on a nonautonomous competitive system subject to stochastic and impulsive perturbations a nonautonomous impulsive stochastic competitive system is considered sufficient conditions for stochastic permanence and extinction are established the influences of stochastic and impulsive perturbations are shown this paper presents a nonautonomous impulsive stochastic differential model of competition between species which contains the classic two species competitive lotka volterra model as a special case with the theory of impulsive stochastic differential equations a good understanding of stochastic permanence and extinction of system is obtained two specific numerical examples are presented to support the analytical results conclusions are also given

a reader s guide to gacs s positive rates paper peter gacs s monograph which follows this article provides a counterexample to the important positive rates conjecture this conjecture which arose in the late 1960 s was based on very plausible arguments some of which come from statistical mechanics during the long gestation period of the gacs example there has been a great deal of skepticism about the validity of his work the construction and verification of gacs s counterexample are unavoidably complex and as a consequence his paper is quite lengthy but because of the novelty of the techniques and the significance of the result his work deserves to become widely known this reader s guide is intended both as a cheap substitute for reading the whole thing as well as a warm up for those who want to plumb its depths

on critical phenomena in interacting growth systems part ii bounded growth this paper completes the classification of some infinite and finite growth systems which was started in part i components whose states are integer numbers interact in a local deterministic way in addition to which every component s state grows by a positive integerk with a probability ek 1 e at every moment of the discrete time proposition 1 says that in the infinite system which starts from the state all zeros percentages of elements whose states exceed a given valuek 0 never exceed ce k wherec const proposition 2 refers to finite systems it states that the same inequalities hold during a time which depends exponentially on the system size

stationary distributions for systems with competing creation annihilation dynamics we have developed a simple systematic method to investigate the existence of stationary probability distributions spds for interacting particle or spin lattice systems exhibiting steady non equilibrium states the latter originate in a competition between several creation annihilation spin flip kinetic mechanisms say each acting presuming a different bath temperature particle spin interactions strength or chemical potential magnetic field it follows the existence of spds for a large class of these systems which may thus be studied by simply applying the techniques of equilibrium theory the method is illustrated with several examples bearing a practical interest

a reaction diffusion equations modelling the effect of fluctuating water levels on prey predator interactions prey predator interactions are influenced by many ecological factors in this paper a prey predator model with spatial diffusion under homogeneous neumann boundary conditions is investigated to study the impact of water level on persistence of two fish populations living in an artificial lake we obtain some important qualitative properties including the existence of the global positive solution the dissipation and permanence of this system the existence of a unique globally stable periodic solution is also presented some numerical examples are presented to verify our results

global stability in a periodic delayed predator prey system abstract in this paper we establish a periodic delayed predator prey system according to two types of single species growth population models with delay by using some analysis methods easily verifiable sufficient conditions for the existence and global asymptotic stability of positive periodic solution of the above system are derived some previous results are generalized and improved biological interpretations on the main results are also given

an almost periodic competitive system subject to impulsive perturbations in this paper an almost periodic system of impulsive differential equations used in modeling two species in competition is considered by the comparison between the solutions of impulsive system and the corresponding non impulsive system and constructing lyapunov function sufficient conditions ensuring the uniformly asymptotic stability of a unique positive almost periodic solution of the system are derived finally an example and its corresponding numerical simulations are presented to explain our theoretical results

periodic duffing delay equations with state dependent impulses conditions are obtained for the duffing equation with delay and state dependent impulses to admit a periodic solution

global stability of a delay differential equations model of plankton allelopathy abstract in this paper we consider a delay differential equations model of plankton allelopathy using an iterative method the global stability of the interior equilibrium point of the system is investigated which shows that toxic substances are harmless for the stability of the interior equilibrium point the results we obtained improve and complement those of chattopadhyay j chattopadhyay effect of toxic substances on a two species competitive system ecol model 84 1996 287 289 samanta g p samanta a two species competitive system under the influence of toxic substances appl math comput 216 2010 291 299 and mukhopadhyay et al a mukhopadhyay j chattopadhyay p k tapaswi a delay differential equations model of plankton allelopathy math biosci 149 1998 167 189 we also present some examples to verify our main result

