parabolic partial differential equations with discrete state dependent delay classical solutions and solution manifold abstract classical solutions to pdes with discrete state dependent delay are studied we prove the well posedness in a set x f which is analogous to the solution manifold used for ordinary differential equations with state dependent delay we prove that the evolution operators are c 1 smooth on the solution manifold

existence of multiple positive periodic solutions for differential equation with state dependent delays by utilizing a fixed point theorem in cones we present some sufficient conditions which guarantee the existence of multiple positive periodic solutions for a class of differential equations with state dependent delays our results extend and improve some previous results

superstability and rigorous asymptotics in singularly perturbed state dependent delay differential equations abstract we study the singularly perturbed state dependent delay differential equation e x t x t k x t r r r x t 1 x t which is a special case of the equation e x t g x t x t r r r x t one knows that for every sufficiently small e 0 eq possesses at least one so called slowly oscillating periodic solution and moreover the graph of every such solution approaches a specific sawtooth like shape as e 0 in this paper we obtain higher order asymptotics of the sawtooth including the location of the minimum and maximum of the solution with the form of the solution near these turning points and as well an asymptotic formula for the period using these and other asymptotic formulas we further show that the solution enjoys the property of superstability namely the nontrivial characteristic multipliers are of size o e for small e this stability property implies that this solution is unique among all slowly oscillating periodic solutions again for small e

global dynamics of a state dependent delay model with unimodal feedback abstract we obtain the global attractivity of nonnegative stationary states for a parameterized state dependent delay equation with unimodal feedback under certain mild conditions we show that the equation with unimodal type nonlinearity can generate rich dynamics as the parameter varies to be specific global attractivity of the positive stationary state is obtained in a set of nonnegative bounded continuous functions when the stationary state is less than a value implicitly determined by a condition on the unimodal feedback the general results of global attractivity are illustrated through two examples arising from population dynamics moreover hopf bifurcations are demonstrated in the examples when the positive stationary states lose global attractivity

exponential stability for a class of state dependent delay equations via the crandall liggett approach abstract in this paper we are concerned with the exponential asymptotic stability of the solution of a class of differential equations with state dependent delays our approach is based on the crandall liggett approximation and the properties of semigroups

a periodic solution of a differential equation with state dependent delay we study a differential equation for delayed negative feedback which models a situation where the delay depends on the present state and becomes effective in the future the main result is existence of a periodic solution in case the equilibrium is linearly unstable the proof employs the ejective fixed point principle on a compact convex set k0 c h 0 r of lipschitz continuous functions and uses that the equation generates a smooth semiflow on an infinite dimensional submanifold of the space c1 h 0 r

a differential equation with state dependent delay from cell population biology we analyze a differential equation describing the maturation of a stem cell population with a state dependent delay which is implicitly defined via the solution of an ode we elaborate smoothness conditions for the model ingredients in particular vital rates that guarantee the existence of a local semiflow and allow to specify the linear variational equation the proofs are based on theoretical results of hartung et al combined with implicit function arguments in infinite dimensions moreover we elaborate a criterion for global existence for differential equations with state dependent delay to prove the result we adapt a theorem by hale and lunel to the c1 topology and use a result on metric spaces from diekmann et al

linearized stability in periodic functional differential equations with state dependent delays in this paper we study stability of periodic solutions of a class of nonlinear functional differential equations fdes with state dependent delays using the method of linearization we show that a periodic solution of the nonlinear fde is exponentially stable if the zero solution of an associated linear periodic linear homogeneous fde is exponentially stable

differentiability of solutions with respect to parameters in neutral differential equations with state dependent delays in this paper we consider a class of nonlinear neutral differential equations with state dependent delays in both the neutral and the retarded terms we study well posedness and continuous dependence issues and differentiability of the parameter map with respect to the initial function and other possibly infinite dimensional parameters in a pointwise sense and also in the c norm

