed reaction diffusion equation this result is obtained by using a generalization of the diekmann kaper theory for a nonlinear convolution equation several applications to the systems of non local reaction diffusion equations with distributed time delay are also considered

traveling waves for nonlinear cellular neural networks with distributed delays abstract in this paper we will establish the existence and nonexistence of traveling waves for nonlinear cellular neural networks with finite or infinite distributed delays the dynamics of each given cell depends on itself and its nearest m left or l right neighborhood cells where delays exist in self feedback and left or right neighborhood interactions our approach is to use schauder s fixed point theorem coupled with upper and lower solutions of the integral equation in a suitable banach space further we obtain the exponential asymptotic behavior in the negative infinity and the existence of traveling waves for the minimal wave speed by the limiting argument our results improve and cover some previous works

slowly oscillating wave solutions of a single species reaction diffusion equation with delay we study positive bounded wave solutions u t x x ct 0 of equation ut t x u t x u t x g u t h x x rm this equation is assumed to have two non negative equilibria u1 0 and u2 0 the birth function g c r r is unimodal and differentiable at 0 and some results also require the feedback condition g s s c our results are based on a detailed geometric description of the wave profile in particular the monotonicity of its leading edge is established we also discuss the uniqueness problem indicating a subclass g of asymmetric tent maps such that given g g there exists exactly one positive travelling front for each fixed admissible speed

domain decomposition methods for a class of spatially heterogeneous delayed reaction diffusion equations abstract we derive and study a class of delayed reaction diffusion equations with spatial heterogeneity which models the population of a single species with different habitats for mature and immature individuals we introduce new solid cones obtain spectral bounds of several spatial heterogeneous operators and establish limiting non negativeness property for the whole space and the eventual comparison principle for bounded domains as a result we develop new domain decomposition methods so that one can compare solutions with those to associated equations from a suitable bounded spatial domain to the whole space then by employing domain decomposition methods and dynamical system approaches we obtain threshold results under the supremum norm these results are greatly different from the existing ones of other evolution equations in unbounded domains or the whole space the main results are applied to two examples with the ricker birth function and with the mackey glass birth function it reveals that the size of the immature habitat can affect the reproduction and spread of the population

on the general linear coupled system for diffusion in media with two diffusivities abstract for the general linear coupled system of partial differential equations arising in the theory of diffusion in media with double diffusivity simple uniqueness criteria and a method of solution of boundary value problems are established the equations studied retain the so called cross terms which have been neglected in all previous investigations moreover these equations arise as generalizations of a number of existing theories for example heat flow in heterogeneous multicomponent systems flow of water in fissured rocks and a model of an arms race the simple inequalities obtained on the various constants of the theory which guarantee uniqueness of solutions and existence of source solutions might serve as guidelines in an experimental determination of these constants the solution procedure involves solving two boundary value problems for the classical diffusion equation and the formulae given mean that closed form expressions can be deduced for a number of commonly occurring boundary value problems the paper emphasizes the general equations without special reference to particular physical applications or boundary value problems

cyclic shifts and an exponential formula abstract with x and y noncommutative indeterminates a natural presentation of z log e x e y as a sum of group conjugates of x and y is shown to depend on the fact that the terms involving the cyclic shifts of any fixed word other than x or y in the power series presentation of z sum to zero

traveling waves in the nonlocal kpp fisher equation different roles of the right and the left interactions abstract we consider the nonlocal kpp fisher equation u t t x u x x t x u t x 1 k u t x which describes the evolution of population density u t x with respect to time t and location x the non locality is expressed in terms of the convolution of u t with kernel k 0 r k s d s 1 the restrictions k s s 0 and k s s 0 are responsible for interactions of an individual with his left and right neighbors respectively we show that these two parts of k play quite different roles as for the existence and uniqueness of traveling fronts to the kpp fisher equation in particular if the left interaction is dominant the uniqueness of fronts can be proved while the dominance of the right interaction can induce the co existence of monotone and oscillating fronts we also present a short proof of the existence of traveling waves without assuming various technical restrictions usually imposed on k