critical attractive spin systems we study a class of attractive spin systems we prove that for these processes the system dies out when the parameters are on the critical surface we also prove that the supercritical process survives with positive probability in a sufficiently thick space time slab

percolation games probabilistic cellular automata and the hard core model let each site of the square lattice mathbb z 2 be independently assigned one of three states a trap with probability p a target with probability q and open with probability 1 p q where 0 p q 1 consider the following game a token starts at the origin and two players take turns to move where a move consists of moving the token from its current site x to either x 0 1 or x 1 0 a player who moves the token to a trap loses the game immediately while a player who moves the token to a target wins the game immediately is there positive probability that the game is drawn with best play i e that neither player can force a win this is equivalent to the question of ergodicity of a certain family of elementary one dimensional probabilistic cellular automata pca these automata have been studied in the contexts of enumeration of directed lattice animals the golden mean subshift and the hard core model and their ergodicity has been noted as an open problem by several authors we prove that these pca are ergodic and correspondingly that the game on mathbb z 2 has no draws on the other hand we prove that certain analogous games do exhibit draws for suitable parameter values on various directed graphs in higher dimensions including an oriented version of the even sublattice of mathbb z d in all d ge 3 this is proved via a dimension reduction to a hard core lattice gas in dimension d 1 we show that draws occur whenever the corresponding hard core model has multiple gibbs distributions we conjecture that draws occur also on the standard oriented lattice mathbb z d for d ge 3 but here our method encounters a fundamental obstacle

the positive rates problem for attractive nearest neighbor spin systems on ℤ it is shown that a spin system on has only one invariant probability measure if it has attractive or repulsive nearest neighbor flip rates which are strictly positive and periodic under translation along

dynamic behaviors of the impulsive periodic multi species predator prey system the dynamic behaviors of an impulsive periodic predator prey model with n preys and m predators are studied in this paper by constructing a suitable lyapunov function and using the comparison theorem of impulsive differential equation sufficient conditions which ensure the permanence and global attractivity of the system are obtained at the same time a set of criteria which guarantee that some species in the system are permanent and globally attractive while the remaining species are driven to extinction is obtained our results show that for the multi species predator prey community impulsivity is one of the important reasons that can change the long time behaviors of species

a stability criterion for attractive nearest neighbor spin systems on mathbb z we study attractive nearest neighbor spin systems on the integers having all 0 s and all 1 s as traps a necessary and sufficient condition for stability under small one sided perturbations of the flip rates is established

optimal impulsive harvesting on non autonomous beverton holt difference equations many recent advances in the theory of the optimal economic exploitation of renewable fish resources have been gained by applying optimal control theory however despite these successes much less is known about how seasonal environments affect the maximum sustainable yield msy or population persistence and any effects of relations between intensity and frequency of harvesting assuming that fish populations follow beverton holt equations we investigated impulsive harvesting in seasonal environments focusing on both economic aspects and resource sustainability we first investigated the existence and stability of a periodic solution and its analytic formula and then showed that the population persistence depends on the intensity and frequency of harvesting with the msy as a management objective we investigated optimal impulsive harvesting policies the optimal harvesting effort that maximizes the sustainable yield the corresponding optimal population level and the msy are obtained by using discrete euler lagrange equations and product formulae and their explicit expressions were obtained in terms of the intrinsic growth rate the carrying capacity and the impulsive moments these results imply that harvest timing is of crucial importance to the msy since impulsive differential equations incorporate elements of continuous and discrete systems we can apply all results obtained for beverton holt equations with impulsive effects to periodic logistic equations with impulsive harvesting