the existence of nonoscillatory solutions to a generalized nonautonomous delay logistic equation sufficient conditions for the existence of nonoscillatory solutions to the delay logistic equation x t r t x t 1 x g t k are obtained where 1

semigroup properties and the crandall liggett approximation for a class of differential equations with state dependent delays abstract we present an approach for the resolution of a class of differential equations with state dependent delays by the theory of strongly continuous nonlinear semigroups we show that this class determines a strongly continuous semigroup in a closed subset of c 0 1 we characterize the infinitesimal generator of this semigroup through its domain finally an approximation of the crandall liggett type for the semigroup is obtained in a dense subset of c as far as we know this approach is new in the context of state dependent delay equations while it is classical in the case of constant delay differential equations

on extreme points of subordination families with a convex majorant abstract let z c z b 0 denote the set of functions analytic in and satisfying z 0 0 suppose f is analytic and univalent in and maps it onto a convex domain d other than a strip a wedge or a half plane let s f fo b 0 let es f denote the set of extreme points of s f d j hallenbeck and t h macgregor asked in 3 whether the set es f ever lies strictly between the minimal fo b 0 lim r 1 re i 1 a e and the maxima fc is an extreme point of b 0 sets we prove that this always happens when the boundary of d contains a line segment let d f be any convex domain and let f be subordinate to f we show that if f is an extreme point of s f then the closure of the boundary values f of f f lim r 1 f re i a e has a nonempty intersection with the set of extreme points of the boundary of d we also completely determine extreme points of s f when the boundary of d consists of finitely many arcs wih positive curvature

stability of a competition model with two stage structure a population model with stage structure which describes the competition of two species for space is considered the model is established under the assumptions that reproductive rates of two species are affected by competition and that the space occupied by one type of juveniles cannot be neglected estimates on the length of delays for which positive equilibrium of the model remains stable are given sufficient conditions are also derived that ensure global stability of positive equilibrium

global hopf bifurcation for differential equations with state dependent delay abstract we develop a global hopf bifurcation theory for a system of functional differential equations with state dependent delay the theory is based on an application of the homotopy invariance of s 1 equivariant degree using the formal linearization of the system at a stationary state our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state dependent delay

global attractivity of nonautonomous stage structured population models with dispersal and harvest the nonautonomous stage structured single species dispersal model with harvesting of mature individuals in an n patch environment is considered in which the individual members of the population have a life history that takes them through two stages immature and mature by using the theory of monotone and concave operators to functional differential equations we establish conditions under which this population dynamical system admits a positive periodic solution which attracts all positive solutions

optimal harvesting policy for single population with stage structure it is the purpose of this paper to analyze the exploitation of the stage structured singleautonomous population model we analyze the harvesting population equation and obtain existence local and global geometric properties of nonnegative equilibria explicit expressions are obtained for the optimal harvesting effort the maximum sustainable yield and the corresponding population level

on state dependent delay partial neutral functional integro differential equations in this paper the existence of mild solutions for a class of abstract neutral integro differential equations with state dependent delay is studied

the solution manifold and c1 smoothness for differential equations with state dependent delay abstract let h 0 u c 1 h 0 r n open and f u r n continuously differentiable if f satisfies two mild additional smoothness conditions then the set x u 0 f is a c1 submanifold of codimension n in c 1 h 0 r n the maximal solutions x of the initial value problems x t f x t x 0 x define a continuous semiflow f on x and all operators f t are continuously differentiable their derivatives d2f t are given in the usual way by solutions v to the variational equation along x with segments vt in the tangent spaces tf t x the additional conditions on f are motivated by properties of differential equations with state dependent delay and are verified for an example

existence of periodic solutions for delay differential equations with state dependent delay abstract in this paper we study the existence of periodic solutions of a delay differential equation with delay depending indirectly on the state a fixed point problem related to a poincare operator is constructed and solved using an ejective fixed point theorem