on the existence of traveling waves for delayed reaction diffusion equations abstract we study the existence of traveling wave solutions for reaction diffusion equations with nonlocal delay where reaction terms are not necessarily monotone the existence of traveling wave solutions for reaction diffusion equations with nonlocal delays is obtained by combining upper and lower solutions for associated integral equations and the schauder fixed point theorem the smoothness of upper and lower solutions is not required in this paper

stability of non monotone critical traveling waves for reaction diffusion equations with time delay abstract this paper is concerned with the stability of critical traveling waves for a kind of non monotone time delayed reaction diffusion equations including nicholson s blowflies equation which models the population dynamics of a single species with maturation delay such delayed reaction diffusion equations possess monotone or oscillatory traveling waves the latter occurs when the birth rate function is non monotone and the time delay is big it has been shown that such traveling waves x c t exist for all c c and are exponentially stable for all wave speed c c 13 where c is called the critical wave speed in this paper we prove that the critical traveling waves x c t monotone or oscillatory are also time asymptotically stable when the initial perturbations are small in a certain weighted sobolev norm the adopted method is the technical weighted energy method with some new flavors to handle the critical oscillatory waves finally numerical simulations for various cases are carried out to support our theoretical results

a matrix approach to the analytic numerical solution of mixed partial differential systems e navarro e ponsoda and l jodar departamento de matem tica aplicada universidad polit cnica de valencia p o box 22012 valencia spain received and accepted march 1994 abstract in this paper analytic numerical solutions for initial boundary value problems related to systems of partial differential equations are proposed given an admissible error e 0 and a finite domain g a numerical approximation is constructed in terms of the data so that the error is uniformly upper bounded by e in g keywords partial differential system eigenvalues bound error bound schur decomposition matrix norms matrix functions

continuous numerical solutions of coupled mixed partial differential systems using fer s factorization in this paper continuous numerical solutions expressed in terms of matrix exponentials are constructed to approximate time dependent systems of the type ut a t uxx b t u 0 0 0 u 0 t u p t 0 u x 0 f x 06 x6p after truncation of an exact series solution the numerical solution is constructed using fer s factorization given 0 and t0 t1 with 0 t0 t1 and d t0 t1 f x t 06x6p t06t6t1g the error of the approximated solution with respect to the exact series solution is less than uniformly in d t0 t1 an algorithm is also included c 1999 elsevier science b v all rights reserved ams classication 65m15 34a50 35c10 35a50

asymptotic patterns of a structured population diffusing in a two dimensional strip abstract in this paper we derive a population model for the growth of a single species on a two dimensional strip with neumann and robin boundary conditions we show that the dynamics of the mature population is governed by a reaction diffusion equation with delayed global interaction using the theory of asymptotic speed of spread and monotone traveling waves for monotone semiflows we obtain the spreading speed c the non existence of traveling waves with wave speed 0 c c and the existence of monotone traveling waves connecting the two equilibria for c c

asymptotic behavior and traveling wave solutions for parabolic functional differential equations on presente une generalisation de la theorie des equations kpp et de diffusion bistables non lineaires on montre que des solutions onde de propagation existent pour des equations fonctionnelles paraboliques non lineaires

minimal wave speed for a class of non cooperative reaction diffusion systems of three equations abstract in this paper we study the traveling wave solutions and minimal wave speed for a class of non cooperative reaction diffusion systems consisting of three equations based on the eigenvalues a pair of upper lower solutions connecting only the invasion free equilibrium are constructed and the schauder s fixed point theorem is applied to show the existence of traveling semi fronts for an auxiliary system then the existence of traveling semi fronts of original system is obtained by limit arguments the traveling semi fronts are proved to connect another equilibrium if natural birth and death rates are not considered and to be persistent if these rates are incorporated then non existence of bounded traveling semi fronts is obtained by two sided laplace transform then the above results are applied to some disease transmission models and a predator prey model

travelling waves for a size and space structured model in population dynamics point to sustained oscillating solution connections abstract this work is devoted to the study of travelling wave solutions for some size structured model in population dynamics the population under consideration is also spatially structured and has a nonlocal spatial reproduction this phenomenon may model the invasion of plants within some empty landscape since the corresponding unspatially structured size structured models may induce oscillating dynamics due to hopf bifurcations the aim of this work is to prove the existence of point to sustained oscillating solution travelling waves for the spatially structured problem from a biological view point such solutions represent the spatial invasion of some species with spatio temporal patterns at the place where the population is established