capacity scaling in mimo wireless systems under correlated fading previous studies have shown that single user systems employing n element antenna arrays at both the transmitter and the receiver can achieve a capacity proportional to n assuming independent rayleigh fading between antenna pairs we explore the capacity of dual antenna array systems under correlated fading via theoretical analysis and ray tracing simulations we derive and compare expressions for the asymptotic growth rate of capacity with n antennas for both independent and correlated fading cases the latter is derived under some assumptions about the scaling of the fading correlation structure in both cases the theoretic capacity growth is linear in n but the growth rate is 10 20 smaller in the presence of correlated fading we analyze our assumption of separable transmit receive correlations via simulations based on a ray tracing propagation model results show that empirical capacities converge to the limit capacity predicted from our asymptotic theory even at moderate n 16 we present results for both the cases when the transmitter does and does not know the channel realization

exact solution of a directed site animals enumeration problem in three dimensions an exact equivalence is established between the directed site animals enumeration dsae problem in d dimensions and a crystal growth model defined on the same lattice in special cases the latter reduces to the calculation of the free energy of a d ensuremath 1 dimensional lattice gas with extended hard cores the author solves a dsae problem with d 3 exactly by using its equivalence to the hard hexagon problem and shows that the exponent ensuremath theta frac 5 6 a new and simpler solution to the dsae problem on the square and triangular lattices is given

permanence and global attractivity of an impulsive ratio dependent predator prey system in a patchy environment this paper studies an impulsive nonautonomous two species predator prey lotka volterra type dispersal system in a patchy environment in which the prey species can disperse among n different patches but the predator species is confined to one patch and cannot disperse on the basis of inequality estimation techniques and the comparison theory of impulsive differential equations sufficient conditions of integrable form for the permanence and extinction are obtained by constructing appropriate lyapunov function conditions for the global attractivity of the system are established

statistical mechanics of probabilistic cellular automata we investigate the behavior of discrete time probabilistic cellular automata pca which are markov processes on spin configurations on ad dimensional lattice from a rigorous statistical mechanics point of view in particular we exploit whenever possible the correspondence between stationary measures on the space time histories of pcas on d and translation invariant gibbs states for a related hamiltonian on d 1 this leads to a simple large deviation formula for the space time histories of the pca and a proof that in a high temperature regime the stationary states of the pca are gibbsian we also obtain results about entropy fluctuations and correlation inequalities and demonstrate uniqueness of the invariant state and exponential decay of correlations in a high noise regime we discuss phase transitions in the low noise or low temperature regime and review toom s proof of nonergodicity of a certain class of pcas

percolation and disorder resistance in cellular automata we rigorously prove a form of disorder resistance for a class of one dimensional cellular automaton rules including some that arise as boundary dynamics of two dimensional solidification rules specifically when started from a random initial seed on an interval of length l with probability tending to one as l the evolution is a replicator that is a region of space time of density one is filled with a spatially and temporally periodic pattern punctuated by a finite set of other finite patterns repeated at a fractal set of locations on the other hand the same rules exhibit provably more complex evolution from some seeds while from other seeds their behavior is apparently chaotic a principal tool is a new variant of percolation theory in the context of additive cellular automata from random initial states

on a stochastic logistic equation with impulsive perturbations a stochastic logistic model with impulsive perturbations is proposed and investigated first we give a new definition of a solution of an impulsive stochastic differential equation isde which is more convenient for use than the existing one using this definition we show that our model has a global and positive solution and obtain its explicit expression then we establish the sufficient conditions for extinction non persistence in the mean weak persistence persistence in the mean and stochastic permanence of the solution the critical value between weak persistence and extinction is obtained in addition the limit of the average in time of the sample path of the solution is estimated by two constants afterwards the lower growth rate and the upper growth rate of the solution are estimated finally sufficient conditions for global attractivity are established

around probabilistic cellular automata we survey probabilistic cellular automata with approaches coming from combinatorics statistical physics and theoretical computer science each bringing a different viewpoint some of the questions studied are specific to a domain and some others are shared most notably the ergodicity problem