wavefronts and global stability in a time delayed population model with stage structure we formulate and study a one dimensional single species diffusive delay population model the time delay is the time taken from birth to maturity without diffusion the delay differential model extends the well known logistic differential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes this delayed model is known to have simple global dynamics similar to that of the logistic equation through the use of a sub supersolution pair method we show that the diffusive delay model continues to generate simple global dynamics this has the important biological implication that quadratically regulated death processes dramatically simplify the growth dynamics we also consider the possibility of travelling wavefront solutions of the scalar equation for the mature population connecting the zero solution of that equation with the positive steady state our main finding here is that our fronts appear to be all monotone regardless of the size of the delay this is in sharp contrast to the frequently reported findings that delay causes a loss of monotonicity with the front developing a prominent hump in some other delay models

metastable periodic patterns in singularly perturbed state dependent delayed equations abstract we consider the scalar delayed differential equation x t x t f x t r where 0 r r x 1 r x and f represents either a monotone positive feedback d f d x 0 or a monotone negative feedback d f d x 0 plus some other hypotheses when the delay is a constant i e r x 1 this equation can support metastable rapidly oscillating solutions that are transients whose duration is of order exp c for some c 0 in this paper combining analytic and numerical techniques we investigate whether this metastable behavior persists when the delay r x depends non trivially on the state variable x more precisely we proceed in three steps first we explore theoretically and numerically the existence of branches of rapid periodic solutions for these equations then to predict conditions for metastability we introduce transition layer equations that depict the asymptotic shape of transient oscillations when tends to zero and finally these predictions are validated by numerical simulations numerical explorations are also performed for other scalings of with our conclusions are i when 0 0 there are no metastable transients ii when 0 0 and 0 is finite for monotone negative feedback the metastable oscillations exist irrespective of choices of f and r iii for monotone positive feedback they require that the feedback f and the delay r satisfy extra conditions such as f being an odd function and the delay r x an even function one novel result is that state dependent delays may lead to metastable dynamics in equations that cannot support such regimes when the delay is constant

boundary layer phenomena for differential delay equations with state dependent time lags i in this paper we begin a study of the differential delay equation mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0jc9yq jc9 vqaqpepm0xbba9pwe9q8fs0 yqaqpepae9pg0firpeps0he9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaaiabew7aljqadi hagaqbaiaacicacawg0bgaaiykaiabg2da9iabgkhitiaadihacagg oagaamidaiaacmcacqghrawkcawgmbgaaiikaiaadihacaggoagaam idaiabgkhitiaadkhacaggpagaaiykaiaacycacaqggagaaeiiaiaa dkhacqgh9aqpcawgybgaaiikaiaadihacaggoagaamidaiaacmcaca ggpaaaaa 5192 varepsilon x t x t f x t r r r x t we prove the existence of periodic solutions for 0 0 where 0 is an optimal positive number we investigate regularity and monotonicity properties of solutions x t which are defined for all t and of associated functions like t t r x t we begin the development of a poincare bendixson theory and phase plane analysis for such equations in a companion paper these results will be used to investigate the limiting profile and corresponding boundary layer phenomena for periodic solutions as approaches zero

smoothness of semiflows for parabolic partial differential equations with state dependent delay in this paper the smoothness properties of semiflows on c1 solution submanifold of a parabolic partial differential equations with state dependent delay are investigated the problem is formulated as an abstract ordinary retarded functional differential equation of the form du t dt au t f ut with a continuously differentiable map g from an open subset u of the space c1 h 0 l2 where a is the infinitesimal generator of a compact c0 semigroup the present study is continuation of a previous work 14 that highlights the classical solutions and c1 smoothness of solution manifold here we further prove the continuous differentiability of the semiflow we finally verify all hypotheses by a biological example which describes a stage structured diffusive model where the delay which is the time taken from birth to maturity is assumed as a function of a immature species population

applications of a zp index theory to periodic solutions for a class of functional differential equations abstract by means of a new geometrical index with z p group actions multiplicity results for a certain class of nonautonomous time periodic functional differential systems are obtained

