the dynamics of a fisher kpp nonlocal diffusion model with free boundaries abstract we introduce and study a class of free boundary models with nonlocal diffusion which are natural extensions of the free boundary models in 16 and elsewhere where local diffusion is used to describe the population dispersal with the free boundary representing the spreading front of the species we show that this nonlocal problem has a unique solution defined for all time and then examine its long time dynamical behavior when the growth function is of fisher kpp type we prove that a spreading vanishing dichotomy holds though for the spreading vanishing criteria significant differences arise from the well known local diffusion model in 16

the berry phase for spin in the majorana representation a continuous cyclic sequence of quantum states has an associated geometric or berry phase for spin j such a sequence is described by a cyclic change in the 2j 1 coefficients of the basis states the berry phase is analysed here for the general case that is the coefficients are allowed to vary in an arbitrary cyclic manner the result is expressed in geometric terms specifically in the democratic representation due to majorana this uniquely characterizes the spin state up to overall phase by the positions of 2j dots on the unit sphere of directions in real space if the positions are denoted by unit vectors where each traces out a parametrized loop on the sphere and the berry phase is given by an integral of combinations of these vectors

generalized transition waves and their properties in this paper we generalize the usual notions of waves fronts and propagation speeds in a very general setting these new notions which cover all usual situations involve uniform limits with respect to the geodesic distance to a family of hypersurfaces that are parametrized by time we prove the existence of new such waves for some time dependent reaction diffusion equations as well as general intrinsic properties some monotonicity properties and some uniqueness results for almost planar fronts the classification results which are obtained under some appropriate assumptions show the robustness of our general definitions 2012 wiley periodicals inc

phase space approach to quantum dynamics replaces the schrodinger equation for the time propagation of states of a quantized 2d spherical phase space by the dynamics of a system of n particles lying in phase space this is done through factorization formulae of an analytic function theory arising in a coherent state representation the particles being the zeroes of the quantum state for linear hamiltonians like a spin in a uniform magnetic field the motion of the particles is classical however nonlinear terms induce interactions between the particles their time propagation is studied and it is shown that contrary to integrable systems for chaotic maps they tend to fill as their classical counterpart the whole phase space in a uniform way

chaotic analytic zero points exact statistics for those of a random spin state a natural statistical ensemble of 2j points on the unit sphere can be associated via the majorana representation with a random quantum state of spin j and an exact expression is obtained here for the general k point correlation function in this ensemble the pair correlation in the large j limit takes the relatively simple form where and is the angular separation of the pair of points on the sphere it appears from the numerical work of others that in this limit these statistics are typical of the zero points of analytic functions associated with chaotic quantum dynamical systems

positive solutions of three point boundary value problems for second order impulsive differential equations with advanced arguments in this paper we discuss three point boundary value problems for second order impulsive differential equations with advanced arguments we establish sufficient conditions under which such problems have positive solutions to obtain the results we use the fixed point index

logistic type attraction repulsion chemotaxis systems with a free boundary or unbounded boundary ii spreading vanishing dichotomy in a domain with a free boundary abstract the current series of research papers is to investigate the asymptotic dynamics in logistic type chemotaxis models in one space dimension with a free boundary or unbounded boundary such a model with a free boundary describes the spreading of a new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front in this first of the series we investigated the dynamical behaviors of logistic type chemotaxis models on the half line r which are formally corresponding limit systems of the free boundary problems in the second of the series we establish the spreading vanishing dichotomy in chemoattraction repulsion systems with a free boundary as well as with double free boundaries

transformation techniques and fixed point theories to establish the positive solutions of second order impulsive differential equations abstract using a new method for dealing with the impulse term of second impulsive differential equations we are concerned with determining values of for which there exist positive solutions of the following boundary value problems in the form of x t f t x t 0 t j t t k x t k x t k c k x t k k 1 2 n a x 0 b x 0 a x 1 b x 1 0 1 h s x t d t where 0 is a positive parameter c k is a real sequence with c k 1 k 1 2 n the dependence of positive solution x t on the parameter is also studied i e lim x or lim x 0 the proof of our main result is based upon transformation techniques and fixed point theories this is probably the first time the existence of positive solutions of second order impulsive differential equations has been studied by applying this technique

spreading vanishing dichotomy in a diffusive logistic model with a free boundary ii abstract we study the diffusive logistic equation with a free boundary in higher space dimensions and heterogeneous environment such a model may be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front for simplicity we assume that the environment and the solution are radially symmetric in the special case of one space dimension and homogeneous environment this free boundary problem was investigated in du and lin 2010 10 we prove that the spreading vanishing dichotomy established in du and lin 2010 10 still holds in the more general and ecologically realistic setting considered here moreover when spreading occurs we obtain best possible upper and lower bounds for the spreading speed of the expanding front when the environment is asymptotically homogeneous at infinity these two bounds coincide our results indicate that the asymptotic spreading speed determined by this model does not depend on the spatial dimension