a two species competitive system under the influence of toxic substances the effect of toxicants on ecological systems is an important issue from mathematical and experimental points of view in this paper we have studied the dynamical behaviours of a two species competitive system affected by toxic substances boundedness local and global stabilities are addressed we have also studied the stability of this system with diffusion around the non trivial stable in the deterministic case equilibrium point in a random environment extensive numerical simulations are performed using analytical results for a hypothetical set of parameter values

global dynamics of the periodic logistic system with periodic impulsive perturbations this paper studies the global behaviors of the periodic logistic system with periodic impulsive perturbations the results of d d bainov and p s simeonov 1993 are extended and dynamics different from the corresponding continuous system are found it is shown that the system may have a unique positive periodic solution which is globally asymptotically stable or go extinct when the two periods are rational dependent when they are rational independent the system has no periodic solutions however still has a global attractor or go extinct under some conditions

existence of periodic solution of order one of planar impulsive autonomous system based on the situation that all the recent research about state dependent impulsive differential equations is focused on systems which have explicit solutions this paper try to consider those systems with state dependent impulsion which have not explicit solutions we get an existence theorem of periodic solution of order one for a general planar autonomous impulsive system and by applying it to a special state dependent impulsive differential equations we get the concrete conditions of existence of one order periodic solution of that special system

perfect simulation of infinite range gibbs measures and coupling with their finite range approximations in this paper we address the questions of perfectly sampling a gibbs measure with infinite range interactions and of perfectly sampling the measure together with its finite range approximations we solve these questions by introducing a perfect simulation algorithm for the measure and for the coupled measures the algorithm works for general gibbsian interaction under requirements on the tails of the interaction as a consequence we obtain an upper bound for the error we make when sampling from a finite range approximation instead of the true infinite range measure

attractor properties for irreversible and reversible interacting particle systems we consider translation invariant interacting particle systems on the lattice with finite local state space admitting at least one gibbs measure as a time stationary measure the dynamics can be irreversible but should satisfy some mild non degeneracy conditions we prove that weak limit points of any trajectory of translation invariant measures satisfying a non nullness condition are gibbs states for the same specification as the time stationary measure this is done under the additional assumption that zero entropy loss of the limiting measure w r t the time stationary measure implies that they are gibbs measures for the same specification we show how to prove the non nullness for a large number of cases and also give an alternate version of the last condition such that the non nullness requirement can be dropped as an application we obtain the attractor property if there is a reversible gibbs measure our method generalizes convergence results using relative entropy techniques to a large class of dynamics including irreversible and non ergodic ones

generic approach to the performance analysis of correlated transmit receive diversity mimo systems with without co channel interference optimal transmit receive diversity trd is one of the most important configurations for wireless multiple input multiple output mimo systems due to its good performance and ease of implementation though investigated intensively the performance of optimal trd in general correlated fading with cochannel interference is still not well understood since the optimal trd s output instantaneous signal to interference plus noise ratio sinr is equal to the largest sample eigenvalue of a quadratic form involving signal and interference channel matrices directly determining the probability density function pdf of this eigenvalue has been a prevailing approach in the literature given the nonlinearity involved in the quadratic form however finding such a pdf is not simple except for some special channel conditions in this paper we formulate the problem in a totally different framework as testing the positive definiteness of a random matrix whereby the theory of matrix variate distributions can be invoked to obtain exact solutions in terms of special functions the solutions are very general including most of existing results as a special case and allowing for the correlation structures of both signal and interferers to be arbitrary at both transmitter and receiver ends numerical results are presented to validate the theoretical analysis

permanence and global stability for nonautonomous n species lotka volterra competitive system with impulses and infinite delays abstract in this paper we consider a class of nonautonomous n species lotka volterra competitive systems with impulses and infinite delays by developing the methods given in teng 2002 25 we give sufficient conditions for permanence and global attractivity of the system