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

existence and nonexistence of monotone traveling waves for the delayed fisher equation abstract in this paper a new approach based on a shooting method in a half line coupled with the technique of upper lower solution pair is used to study the existence and nonexistence of monotone waves for one form of the delayed fisher equation that does not have the quasimonotonicity property a necessary and sufficient condition is provided this new method can be extended to investigate many other nonlocal and non monotone delayed reaction diffusion equations

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

existence and uniqueness of traveling waves for non monotone integral equations with applications abstract a class of integral equations without monotonicity is investigated it is shown that there is a spreading speed c 0 for such an integral equation and that its limiting integral equation admits a unique traveling wave up to translation with speed c c and no traveling wave with c c these results are also applied to some nonlocal reaction diffusion population models

persistence of wavefronts in delayed nonlocal reaction diffusion equations we develop a perturbation argument based on existing results on asymptotic autonomous systems and the fredholm alternative theory that yields the persistence of traveling wavefronts for reaction diffusion equations with nonlocal and delayed nonlinearities when the time lag is relatively small this persistence result holds when the nonlinearity of the corresponding ordinary reaction diffusion system is either monostable or bistable we then illustrate this general result using five different models from population biology epidemiology and bio reactors

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

separation dichotomy and wavefronts for a nonlinear convolution equation abstract this paper is concerned with a scalar nonlinear convolution equation which appears naturally in the theory of traveling waves for monostable evolution models first we prove that at each end of the real line every bounded positive solution of the convolution equation should either be separated from zero or be exponentially converging to zero this dichotomy principle is then used to establish a general theorem guaranteeing the uniform persistence and existence of semi wavefront solutions to the convolution equation finally we apply our theoretical results to several well studied classes of evolution equations with asymmetric non local and non monotone response we show that contrary to the symmetric case these equations can possess simultaneously stationary expansion and extinction waves

monotone semiflows in scalar non quasi monotone functional differential equations abstract by introducing a stronger than pointwise ordering conditions are found under which scalar functional differential equations generate monotone semiflows even if they are not quasi monotone typically the maximum delay must be the smaller the more quasi monotonicity is violated the theory of monotone semiflows is used to show that most solutions converge to equilibrium and that stability of equilibria is essentially the same as for ordinary differential equations

perron theorem in the monotone iteration method for traveling waves in delayed reaction diffusion equations abstract in this paper we revisit the existence of traveling waves for delayed reaction diffusion equations by the monotone iteration method we show that perron theorem on existence of bounded solution provides a rigorous and constructive framework to find traveling wave solutions of reaction diffusion systems with time delay the method is tried out on two classical examples with delay the predator prey and belousov zhabotinskii models

some open problems in matrix theory arising in linear systems and control control theory has long provided a rich source of motivation for developments in matrix theory accordingly we discuss some open problems in matrix theory arising from theoretical and practical issues in linear systems theory and feedback control the problems discussed include robust stability matrix exponent induced norms stabilizability and pole assignability and nonstandard matrix equations a substantial number of references are included to acquaint matrix theorists with problems and trends in this application area feedback control theory has long provided a rich source of motivation for developments in matrix theory the purpose of this paper is to discuss several open problems in matrix theory that arise from theoretical and practical issues in feedback control theory and the associated area of linear systems theory many of these problems are remarkably simple to state are of intense interest in control theory and applications and yet remain unsolved besides leading to the resolution of these problems it is hoped that this paper will help to stimulate increased interaction between matrix and control theorists accord

shadowing lemma and singularly perturbed boundary value problems a complete procedure is given to determine the outer and inner expansions of a singularly perturbed boundary value problem in mathbb r n the validity of such expansions is deduced from a gene