differential equations with discrete state dependent delay uniqueness and well posedness in the space of continuous functions abstract partial differential equations with discrete concentrated state dependent delays in the space of continuous functions are investigated in general the corresponding initial value problem is not well posed so we find an additional assumption on the state dependent delay function to guarantee the well posedness for the constructed dynamical system we study the long time asymptotic behavior and prove the existence of a compact global attractor

a non local pde model for population dynamics with state selective delay local theory and global attractors we propose a non local pde model for the evolution of a single species population that involves delayed feedback where the delay such as the maturation time in the delayed birth rate is selective and the selection depends on the status of the system this delay selection in contrast with the usual state dependent delay widely used in ordinary delay differential equation ensures the lipschitz continuity of the nonlinear functional in the classical phase space we also develop the local theory and the existence and upper semi continuity of the global attractor with respect to parameters

the effect of state dependent delay and harvesting on a stage structured predator prey model we propose and analyze a stage structured predator prey model in which the time from birth to maturity is directly related to the number of individuals present individuals mature more quickly when there are fewer of them around the state dependent time delay is taken to be an increasing differentiable bounded function in this research we study the dynamics of our model analytically we present results on positivity and boundedness of all populations criteria for the existence of all equilibria and uniqueness of a positive equilibrium are given in order to observe the effect of state dependent maturation delay local stability analysis around all equilibria of the proposed model is discussed due to variation of maturation delay also global stability of trivial and the boundary equilibria is investigated using liapunov functional and lasalle invariant principle to investigate the effect of state dependent maturation delay and the harvesting effort of all species we carried out numerical simulations

a condition on delay for differential equations with discrete state dependent delay abstract parabolic differential equations with discrete state dependent delay are studied the approach based on an additional condition on the delay function introduced in a v rezounenko differential equations with discrete state dependent delay uniqueness and well posedness in the space of continuous functions nonlinear anal 70 11 2009 3978 3986 is developed we propose and study an analogue of the condition which is sufficient for the well posedness of the corresponding initial value problem on the whole space of continuous functions c the dynamical system is constructed in c and the existence of a compact global attractor is proved

boundedness global existence and continuous dependence for nonlinear dynamical systems describing physiologically structured populations abstract the paper is aimed as a contribution to the general theory of nonlinear infinite dimensional dynamical systems describing interacting physiologically structured populations we carry out continuation of local solutions to maximal solutions in a functional analytic setting for maximal solutions we establish global existence via exponential boundedness and by a contraction argument adapted to derive uniform existence time moreover within the setting of dual banach spaces we derive results on continuous dependence with respect to time and initial state to achieve generality the paper is organized top down in the way that we first treat abstract nonlinear dynamical systems under very few but rather strong hypotheses and thereafter work our way down towards verifiable assumptions in terms of more basic biological modelling ingredients that guarantee that the high level hypotheses hold

fluctuations in a sis epidemic model with variable size population in an epidemiological model time spent in one compartment is often modeled by a delay in the model in general the presence of delay in differential equations can change the stability of an equilibrium to instability and causes the appearance of oscillatory solutions in this paper we consider a sis epidemiological model with demographic effects birth mortality and mortality caused by infection the delay is the period of infection we define the concept of oscillation in the sense that solutions of the model studied fluctuate around a steady state our goal is to show that in this model there are oscillating solutions for certain parameters values we determine a large set of initial data for which solutions of this model are slowly oscillating