positivity of entropy production we discuss the positivity of the mean entropy production for stochastic systems driven from equilibrium as it was defined in refs 7 and 8 non zero entropy production is closely linked with violation of the detailed balance condition this connection is rigorously obtained for spinflip dynamics we remark that the positivity of entropy production depends on the choice of time reversal transformation hence on the choice of the dynamical variables in the system of interest

periodicity in a generalized semi ratio dependent predator prey system with time delays and impulses abstract with the help of a continuation theorem based on coincidence degree theory we establish necessary and sufficient conditions for the existence of positive periodic solutions in a generalized semi ratio dependent predator prey system with time delays and impulses which covers many models appeared in the literature when the results reduce to the semi ratio dependent predator prey system without impulses they generalize and improve some known ones

projections of gibbs measures may be non gibbsian consider the phase of the two dimensional nearest neighbor ferromagnetic ising model at a temperature belowt c let be the restriction of this measure to a coordinate axis we prove that there is no one dimensional translation invariant summable interaction for which is a gibbs measure this is proven by showing that if such an interaction existed would have large deviation properties different from those it actually has percolation methods are used in the proof

the restriction of the ising model to a layer we discuss the status of recent gibbsian descriptions of the restriction projection of the ising phases to a layer we concentrate on the projection of the two dimensional low temperature ising phases for which we prove a variational principle

ergodicity for spin systems with stirrings or 1 each site after an exponentially distributed random time is updated by a rule that depends on the configuration in the site s neighborhood such processes have been used to study the gibbs states associated with some potential reversible spin flip systems having as invariant measure a gibbs state related to some potential are called stochastic ising models a particle system is ergodic if a there exists a unique invariant measure and b starting from any initial measure the process converges to that invariant measure ergodicity for an attractive ising model is equivalent to absence of phase transition for the related gibbs state indeed the study of gibbs states allowed the proof of ergodicity of attractive short range onedimensional ising models and the absence of ergodicity for nearest neighbor two dimensional ising models two of the most important results on the stochastic ising model a criterion for the ergodicity of spin systems has been given by dobrushin 4 dobrushin established that m e is a sufficient condition for the ergodicity of a spin flip system where m represents the maximal influence of other sites on the spin flip rate at any site and e is in some sense the minimum spin flip rate at any site this approach was also studied by gray and griffeath 11 sullivan 19 and holley and stroock 14 15 a review can be found in liggett 17 recent related results are obtained by aizenman and holley 1 in this paper we exploit a generalized graphical representation of the systems to obtain some results for spin systems and combinations of spin systems with stirring processes in theorem 2 5 we show that m 28 is a sufficient condition for the ergodicity of the one dimensional nearest neighbor ising model the result is interesting because no attractiveness conditions are imposed in the attractive case gray 7 proved that any one dimensional

periodic lienard type delay equations with state dependent impulses abstract conditions are obtained for lienard type equations with delay and state dependent impulses to admit an absolutely continuous periodic solution with first derivative of bounded variation and consequently with lebesgue integrable second derivative the results are applied to josephson s equation and the nonconservative forced pendulum equation

globally asymptotic stability in two periodic delayed competitive systems based on two types of well known periodic single species population growth models with time delay we propose two corresponding periodic competitive systems with multiple delays by using some new analysis techniques we derive the same criteria for the existence and globally asymptotic stability of positive periodic solutions of the above two competitive systems the easily verifiable criteria show that the existence and globally asymptotic stability of positive periodic solutions of two delayed nonautonomous competitive systems correspond to the existence and global stability of positive equilibrium of corresponding undelayed autonomous system as an application some special cases of the above systems are examined and some earlier results are extended and improved biological interpretations on the main results are also given

reformulation of gray s duality for attractive spin systems and its applications duality relations which associate two markov processes in different state spaces have been useful tools in the study of the long term behaviour of the stochastic processes of spin systems in 1986 gray introduced a new duality theory which is applicable to general spin systems with attractive transition rates the theory was developed by making full use of graphical representations in the present paper gray s duality is reformulated by studying the action of the formal generators on a newly chosen duality function and the dual processes defined in the state space upsilon a a is a finite subset of y are discussed where y is a collection of finite subsets of zd as applications of the submodularity of the survival probability sigma a of the dual processes rigorous lower bounds of the critical values are derived for the theta contact process the multi particle creation model and the sexual reproduction process