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

uniqueness of fast travelling fronts in reaction diffusion equations with delay we consider positive travelling fronts u t x xct 0 of the equation ut t x u t x u t x g u th x xm this equation is assumed to have exactly two non negative equilibria u10 and u2 0

travelling waves of a diffusive kermack mckendrick epidemic model with non local delayed transmission we obtain full information about the existence and non existence of travelling wave solutions for a general class of diffusive kermack mckendrick sir models with non local and delayed disease transmission we show that this information is determined by the basic reproduction number of the corresponding ordinary differential model and the minimal wave speed is explicitly determined by the delay such as the latent period and non locality in disease transmission and the spatial movement pattern of the infected individuals the difficulty is the lack of order preserving property of the general system and we obtain the threshold dynamics for spatial spread of the disease by constructing an invariant cone and applying schauder s fixed point theorem

traveling waves in a nonlocal dispersal population model with age structure abstract this paper is concerned with the traveling waves in a single species population model which is derived by considering the nonlocal dispersal and age structure if the birth function is monotone then the existence of traveling wavefront is reduced to the existence of a pair of super and subsolutions without the requirement of smoothness it is proved that the traveling wavefront is strictly increasing and unique up to a translation the asymptotic behavior of traveling wavefronts is also obtained if the birth function is not monotone the existence of traveling wave solution is affirmed by introducing two auxiliary nonlocal dispersal equations with quasi monotonicity

transversal heteroclinic and homoclinic orbits in singular perturbation problems abstract in this paper we give a geometric construction of heteroclinic and homoclinic orbits for singularly perturbed differential equations by using methods from invariant manifold theory we show that transversal intersection of stable and unstable manifolds of the reduced problem implies the existence of transversal heteroclinic or homoclinic orbits of the singularly perturbed problem we derive analytical conditions for transversality we show how these results can be used to prove the existence of heteroclinic and homoclinic orbits in singularly perturbed problems which depend on additional parameters we describe a configuration which implies transversal intersection of the stable and unstable manifolds of periodic orbits and the associated chaotic dynamics

existence of fast positive wavefronts for a non local delayed reaction diffusion equation abstract we establish the existence of a continuous family of fast positive wavefronts u t x x c t 0 for the non local delayed reaction diffusion equation u t t x u x x t x u t x r k x w g u t h w d w here 0 and 0 are fixed points of g c 2 r r and the non negative k is such that r k w e w d w is finite for every real we also prove that the fast wavefronts are non monotone if g h e h 1 1

entire solutions for nonlocal dispersal equations with spatio temporal delay monostable case abstract this paper deals with entire solutions for a general nonlocal dispersal monostable equation with spatio temporal delay i e solutions that are defined in the whole space and for all time t r we first derive a particular model for a single species and show how such systems arise from population biology then we construct some new types of entire solutions other than traveling wave solutions and equilibrium solutions of the equation under consideration with quasi monotone and non quasi monotone nonlinearities various qualitative properties of the entire solutions are also investigated in particular the relationship between the entire solutions and the traveling wave fronts which they originated is considered our main arguments are based on the comparison principle the method of super and sub solutions and the construction of auxiliary control systems

on abstract differential equations with state dependent non local conditions abstract in this paper we introduce the concept of state dependent nonlocal condition and study the existence and uniqueness of solution for a general class of abstract differential equations with state dependent delay subject to this class of conditions in this type of problem the nonlocal condition can be dependent on times or time intervals determined by the unknown solution some examples on partial differential equations with state dependent delay arising in population dynamics are presented

spatial structures and periodic travelling waves in an integro differential reaction diffusion population model an integro differential reaction diffusion equation is proposed as a model for populations where local aggregation is advantageous but intraspecific competition increases as global populations increase it is claimed that this is inherently more realistic than the usual kind of reaction diffusion model for mobile populations three kinds of bifurcation from the uniform steady state solution are considered i to steady spatially periodic structures ii to periodic standing wave solutions and iii to periodic travelling wave solutions these correspond to aggregation and motion of populations

