dynamical behavior of a one predator and two independent preys system with stochastic perturbations abstract a one predator and two independent preys system with stochastic perturbations is considered first we show that there is a positive solution of this system then we investigate the long time behavior of this system sufficient criteria for extinction and persistence in time average are established moreover it is also shown that this system has a stationary distribution and it is ergodicity finally examples and simulations are carried on to verify these results

border collision period doubling scenario using a one dimensional dynamical system representing a poincare return map for dynamical systems ofthe lorenz type we investigate the border collision period doubling bifurcation scenario in contrast to theclassical period doubling scenario this scenario is formed by a sequence of pairs of bifurcations whereby eachpair consists of a border collision bifurcation and a pitchfork bifurcation the characteristic properties of thisscenario like symmetry breaking and symmetry recovering as well as emergence of coexisting attractors andnoninvariant attractive sets are investigated doi 10 1103 physreve 70 026222 pacs number s 05 45 ac 82 40 bj 95 10 fhi introduction

dynamics and spatial organization in two species competition the authors investigate the time evolution of a prototypical population biology reaction which involves reproduction self regulation and competitive annihilation of two distinct species in one dimension the authors use a quasistatic analysis to argue that for a system with equal initial densities of two strongly competing species an alternating pattern of domains forms whose lengths grow logarithmically with time a scaling analysis of the underlying master equation as well as numerical integration of the reaction diffusion equations support this result for unequal initial densities the concentration of the minority species undergoes a power law decay with a non universal exponent the authors generalize the model by allowing for a nonlinear self regulation term in the rate equations as a function of the exponent of this nonlinearity the typical domain size may grow either as a power law with time or saturate at a finite value this general approach also suggests that a coarsening domain mosaic occurs in arbitrary spatial dimensions in two dimensions numerical integration of the reaction diffusion equations indicates that the average domain area grows approximately as t0 84

nonequilibrium phase transition due to isolation of communities we introduce a simple model of a growing system with m competing communities the model corresponds to the phenomenon of defeats suffered by social groups living in isolation a nonequilibrium phase transition is observed when at critical time t c the first isolated cluster occurs in the one dimensional system the volume of the new phase i e the number of the isolated individuals increases with time as z ensuremath sim t 3 for a large number of possible communities the critical density of filled space is equal to ensuremath rho c m n 1 3 where n is the system size a similar transition is observed for erd ifmmode mbox h o else h o fi s r enyi random graphs and barab asi albert scale free networks analytical results are in agreement with numerical simulations

phase ordering and persistence in a class of stochastic processes interpolating between the ising and voter models we study the dynamics of a class of two dimensional stochastic processes depending on two parameters which may be interpreted as two different temperatures respectively associated to interfacial and to bulk noise special lines in the plane of parameters correspond to the ising model voter model and majority vote model the dynamics of this class of models may be described formally in terms of reaction diffusion processes for a set of coalescing annihilating and branching random walkers we use the freedom allowed by the space of parameters to measure by numerical simulations the persistence probability of a generic model in the low temperature phase where the system coarsens this probability is found to decay at large times as a power law with a seemingly constant exponent we also discuss the connection between persistence and the nature of the interfaces between domains

discrete opinion dynamics on networks based on social influence a model of opinion dynamics based on social influence on networks was studied the opinion of each agent can have integer values i 1 2 i and opinion exchanges are restricted to connected agents it was found that for any i 2 and self confidence parameter 0 u 1 when u is a degree independent constant the weighted proportion qi of the population that hold a given opinion i is a martingale and the fraction qi of opinion i will gradually converge to qi the tendency can slow down with the increase of degree assortativity of networks when u is degree dependent qi does not possess the martingale property however qi still converges to it in both cases for a finite network the states of all agents will finally reach consensus further if there exist stubborn persons in the population whose opinions do not change over time it was found that for degree independent constant u both qi and qi will converge to fixed proportions which only depend on the distribution of initial obstinate persons and naturally the final equilibrium state will be the coexistence of diverse opinions held by the stubborn people the analytical results were verified by numerical simulations on barabasi albert ba networks the model highlights the influence of high degree agents on the final consensus or coexistence state and captures some realistic features of the diffusion of opinions in social networks

real space renormalization group and totalitarian paradox of majority rule voting abstract the effect of majority rule voting in hierarchical structures is studied using the basic concepts from real space renormalization group it shows in particular that a huge majority can be self eliminated while climbing up the hierarchy levels this majority democratic self elimination articulates around the existence of fixed points in the voting flow an unstable fixed point determines the critical threshold to full and total power it can be varied from 50 up to 77 of initial support our model could shed new light on the last century eastern european communist collapse