simple one dimensional interaction systems with superexponential relaxation times finite one dimensional random processes with local interaction are presented which keep some information of a topological nature about their initial conditions during time the logarithm of whose expectation grows asymptotically at least asm3 wherem is the size of the setr m of states of one component actuallyr m is a circle of lengthm at every moment of the discrete time every component turns into some kind of average of its neighbors after which it makes a random step along this circle all these steps are mutually independent and identically distributed in the present version the absolute values of the steps never exceed a constant the processes are uniform in space time and the set of states this estimation contributes to our awareness of what kind of stable behavior one can expect from one dimensional random processes with local interaction

tails in harnesses we consider space and time uniformd dimensional random processes with linear local interaction which we call harnesses and which may be used as discrete mathematical models of random interfaces their components are rea random variablesast wheres zd andt 0 1 2 at every time step two events occur first every component turns into a linear combination of itsn neighbors and second a symmetric random i i d noise v is added to every component for any zd define a s as follows if 0 0 0 0 ast ast then by induction delta sigma e 1 a s t delta sigma a s e 1 t delta sigma a s t whereei is thed dimensional vector whoseith component is one and other components are zeros denote the sum of components of call a real random variable symmetric if it is distributed as for any symmetric random variable power decay or p decay is defined as the supremum of thoser for which therth absolute moment of is finite convergence a s in probability and in law whent is examined in terms of p decay v ifd 1 0 ord 2 0 0 ast diverges in all the other cases if p decay v d 2 d ast converges and p decay for any symmetric random variable exponential decay or e decay is defined as the supremum of thoser for which the expectation of exp x r is finite let e decay v 0 whenever ast converges that is ifd 2 or 0 ifd 2 e decay limast min e decay v d 2 2 if 1 e decay lim ast min e decay d 2 if e decay lim ast e decay

selective harvesting of two competing fish species in the presence of toxicity with time delay abstract in this paper we have considered a two competing fish species with the delay due to gestation each of which obeys the law of logistic growth and releases a substance toxic to the other we also consider the selective harvesting of fishes above a certain age or size by incorporating a time delay in the harvesting term the dynamical behavior of the exploited system is examined in the presence and absence of delay boundedness local and global stabilities are addressed the possibility of the existence of a dynamic equilibrium and the optimal harvesting policy is studied in the absence of delay it is shown that the time delay can cause a stable equilibrium to become unstable and even a switching of stabilities extensive numerical simulations are performed using analytical results for a hypothetical set of parameter values

pseudo free energies and large deviations for non gibbsian fkg measures a large deviation theorem for the invariant measures of translation invariant attractive interacting particle systems on 0 1 zd is proven in this way a pseudo free energy and pressure is defined for ergodic systems the large deviations property holds with the usual scaling the case of non ergodic systems is also discussed a similar result holds for occupation times the perturbation by an external field is treated

global existence results for impulsive functional differential equations abstract in this paper we investigate the existence of solutions of impulsive delay differential equation x t f t x t t 0 t k k n x t i k x t t k k n new global existence theorems are established without assuming the global existence of solutions of the corresponding continuous equation

transformations of gibbs measures we study local transformations of gibbs measures we establish sufficient conditions for the quasilocality of the images and obtain results on the existence and continuity properties of their relative energies general results are illustrated by simple examples