haar measure and the artin conductor let g be a connected reductive group defined over a local non archimedean field k the group g k is locally compact and unimodular in gr a haar measure g was defined on g k using the theory of bruhat and tits in this note we give another construction of the measure g using the artin conductor of the motive m of g over k the equivalence of the two constructions is deduced from a result of prasad p 1 the root datum and motive of g in this section k is an arbitrary field and g is a connected reductive group over k we let k be an algebraic closure of k ks the separable closure of k in k and gal ks k let t b g be a maximal torus contained in a borel subgroup defined over ks let g b t be the based root datum defined by this choice we recall cf sp pg3 12 that x t t b x t t b 1 1 with x t and x t the character and cocharacter groups of t respectively and and the simple roots and coroots determined by b respectively let w ng t t be the weyl group of the finite group w acts as automorphisms of x t and is generated by the reflections s x x x 1 2 for the galois group acts as automorphisms of ie as automorphisms of the group x t preserving the finite set as follows if then we can find g g ks such that int g t g t g t and int g b g b g b with g well defined up to left multiplication by t ks hence it induces a well defined automorphism x t x t date 14th january 1997 1 2 benedict h gross and wee teck gan preserving hence we get a group homomorphism aut via acts on aut by inner automorphisms similarly if f g g is any automorphism of g over ks it induces an automorphism f of which depends only on the image of f in the quotient group outks g of outer automorphisms the resulting map outks g aut is an isomorphism which respects the respective galois actions on the two groups cf sp pg10 the galois group also acts on w via the formula s s 1 3 and the semi direct product w acts on the rational vector space e x t q 1 4 let r sym e w which is a graded q module let r be the ideal of elements of positive degree in r and define v r r 2 d 1vd 1 5 this is a graded q module and chevalley proved that dim v dim e cf ch steinberg extended the proof to show that e and v are isomorphic modules cf st pg22 we sketch a proof of this result that does not involve the classification of irreducible root systems proposition 1 6 the q modules e and v are isomorphic proof by the criterion in se pg 104 it suffices to show that for all the fixed spaces e and v have the same dimension for any graded module a am we define the poincare series of by p a t tr am t then p a b p a p b steinberg showed that there is an isomorphism of graded modules s e s vd a here a is finite dimensional with basis bw w w and action given by bw b w the degree of bw is the length l w of w with respect to the generators s furnished by this isomorphism yields the following identity of poincare series det 1 t e d 1 det 1 t vd

slowly oscillating periodic solutions of autonomous state dependent delay equations more than a dozen years since the peak of activity on the question of the existence of periodic slowly oscillating solutions of autonomous delay differential equations following the early work of jones l wright 2 and grafton 3 the work of nussbaum 4 51 is to be specially noted for providing several new fixed point results and a global bifurcation theorem which are particularly useful for proving the existence of periodic solutions other important works include those of hadeler and tomiuk 6 kaplan and yorke 7 chow 8 chow and hale 9 alt lo 111 and walther 12 131 see hale s book 14 for an overview to the best of our knowledge only nussbaum 5 and alt 1 l considered the question of the existence of periodic solutions of autonomous state dependent delay differential equations nussbaum considered a special equation 0 3 below in 5 but did not prove a general result alt 1 l considered more complicated integral threshold type state dependent delays which have arisen in various models in epidemiology and structured population models alt obtained a theorem for a general class of state dependent equations on which we comment below the lack of results on periodic solutions for state dependent equations is probably explained by the fact that it is not clear what kinds of state dependent delays are interesting or natural and on a lack of compelling examples of such equations arising from plausible mathematical models however some recent papers of belair and mackey 15 and belair 16 contain some interesting state dependent delay equations arising in economics and population biology state dependent delays of threshold type arise naturally in structured population models see lo 111 the motivation for this paper stems from consideration of the innocent looking equation x t kx t r x t 0 1 where r x is bell shaped e g r x 01 emxz 1 01 0 i 01 5 1 if k 7r 2 and r o 1 then the zero solution of 0 1 is unstable a formal linearization would yield z t kz t 1 which is known to be unstable for

stability and hopf bifurcation analysis of an eco epidemic model with a stage structure abstract in this paper an eco epidemiological model with a stage structure is considered the asymptotical stability of the five equilibria the existence of stability switches about positive equilibrium is investigated it is found that hopf bifurcation occurs when the delay passes though a critical value some explicit formulae determining the stability and the direction of the hopf bifurcation periodic solutions bifurcating from hopf bifurcations are obtained by using the normal form theory and center manifold theory some numerical simulations for justifying the theoretical analysis are also provided finally biological explanations and main conclusions are given