traveling wave fronts for generalized fisher equations with spatio temporal delays we study the existence of traveling wave fronts for a reaction diffusion equation with spatio temporal delays and small parameters the equation reduces to a generalized fisher equation if small parameters are zero we present two results in the first one we deal with the equation with very general kernels and show the persistence of fisher wave fronts for all sufficiently small parameters in the second one we deal with some particular kernels with which the nonlocal equation can be reduced to a system of singularly perturbed odes and we are then able to apply the geometric singular perturbation theory and phase plane arguments to this system to show the existence of the minimal wave speed the existence of a continuum of wave fronts and the global uniqueness of the physical wave front with each wave speed

nonmonotone travelling waves in a single species reaction diffusion equation with delay abstract we prove the existence of a continuous family of positive and generally nonmonotone travelling fronts for delayed reaction diffusion equations u t t x u t x u t x g u t h x when g c 2 r r has exactly two fixed points x 1 0 and x 2 k 0 recently nonmonotonic waves were observed in numerical simulations by various authors here for a wide range of parameters we explain why such waves appear naturally as the delay h increases for the case of g with negative schwarzian our conditions are rather optimal we observe that the well known mackey glass type equations with diffusion fall within this subclass of as an example we consider the diffusive nicholson s blowflies equation

traveling waves for non local delayed diffusion equations via auxiliary equations in this paper we study the existence of traveling wave solutions for a class of delayed non local reaction diffusion equations without quasi monotonicity the approach is based on the construction of two associated auxiliary reaction diffusion equations with quasi monotonicity and a profile set in a suitable banach space by using the traveling wavefronts of the auxiliary equations under monostable assumption by using the schauder s fixed point theorem we then show that there exists a constant c 0 such that for each c c the equation under consideration admits a traveling wavefront solution with speed c which is not necessary to be monotonic

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

entire solutions in monostable reaction diffusion equations with delayed nonlinearity abstract entire solutions for monostable reaction diffusion equations with nonlocal delay in one dimensional spatial domain are considered a comparison argument is employed to prove the existence of entire solutions which behave as two traveling wave solutions coming from both directions some new entire solutions are also constructed by mixing traveling wave solutions with heteroclinic orbits of the spatially averaged ordinary differential equations and the existence of such a heteroclinic orbit is established using the monotone dynamical systems theory key techniques include the characterization of the asymptotic behaviors of solutions as t in term of appropriate subsolutions and supersolutions two models of reaction diffusion equations with nonlocal delay arising from mathematical biology are given to illustrate main results

bistable wavefronts in a diffusive and competitive lotka volterra type system with nonlocal delays abstract a diffusive lotka volterra type model with nonlocal delays for two competitive species is considered the existence of a traveling wavefront analogous to a bistable wavefront for a single species is proved by transforming the system with nonlocal delays to a four dimensional system without delay furthermore in order to prove the asymptotic stability up to translation of bistable wavefronts of the system the existence regularity and comparison theorem of solutions of the corresponding cauchy problem are first established for the systems on r by appealing to the theory of abstract functional differential equations the asymptotic stability up to translation of bistable wavefronts are then proved by spectral methods in particular we also prove that the spreading speed is unique by upper and lower solutions technique from the point of view of ecology our results indicate that the nonlocal delays appeared in the interaction terms are not sensitive to the invasion of species of spatial isolation

analytic solution with a priori error bounds for a class of mixed coupled partial differential equations in this paper the existence of a sequence of stable solutions xn of the parametric matrix equation z2 az n2b 0 so that xnn converges to a stable matrix is studied the result is used for solving mixed problems related to the coupled second order partial differential system utt auxx but 0

positive heteroclinics and traveling waves for scalar population models with a single delay the existence of positive heteroclinic solutions is proven for a class of scalar population models with one discrete delay traveling wave solutions for scalar delayed reaction diusion equations are also obtained as perturbations of heteroclinic solutions of the associated equation without diusion as an illustration the results are applied to the nicholson s blowflies equation with diusion n t t x d 2n x2 t x n t x pn t x e an t x in the case of p e for which the nonlinearity is non monotone