stochastic dynamics of invasion and fixation we study evolutionary game dynamics in finite populations we analyze an evolutionary process which we call pairwise comparison for which we adopt the ubiquitous fermi distribution function from statistical mechanics the inverse temperature in this process controls the intensity of selection leading to a unified framework for evolutionary dynamics at all intensities of selection from random drift to imitation dynamics we derive a simple closed formula that determines the feasibility of cooperation in finite populations whenever cooperation is modeled in terms of any symmetric two person game in contrast with previous results the present formula is valid at all intensities of selection and for any initial condition we investigate the evolutionary dynamics of cooperators in finite populations and study the interplay between intensity of selection and the remnants of interior fixed points in infinite populations as a function of a given initial number of cooperators showing how this interplay strongly affects the approach to fixation of a given trait in finite populations leading to counterintuitive results at different intensities of selection

optimal linear glauber model contrary to the actual nonlinear glauber model the linear glauber model lgm is exactly solvable although the detailed balance condition is not generally satisfied this motivates us to address the issue of writing the transition rate w j in a best possible linear form such that the mean squared error in satisfying the detailed balance condition is least the advantage of this work is that by studying the lgm analytically we will be able to anticipate how the kinetic properties of an arbitrary ising system depend on the temperature and the coupling constants the analytical expressions for the optimal values of the parameters involved in the linear w j are obtained using a simple moore penrose pseudoinverse matrix this approach is quite general in principle applicable to any system and can reproduce the exact results for one dimensional ising system in the continuum limit we get a linear time dependent ginzburg landau equation from the glauber s microscopic model of non conservative dynamics we analyze the critical and dynamic properties of the model and show that most of the important results obtained in different studies can be reproduced by our new mathematical approach we will also show in this paper that the effect of magnetic field can easily be studied within our approach in particular we show that the inverse of relaxation time changes quadratically with weak magnetic field and that the fluctuation dissipation theorem is valid for our model

voting and catalytic processes with inhomogeneities we consider the dynamics of the voter model and of the monomer monomer catalytic process in the presence of many competing inhomogeneities and show through exact calculations and numerical simulations that their presence results in a nontrivial fluctuating steady state whose properties are studied and turn out to specifically depend on the dimensionality of the system the strength of the inhomogeneities and their separating distances in fact in arbitrary dimensions we obtain an exact yet formal expression of the order parameters magnetization and concentration of adsorbed particles in the presence of an arbitrary number n of inhomogeneities zealots in the voter language and formal similarities with suitable electrostatic systems are pointed out in the nontrivial cases n 1 2 we explicitly compute the static and long time properties of the order parameters and therefore capture the generic features of the systems when ng2 the problems are studied through numerical simulations in one spatial dimension we also compute the expressions of the stationary order parameters in the completely disordered case where n is arbitrary large particular attention is paid to the spatial dependence of the stationary order parameters and formal connections with electrostatics

solution of voter model dynamics on annealed small world networks an analytical study of the behavior of the voter model on the small world topology is performed in order to solve the equations for the dynamics we consider an annealed version of the watts strogatz ws network where long range connections are randomly chosen at each time step the resulting dynamics is as rich as on the original ws network a temporal scale tau separates a quasistationary disordered state with coexisting domains from a fully ordered frozen configuration tau is proportional to the number of nodes in the network so that the system remains asymptotically disordered in the thermodynamic limit

cooperation in aspiration based n person prisoner s dilemmas we propose a mathematical model of the n person prisoner s dilemma game played by a continuous population of agents with a time dependent aspiration level the model a system of differential equations takes into account the evolution of the aspiration level and of the mean frequency of the cooperators in the population the dependence of the asymptotic level of cooperation on the individual payoffs and on the transition rates determining the agent s reaction to the received payoffs is studied in general the existence and the magnitude of the asymptotic level of cooperation depends on n the payoffs and the transition rates and decreases with increasing n

weighted extended b spline approximation of dirichlet problems we describe a new finite element method which uses weighted extended b splines on a regular grid as basis functions for solving dirichlet problems on bounded domains in arbitrary dimensions this web method does not require any grid generation and can be implemented very efficiently it yields smooth high order accurate approximations with relatively low dimensional subspaces