existence of positive periodic solutions for an impulsive differential equation abstract in this paper we consider the impulsive differential equation x t a t x t p t f t x t 0 t x t 1 t x t n t a e t 0 t t k x t k x t k b k x t k k 1 2 by using a fixed point theorem some criteria are established for the existence of a positive periodic solution of the above equation our work extends and improves some known results

transition semi wave solutions of reaction diffusion equations with free boundaries abstract in this paper we define the transition semi wave solution cf definition 1 1 of the following reaction diffusion equation with free boundaries 1 u t u x x f t x u t r x h t u t h t 0 t r h t u x t h t t r in the homogeneous case i e f t x u f u under the hypothesis f u c 1 0 1 f 0 f 1 0 f 1 0 f u 0 for u 1 we prove that the semi wave connecting 1 and 0 of 1 is unique provided it exists furthermore we prove that any bounded transition semi wave connecting 1 and 0 is exactly the semi wave in the cases where f is kpp fisher type and almost periodic in time space i e f t x u u c t u resp u a x u with c t resp a x being almost periodic applying totally different method we also prove any bounded transition semi wave connecting the unique almost periodic positive solution of u t u c t u resp u x x u a x u 0 and 0 is exactly the unique almost periodic semi wave of 1 finally we provide an example of the heterogeneous equation to show the existence of the transition semi wave without any global mean speeds

a diffusive logistic model with a free boundary in time periodic environment abstract we study the diffusive logistic equation with a free boundary in time periodic environment such a model may be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front for time independent environment in the cases of one space dimension and higher space dimensions with radial symmetry this free boundary problem has been studied in du and lin 2010 12 du and guo 2011 9 in both cases a spreading vanishing dichotomy was established and when spreading occurs the asymptotic spreading speed was determined in this paper we show that the spreading vanishing dichotomy is retained in time periodic environment and we also determine the spreading speed the former is achieved by further developing the earlier techniques and the latter is proved by introducing new ideas and methods

existence and global attractivity of positive periodic solution for an impulsive lasota wazewska model abstract sufficient conditions are obtained for the existence and global attractivity of periodic positive solution of an impulsive lasota wazewska model for the survival of red blood cells

positive periodic solutions of functional differential equations abstract we consider the existence multiplicity and nonexistence of positive periodic solutions for the periodic equation x t a t g x x t b t f x t t where a b c r 0 are periodic 0 a t dt 0 0 b t dt 0 f g c 0 0 and f u 0 for u 0 g x is bounded t is a continuous periodic function define f 0 lim u 0 f u u f lim u f u u i0 number of zeros in the set f 0 f and i number of infinities in the set f 0 f we show that the equation has i0 or i positive periodic solution s for sufficiently large or small 0 respectively

asymptotic spreading in heterogeneous diffusive excitable media abstract we establish propagation and spreading properties for nonnegative solutions of nonhomogeneous reaction diffusion equations of the type t u a t x u q t x u f t x u with compactly supported initial conditions at t 0 here a q f have a general dependence in t r and x r n we establish properties of families of propagation sets which are defined as families of subsets s t t 0 of r n such that lim inf t inf x s t u t x 0 the aim is to characterize such families as sharply as possible in particular we give some conditions under which 1 a given path t t 0 where t r n forms a family of propagation sets or 2 one can find such a family with s t x r n x r t and lim t r t this second property is called here complete spreading furthermore in the case q 0 and inf t x r r n f u t x 0 0 as well as under some more general assumptions we show that there is a positive spreading speed that is r t can be chosen so that lim inf t r t t 0 in the general case we also show the existence of an explicit upper bound c 0 such that lim sup t r t t c on the other hand we provide explicit examples of reaction diffusion equations such that for an arbitrary e 0 any family of propagation sets s t t 0 has to satisfy s t x r n x e t for large t in connection with spreading properties we derive some new uniqueness results for the entire solutions of this type of equations lastly in the case of space time periodic media we develop a new approach to characterize the largest propagation sets in terms of eigenvalues associated with the linearized equation in the neighborhood of zero

positive solutions to second order four point boundary value problems for impulsive differential equations abstract in this paper by using the leggett williams theorem we present sufficient conditions which guarantee the existence of three positive solutions to second order four point boundary value problems for impulsive differential equations

positive solutions of a nonlinear impulsive equation with periodic boundary conditions abstract in this paper we investigate nonlinear second order differential equations subject to linear impulse conditions and periodic boundary conditions sign properties of an associated green s function are exploited to get existence results for positive solutions of the nonlinear boundary value problem with impulse upper and lower bounds for positive solutions are also given the results obtained yield periodic positive solutions of the corresponding periodic impulsive nonlinear differential equation on the whole real axis

existence and global attractivity of positive periodic solutions of functional differential equations with impulses abstract sufficient and realistic conditions are obtained for the existence and global attractivity of positive periodic solutions of the delay differential system with impulses d y t d t y t f t y t 1 t y t n t a e t 0 t t k y t k y t k b k y t k k 1 2 the method involves the application of the gaines and mawhin s coincidence degree theory the constructing suitable lyapunov functionals and estimations of uniform upper bounds on solutions when these results are applied to some special delay population models some new results are obtained and some known results are generalized in particular our results indicate that under the appropriate linear periodic impulsive perturbations the above impulsive delay differential equation preserves the original periodicity and global attractivity of the nonimpulsive delay differential equation d y t d t y t f t y t 1 t y t n t