periodic solutions for differential equations with state dependent delay and positive feedback existence of a nontrivial periodic orbit and a homoclinic orbit connecting 0 to the periodic orbit is shown for the differential equation x t x t f x t r r r x t with state dependent delay and positive feedback here 0 f and r are smooth real functions with r 0 1 and f 0

analysis of a model representing stage structured population growth with state dependent time delay a stage structured model of population growth is proposed where the time to ma turity is itself state dependent it is shown that under appropriate assumptions all solutions are positive and bounded criteria for the existence of positive equilibria and further conditions for the uniqueness of the equilibria are given the stability of the equilibria are also discussed in addition an attracting region is determined for solutions such that this region collapses to the unique positive equilibrium in the state independent case

on evolutionary volterra equations with state dependent delay abstract in this work we study topological properties of the solution set of abstract volterra equations with state dependent delay particularly we ensure that such a set is a nonempty compact and connected set as application we consider our abstract results in the framework of integro differential equations coming from viscoelasticity theory

epipelagic representations and invariant theory letg be a reductive p adic group we give a new construction of small depth epipelagic supercuspidal representations ofg k using stable orbits in geometric invariant theory git in contrast to previously known methods this construction works without any restrictions on the residue characteristic p we construct appropriate langlands parameters for these representations with some restrictions on p the git arises from graded moy prasad filtrations which we show are isomorphic to the graded lie algebras whose git was analyzed by vinberg and levy this leads to a classification of stable orbits in moy prasad filtrations as well as epipelagic supercuspidal representations in terms of z regular elliptic automorphisms of the absolute root system of g transferred to the l group of g generates the image of tame inertia under the corresponding wild langlands parameter for unramified groups and sufficiently large p we also classify the moy prasad filtrations which have semi stable orbits which solves the long outstanding problem of classifying non degenerate k types for g k

existence results for an impulsive abstract partial differential equation with state dependent delay in this paper we establish the existence of mild solutions for a class of impulsive abstract partial functional differential equation with state dependent delay

multiple positive periodic solutions for a class of nonlinear functional differential equations abstract by employing avery henderson fixed point theorem in cones we obtain some sufficient conditions which guarantee the existence of multiple positive periodic solutions for a class of differential equations with state dependent delays

boundary layer phenomena for differential delay equations with state dependent time lags iii in this paper we begin a study of the differential delay equation varepsilon x t x t f x t r r r x t

stage structured models of intra and inter specific competition within age classes abstract in some species larvae and adults experience competition in completely different ways simple stage structured models without larval competition usually yield a single delay equation for the adults using an age structured system incorporating competition among both larvae and adults we derive a system of distributed delay equations for the numbers of larvae and adults the system is neither cooperative nor reducible to a single equation for either variable positivity boundedness and uniform strong persistence are established linear stability analysis of equilibria is difficult due to the strong coupling but results are proved for small delays using monotone systems theory and exponential ordering for small delay we prove a theorem on generic convergence to equilibria which does not directly follow from standard theory but can be proved indirectly using comparison arguments finally we consider an extension to two strain competition and prove theorems on the linear stability of the boundary equilibria

stability and hopf bifurcation for a cell population model with state dependent delay we propose a mathematical model describing the dynamics of a hematopoietic stem cell population the method of characteristics reduces the age structured model to a system of differential equations with a state dependent delay a detailed stability analysis is performed a sufficient condition for the global asymptotic stability of the trivial steady state is obtained using a lyapunov razumikhin function a unique positive steady state is shown to appear through a transcritical bifurcation of the trivial steady state the analysis of the positive steady state behavior through the study of a first order exponential polynomial characteristic equation concludes the existence of a hopf bifurcation and gives criteria for stability switches a numerical analysis confirms the results and stresses the role of each parameter involved in the system on the stability of the positive steady state