bouncing ball orbits and symmetry breaking effects in a three dimensional chaotic billiard we study the classical and quantum mechanics of a three dimensional stadium billiard it consists of two quarter cylinders that are rotated with respect to each other by 90 ifmmode circ else textdegree fi and it is classically chaotic the billiard exhibits only a few families of nongeneric periodic orbits we introduce an analytic method for their treatment the length spectrum can be understood in terms of the nongeneric and unstable periodic orbits for unequal radii of the quarter cylinders the level statistics agree well with predictions from random matrix theory for equal radii the billiard exhibits an additional symmetry we investigated the effects of symmetry breaking on spectral properties moreover for equal radii we observe a small deviation of the level statistics from random matrix theory this led to the discovery of stable and marginally stable orbits which are absent for unequal radii

local interaction scale controls the existence of a nontrivial optimal critical mass in opinion spreading we study a model of opinion formation where the collective decision of a group is said to happen if the fraction of agents having the most common opinion exceeds a threshold value a critical mass we find that there exists a unique nontrivial critical mass giving the most efficient convergence to consensus in addition we observe that for small critical masses the characteristic time scale for the relaxation to consensus splits into two the shorter time scale corresponds to a direct relaxation and the longer one can be explained by the existence of intermediate metastable states similar to those found in p chen and s redner phys rev e 71 036101 2005 this longer time scale is dependent on the precise condition for consensus with a modification of the condition it can go away

dynamics of influence on hierarchical structures dichotomous spin dynamics on a pyramidal hierarchical structure the bethe lattice are studied the system embodies a number of classes where a class comprises nodes that are equidistant from the root head node weighted links exist between nodes from the same and different classes the spin hereafter state of the head node is fixed we solve for the dynamics of the system for different boundary conditions we find necessary conditions so that the classes eventually repudiate or acquiesce in the state imposed by the head node the results indicate that to reach unanimity across the hierarchy it suffices that the bottommost class adopts the same state as the head node then the rest of the hierarchy will inevitably comply this also sheds light on the importance of mass media as a means of synchronization between the topmost and bottommost classes surprisingly in the case of discord between the head node and the bottommost classes the average state over all nodes inclines towards that of the bottommost class regardless of the link weights and intraclass configurations hence the role of the bottommost class is signified

efficiency of competitions league competition is investigated using random processes and scaling techniques in our model a weak team can upset a strong team with a fixed probability teams play an equal number of head to head matches and the team with the largest number of wins is declared to be the champion the total number of games needed for the best team to win the championship with high certainty t grows as the cube of the number of teams n i e t ensuremath sim n 3 this number can be substantially reduced using preliminary rounds where teams play a small number of games and subsequently only the top teams advance to the next round when there are k rounds the total number of games needed for the best team to emerge as champion t k scales as follows t k ensuremath sim n ensuremath gamma k with ensuremath gamma k 1 ensuremath 2 3 k 1 ensuremath 1 for example ensuremath gamma k 9 5 27 19 81 65 for k 1 2 3 these results suggest an algorithm for how to infer the best team using a schedule that is linear in n we conclude that league format is an ineffective method of determining the best team and that sequential elimination from the bottom up is fair and efficient

electoral surveys influence on the voting processes a cellular automata model abstract nowadays in societies threatened by atomization selfishness short term thinking and alienation from political life there is a renewed debate about classical questions concerning the quality of democratic decision making in this work a cellular automata model for the dynamics of free elections based on the social impact theory is proposed by using computer simulations power law distributions for the size of electoral clusters and decision time have been obtained the major role of broadcasted electoral surveys in guiding opinion formation and stabilizing the status quo was demonstrated furthermore it was shown that in societies where these surveys are manipulated within the universally accepted statistical error bars even a majoritary opposition could be hindered from reaching power through the electoral path

social paradoxes of majority rule voting and renormalization group real space renormalization group ideas are used to study a voting problem in political science a model to construct self directed pyramidal structures from bottom up to the top is presented using majority rules it is shown that a minority and even a majority can be systematically self eliminated from top leadership provided the hierarchy has a minimal number of levels in some cases 70 of the population is found to have zero representation after six hierarchical levels results are discussed with respect to the internal operation of political organizations