optimized diversity combining with imperfect channel estimation in a communication system using receive diversity and linear combining in the presence of cochannel interference cci optimum combining oc is known to give the best error performance since it maximizes the instantaneous signal to interference plus noise ratio of the combiner output and consequently in the presence of gaussian interference plus noise it minimizes the error rate however this is based on the assumption that a perfect estimate of the channel is available channel estimation methods in reality use some overhead when the channel is time invariant the estimation error decreases with increase in the amount of overhead like the number of pilot symbols with the growing need for high data rate applications the amount of overhead that can be allocated for the estimation of the channel needs to be reduced and the channel estimation error cannot be ignored in this situation replacing the channel by its imperfect estimate in the oc weight vector no longer results in an optimum scheme we have to find an optimum scheme based on the channel estimation method and the detection criterion which results in what we call optimized diversity combining odc here we focus on odc resulting from a pilot symbol based maximum likelihood ml channel estimation method applied to a correlated flat rayleigh fading channel in the presence of cci and additive noise the channel is randomly time invariant during the reception of pilot and data symbols the decision rule which is optimum in the ml sense is derived using concepts of gaussian and wishart statistics numerical results show that odc can perform significantly better than oc with imperfect channnel estimates by appropriate choice of system parameters

stochastic approach to equilibrium and nonequilibrium thermodynamics we develop the stochastic approach to thermodynamics based on stochastic dynamics which can be discrete master equation and continuous fokker planck equation and on two assumptions concerning entropy the first is the definition of entropy itself and the second the definition of entropy production rate which is non negative and vanishes in thermodynamic equilibrium based on these assumptions we study interacting systems with many degrees of freedom in equilibrium or out of thermodynamic equilibrium and how the macroscopic laws are derived from the stochastic dynamics these studies include the quasiequilibrium processes the convexity of the equilibrium surface the monotonic time behavior of thermodynamic potentials including entropy the bilinear form of the entropy production rate the onsager coefficients and reciprocal relations and the nonequilibrium steady states of chemical reactions

on critical phenomena in interacting growth systems part i general components which are placed in a finite or infinite space have integer numbers as possible states they interact in a discrete time in a local deterministic way in addition to which all the components states are incremented at every time step by independent identically distributed random variables we assume that the deterministic interaction function is translation invariant and monotonic and that its values are between the minimum and the maximum of its arguments theorems 1 and 2 based on propositions which we give in a separate part ii give sufficient conditions for a system to have an invariant distribution or a bounded mean other statements proved herein provide background for them by giving conditions when a system has no invariant distribution or the mean of its components states tends to infinity all our main results use one and the same geometrical criterion

non reversible stationary measures for infinite interacting particle systems two results concerning the local conditional distributions of a stationary measure for a spin flip process with strictly positive and continuous rates are obtained 1 the local conditional distributions and the rates of the reversed process determine each other 2 either all shift invariant stationary measures are gibbs with the same potential or no shift invariant stationary gibbs measure exist

the survival of branching annihilating random walk branching annihilating random walk is an interacting particle system on as time evolves particles execute random walks and branch and disappear when they meet other particles it is shown here that starting from a finite number of particles the system will survive with positive probability if the random walk rate is low enough relative to the branching rate but will die out with probability one if the random walk rate is high since the branching annihilating random walk is non attractive standard techniques usually employed for interacting particle systems are not applicable instead a modification of a contour argument by gray and griffeath is used

bounds for the critical line of the theta contact processes with 1 or theta or 2 we study a family of the one dimensional contact processes introduced by durrett and griffeath 1983 which is parametrized by theta for each theta or 1 there is a unique critical value lambda c theta so that any process becomes extinct with probability 1 for lambda lambda c theta in this paper we give rigorous upper and lower bounds for the critical line lambda lambda c theta for 1 or theta or 2 in order to obtain the upper bound we extend the holley liggett argument which was originally given for the case theta 2 the basic contact process we construct a new class of attractive renewal measures with positive densities and the upper bound of lambda c theta is given as the largest root of a cubic equation theta lambda 3 3 theta 2 lambda 2 3 2 theta lambda 2 theta 0 recently liggett 1991 reported an upper bound of the critical value for the case theta 1 the threshold contact process by a modified version of the holley liggett argument our result includes these previous results as special cases