stabilization in a state dependent model of turning processes we consider a two degree of freedom model for turning processes which involves a system of differential equations with state dependent delay depending on process parameters e g spindle speed depth of cut the cutting tool can exhibit unwanted vibrations resulting in a nonsmooth surface of the workpiece in this paper we propose a feedback law to stabilize the turning process for a large range of system parameters the feedback law introduces a generic nonhyperbolic stationary point into the model which generates the main technical challenge of this work we establish the stability equivalence between the differential equations with state dependent delay and a corresponding nonlinear system with the delay fixed at its stationary value then we show the stability of that nonlinear system with constant delay by computing its normal form finally we obtain conditions on system parameters which guarantee the stability of the state dependent delay model at the nonhyperbolic stationary point

classes of analytic functions subordinate to convex functions and extreme points in this paper we prove two sufficient conditions for an analytic function f to be an extreme point of the set of functions subordinate to a given convex mapping f when the image of the unit disk under f is a convex domain other than a half plane a strip or an infinite wedge

convergence of solutions for an equation with state dependent delay abstract a result of smith and thieme shows that if a semiflow is strongly order preserving then a typical orbit converges to the set of equilibria for the equation with state dependent delay t x t f x t r x t where 0 and f and r are smooth real functions with f 0 0 and f 0 we construct a semiflow which is monotone but not strongly order preserving we prove a convergence result under a monotonicity condition different from the strong order preserving property and apply it to the above equation to obtain generic convergence

permanence and stability of a predator prey system with stage structure for predator a stage structured predator prey system with delays for prey and predator respectively is proposed and analyzed mathematical analysis of the model equations with regard to boundedness of solutions permanence and stability are analyzed some sufficient conditions which guarantee the permanence of the system and the global asymptotic stability of the boundary and positive equilibrium respectively are obtained

the asymptotic behavior of the strong solutions for a non autonomous non local pde model with delay abstract in this paper we study the asymptotic behavior of the strong solutions of a non autonomous non local pde model with time delay we present the existence and structure of the uniform attractor by constructing the skew product flow of the family of processes generated by the strong solutions in order to obtain the existence of the uniform attractor we prove the family of processes satisfies uniform condition c by using some special technique of phase space decomposition additionally it is shown that all the bounded complete trajectories are globally asymptotic stable under some assumptions as the application of our result we obtain a globally asymptotic stable nontrivial strong periodic solution of a non local pde model

existence results for fractional neutral integro differential equations with state dependent delay in this paper we study the existence of mild solutions for a class of abstract fractional neutral integro differential equations with state dependent delay the results are obtained by using the leray schauder alternative fixed point theorem an example is provided to illustrate the main results

properties of the semigroup and the approximation of crandall liggett associated with a class of differential equations with delays depending implicitly on the state abstract in this paper we are concerned with the semigroup properties and the crandall liggett approximation associated with a class of differential equations with delays depending implicitly on the state the existence of a strongly continuous nonlinear semigroup associated with the solutions of the equations is proved various properties of this semigroup have been established such as the continuous dependence the compactness and the infinitesimal generators an approximation of crandall liggett type has also been established

a semigroup approach to nonlinear nonautonomous neutral functional differential equations abstract a class of nonlinear nonautonomous neutral functional differential equations is studied by associating with it an evolution equation in the space of initial data the space w1 1 existence and regularity results are proved

the dynamics of a mutual interference age structured predator prey model with time delay and impulsive perturbations on predators in this paper we introduce a mutual interference age structured predator prey natural enemy pest model with constant maturation time delay for the prey and then propose a pest management strategy by constant periodic releasing for the predator we show that there exists a global attractive pest eradication periodic solution when the periodic releasing amount m 1 and m 2 are lager than some critical value further to obtain a more effective pest control strategy we give the conditions involving the estimate of m 1 and m 2 in which the model is uniformly permanent and the pest population is under the economic threshold level we believe that the results will provide reliable tactic basis for the practical pest management