non generic spectral statistics in the quantized stadium billiard we consider the effect of a continuous family of neutral bouncing ball orbits on the energy spectrum of the quantized stadium billiard using a semiclassical approximation we derive analytic expressions for standard two point spectral measures the corrections due to the bouncing ball orbits account for some of the non generic features observed in the analysis of the spectrum of a stadium cavity which was recently measured once the bouncing ball contributions are subtracted the spectrum is shown to be well reproduced by the semi classical trace formula based on unstable periodic orbits we also study special patterns in the spectrum which are due to other non generic features such as edge effects and whispering gallery trajectories

an opinion formation model with dissipative structure abstract we propose an opinion formation model which includes both the influence of internal motivation and dissipation active agents adopt their neighbors opinions according to a simplified set of rules but they may gradually lose their interests in the discussion and drop out of it on the other hand inert agents can become active by initial activation or internal motivation from neighbors and participate in the discussion the initial activation is usually due to occurrence of a social event the internal motivation opinion update and dissipation procedure take place simultaneously we apply the voter rule in our model and carry out analysis and numerical simulations results show if nonzero dissipation stays below a threshold value the system evolves to a balance state where the average concentration of one opinion is equal to that of the other with dissipation a number of small size opinion clusters exist in the end but the system gains a fast relaxation rate the final average opinion is closely related to the dissipation intensity and the length of time for initial activation

the effect of zealots on the rate of consensus achievement in complex networks abstract in this study we investigate the role of zealots on the result of voting process on both scale free and watts strogatz networks we observe that inflexible individuals are very effective in consensus achievement and also in the rate of ordering process in complex networks zealots make the magnetization of the system to vary exponentially with time we obtain that on sf networks increasing the zealots population z exponentially increases the rate of consensus achievement the time needed for the system to reach a desired magnetization shows a power law dependence on z as well we obtain that the decay time of the order parameter shows a power law dependence on z we also investigate the role of zealots degree on the rate of ordering process and finally we analyze the effect of network s randomness on the efficiency of zealots moving from a regular to a random network the re wiring probability p r w increases we show that with increasing p r w the efficiency of zealots for reducing the consensus achievement time increases the rate of consensus is compared with the rate of ordering for different re wiring probabilities of ws networks

phase transition in a coevolving network of conformist and contrarian voters in the coevolving voter model each voter has one of two diametrically opposite opinions and a voter encountering a neighbor with the opposite opinion may either adopt it or rewire the connection to another randomly chosen voter sharing the same opinion as we smoothly change the relative frequency of rewiring compared to that of adoption there occurs a phase transition between an active phase and a frozen phase by performing extensive monte carlo calculations we show that the phase transition is characterized by critical exponents ensuremath beta 0 54 1 and overline ensuremath nu 1 5 1 which differ from the existing mean field type prediction we furthermore extend the model by introducing a contrarian type that tries to have neighbors with the opposite opinion and show that the critical behavior still belongs to the same universality class irrespective of such contrarians fraction

scaling behavior in a proportional voting process we perform a statistical analysis on the proportional elections held in brazil in october 1998 we show that the distribution of votes among candidates for the whole country follows a power law n v ensuremath propto v ensuremath ensuremath alpha with ensuremath alpha 1 00 ifmmode pm else textpm fi 0 02 extending over two orders of magnitude the voting distributions for several states of the federation also display scale invariant behavior with ensuremath alpha ensuremath approx 1 we argue that this particular voting system can be modeled as a typical multiplicative process in which the choice of the candidate is governed by a product of probabilities

border collision bifurcations an explanation for observed bifurcation phenomena recently physical and computer experiments involving systems describable by continuous maps that are nondifferentiable on some surface in phase space have revealed novel bifurcation phenomena these phenomena are part of a rich new class of bifurcations which we call border collision bifurcations a general criterion for the occurrence of border collision bifurcations is given illustrative numerical results including transitions to chaotic attractors are presented these border collision bifurcations are found in a variety of physical experiments

numerical study of a three dimensional generalized stadium billiard we study a generalized three dimensional stadium billiard and present strong numerical evidence that this system is completely chaotic in this convex billiard chaos is generated by the defocusing mechanism the construction of this billiard uses cylindrical components as the focusing elements and thereby differs from the recent approach pioneered by bunimovich and rehacek commun math phys 189 729 1997 we investigate the stability of lower dimensional invariant manifolds and discuss bouncing ball modes c 2000 the american physical society

some demographic crashes seen as phase transitions abstract the purpose of this paper is the application of a usual method of statistical mechanics the renormalization based on wilson s recursive relations in order to study the critical behavior of a social index namely the live births per 1000 population the drastic decreases of this index on certain periods have the specific features of the phase transitions as they follow approximately power laws and also they lead to the complete change of the population age structure the values of the critical exponents that are obtained by fitting the experimental data referring to some east european countries are in agreement with the value resulting from the theoretical approach thus showing the universality of the power law behavior in the vicinity of the critical points for complex social systems