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

on the dynamic consistency of a two species competitive discrete system with toxicity local and global analysis abstract discretization of a highly nonlinear system is inevitable for good approximation of its solutions due to lack of its analytical solution however a discrete model constructed by standard discretization technique frequently produces spurious dynamics which are not observed in its constituent continuous model it is therefore important to construct a discrete model that produces the exact dynamics of its continuous counterpart here we transform a continuous time two species nonlinear competition model with the effect of toxic substances into a discrete time system to show its dynamic consistencies with the continuous system by employing nonstandard finite difference scheme linear stability property of each fixed point of the derived model has been proven to be identical with those of the continuous system and the dynamics has been shown to be independent of the step size moreover we prove the global stability of the coexisting equilibrium point of our discrete model comparative simulation results demonstrate that the standard discrete system shows complex dynamics like flip bifurcation and chaos depending on the step size but the proposed discrete model does not show any such spurious dynamics

the gacs kurdyumov levin automaton revisited we study a one dimensional cellular automaton that was originally proposed as a candidate for exhibiting nonergodic behavior under noise we prove that the deterministic model has the eroder property for two and only two invariant states moreover we give the best possible estimates for the corresponding erosion times we then review the results we have obtained from extensive computer simulations for the stochastic model and for a mixed model and argue that they suggest numerical and heuristic evidence in favour of ergodic behavior for all nonzero values of the noise parameter

qualitative analysis on a diffusive predator prey model with toxins abstract this paper is concerned with the diffusive predator prey model with toxins subject to dirichlet boundary conditions the uniform persistence of positive solution is given under certain conditions in addition by the liapunov schmidt method the existence and stability of the bifurcation solution from a double eigenvalues are investigated moreover by the fixed point index theory and perturbation theory of eigenvalues the uniqueness stability and multiplicity of coexistence states are analyzed when some key parameter changes finally some numerical simulations are presented to verify the theoretical conclusions and further to reflect the importance of parameters to the number of coexistence states

duality for general attractive spin systems with applications in one dimension on developpe une theorie des processus duaux qui s applique aux systemes de spin generaux a taux de transition attractifs on illustre l applicabilite de ces processus en generalisant les resultats de durrett et griffeath 1983

decay of correlations under dobrushin s uniqueness condition and its applications an estimate on the correlation of functionals of gibbs fields satisfying dobrushin s uniqueness condition is given as a consequence a result of gross saying that the truncated pair correlation function decays in the same weighted summability sense as the potential can be extended to the whole dobrushin uniqueness region applications to the central limit theorem and the second derivative of the pressure are also given

a simple proof of the stability criterion of gray and griffeath gray and griffeath studied attractive nearest neighbor spin systems on the integers having all 0 s and all 1 s as traps using the contour method they established a necessary and sufficient condition for the stability of the all 1 s equilibrium under small perturbations in this paper we use a renormalized site construction to give a much simpler proof our new approach can be used in many situations as a substitute for the contour method

periodicity and global dynamics of an impulsive delay lasota wazewska model abstract the periodicity of an impulsive delay lasota wazewska model is discussed sufficient and necessary condition for the existence of a positive periodic solution is established sufficient conditions for its global attractivity are also obtained via the methods of lyapunov function and comparison

parallel iterative regularization methods for solving systems of ill posed equations in this note two parallel iterative regularization methods for finding a minimal norm solution to a system of ill posed equations involving the so called strongly inverse monotone operators have been investigated some applications of the proposed methods are considered and numerical experiments are discussed

on the definition of entropy production via examples we present a definition of entropy production rate for classes of deterministic and stochastic dynamics the point of departure is a gibbsian representation of the steady state path space measure for which the density is determined with respect to the time reversed process the gibbs formalism is used as a unifying algorithm capable of incorporating basic properties of entropy production in nonequilibrium systems our definition is motivated by recent work on the gallavotti cohen local fluctuation theorem and it is illustrated via a number of examples