the periodic solutions for general periodic impulsive population systems of functional differential equations and its applications in this paper the general periodic impulsive population systems of functional differential equations are investigated by using the method of poincare map and horn s fixed point theorem we prove that the ultimate boundedness of all solutions implies the existence of periodic solutions as applications of this result the existence of positive periodic solutions for the general periodic impulsive kolmogorov type population dynamical systems are discussed we further prove that as long as the system is permanent there must exist at least one positive periodic solution in addition the permanence and existence of positive periodic solutions are discussed for the periodic impulsive single species logistic models and the periodic impulsive n species lotka volterra competitive models with delays

positive periodic solutions of first order functional differential equations with parameter we prove the existence and multiplicity of positive t periodic solution s for t periodic equation x t h t x lb t f x t t t by krasnoselskii fixed point theorem where f x may be singular at x 0 our results improve some recent results in previous literature

spreading speeds and traveling waves for abstract monostable evolution systems abstract this paper is devoted to the development of the theory of spreading speeds and traveling waves for abstract monostable evolution systems with spatial structure under appropriate assumptions we show that the spreading speeds coincide with the minimal wave speeds for monotone traveling waves in the positive and negative directions then we use this theory to study the spatial dynamics of a parabolic equation in a periodic cylinder with the dirichlet boundary condition a reaction diffusion model with a quiescent stage a porous medium equation in a tube and a lattice system in a periodic habitat

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

exponential separation and principal lyapunov exponent spectrum for random nonautonomous parabolic equations abstract the purpose of the paper is to extend the principal eigenvalue and principal eigenfunction theory for time independent and periodic parabolic equations to random and general nonautonomous ones in the random case a notion of principal lyapunov exponent serving as an analog of principal eigenvalue is introduced it is shown that the principal lyapunov exponent is deterministic and of simple multiplicity it is also shown that there is a one dimensional invariant random subbundle corresponding to the solutions that are globally defined and of the same sign which serves as an analog of principal eigenfunction in addition monotonicity of the principal lyapunov exponent with respect to the zero order terms both in the equation and in the boundary condition is proved when the second and first order terms are deterministic it is proved that the principal lyapunov exponent is greater than or equal to the principal eigenvalue of the associated time averaged equation in the general nonautonomous case the concepts of principal spectrum which serves as an analog of principal eigenvalue and principal lyapunov exponents are introduced as is known the principal spectrum is a compact interval it is proved in the paper that the principal spectrum contains all the principal lyapunov exponents when the second and first order terms are time independent a lower estimate of the infimum of the principal spectrum is given in terms of an associated time averaged equation

existence of positive periodic solutions of impulsive functional differential equations with two parameters in this paper we employ a well known fixed point index theorem to study the existence and non existence of positive periodic solutions for the periodic impulsive functional differential equations with two parameters several existence and non existence results are established

existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects abstract this paper deals with a new existence theory for single and multiple positive periodic solutions to a kind of nonautonomous functional differential equations with impulse actions at fixed moments by employing a fixed point theorem in cones easily verifiable sufficient criteria are established the paper extends some previous results and reports some new results about impulsive functional differential equations

diffusive kpp equations with free boundaries in time almost periodic environments ii spreading speeds and semi wave solutions abstract in this series of papers we investigate the spreading and vanishing dynamics of time almost periodic diffusive kpp equations with free boundaries such equations are used to characterize the spreading of a new species in time almost periodic environments with free boundaries representing the spreading fronts in the first part of the series we showed that a spreading vanishing dichotomy occurs for such free boundary problems see 16 in this second part of the series we investigate the spreading speeds of such free boundary problems in the case that the spreading occurs we first prove the existence of a unique time almost periodic semi wave solution associated to such a free boundary problem using the semi wave solution we then prove that the free boundary problem has a unique spreading speed

continuum and lattice models analysis of the aggregation diffusion cell movement abstract the process by which one may take lattice models of a biophysical process and construct continuous models based on it is of mathematical interest as well as being of practical use such a process makes it possible to investigate how processes at the single cell level affect collective dynamics in this paper we first study the singular limit of a class of reinforced random walks on a lattice which simulates cell migration on discrete points and a complete analysis of the existence and stability of solutions is possible combined with the diffusion approximation assumption a continuous aggregation diffusion or backward forward parabolic equation is derived then the regularity estimate the nonexistence of the solution with small initial data and asymptotic behavior of the solution with large initial data are investigated in contrast to the continuous model the boundedness asymptotic behaviors of the lattice model solution in the diffusion region with monotone initial date and the interface behaviors of the aggregation diffusion regions are explored the numerical simulations agree well with theoretical results and biological observations