comparison of voter and glauber ordering dynamics on networks we study numerically the ordering process of two very simple dynamical models for a two state variable on several topologies with increasing levels of heterogeneity in the degree distribution we find that the zero temperature glauber dynamics for the ising model may get trapped in sets of partially ordered metastable states even for finite system size and this becomes more probable as the size increases voter dynamics instead always converges to full order on finite networks even if this does not occur via coherent growth of domains the time needed for order to be reached diverges with the system size in both cases the ordering process is rather insensitive to the variation of the degreee distribution from sharply peaked to scale free

periodic solutions and stationary distribution for a stochastic predator prey system with impulsive perturbations abstract in this paper we consider a stochastic predator prey system with beddington deangelis functional response and impulsive perturbations first we prove that the model admits a unique global positive solution by constructing the equivalent system without impulsive perturbations second we establish a sufficient condition which allows the existence of a positive periodic solution using stochastic lyapunov functions finally for the predator prey system with beddington deangelis functional response disturbed by both white noise and telephone noise we give the sufficient conditions for the stationary distribution which is ergodic and positive recurrent of the solution the conclusion implies that the stochastic system has a positive t periodic markov process in a certain condition when the corresponding deterministic system has at least one positive t periodic solution

semi focusing billiards hyperbolicity in this paper we answer affirmatively the question concerning the existence of hyperbolic billiards in convex domains of 3 we also prove that a related class of semi focusing billiards has mixed dynamics i e their phase space is an union of two invariant sets of positive measure such that the dynamics is integrable on one set and is hyperbolic on the other these billiards are the first rigorous examples of billiards in domains of 3 with divided phase space

stationary distribution of a regime switching predator prey model with anti predator behaviour and higher order perturbations abstract in this paper we study a regime switching predator prey model with anti predator behaviour and higher order perturbations we obtain the ergodic property by constructing a suitable stochastic lyapunov function with regime switching which provides us a biological perspective of cycling phenomena of a population system and can better describe the stochastic persistence of a population system in practice we find that these restrictive assumptions on the functional response are relative weak and valid for many types of response functions

stochastic bifurcations in the nonlinear parallel ising model we investigate the phase transitions of a nonlinear parallel version of the ising model characterized by an antiferromagnetic linear coupling and ferromagnetic nonlinear one this model arises in problems of opinion formation the mean field approximation shows chaotic oscillations by changing the couplings or the connectivity the spatial model shows bifurcations in the average magnetization similar to that seen in the mean field approximation induced by the change of the topology after rewiring short range to long range connection as predicted by the small world effect these coherent periodic and chaotic oscillations of the magnetization reflect a certain degree of synchronization of the spins induced by long range couplings similar bifurcations may be induced in the randomly connected model by changing the couplings or the connectivity and also the dilution degree of asynchronism of the updating we also examined the effects of inhomogeneity mixing ferromagnetic and antiferromagnetic coupling which induces an unexpected bifurcation diagram with a bubbling behavior as also happens for dilution

border collision bifurcations including period two to period three for piecewise smooth systems abstract we examine bifurcation phenomena for maps that are piecewise smooth and depend continuously on a parameter in the simplest case there is a surface in phase space along which the map has no derivative or has two one sided derivatives is the border of two regions in which the map is smooth as the parameter is varied a fixed point e may collide with the border and we may assume that this collision occurs at 0 a variety of bifurcations occur frequently in such situations but never or almost never occur in smooth systems in particular e may cross the border and so will exist for 0 but it may be a saddle in one case say 0 for 0 there may be a stable period 3 orbit which similarly shrinks to e0 as 0 hence one observes the following stable periodic orbits a stable period 2 orbit collapses to a point and is reborn as a stable period 3 orbit we also see analogously stable period 2 to stable period p orbit bifurcations with p 5 11 52 or period 2 to quasi periodic or even to a chaotic attractor we believe this phenomenon will be seen in many applications

model of cluster growth and phase separation exact results in one dimension we present exact results for a lattice model of cluster growth in one dimension the growth mechanism involves interface hopping and pairwise annihilation supplemented by spontaneous creation of the stable phase 1 regions by overturning the unstable phase 1 spins with probabilityp for cluster coarsening at phase coexistence p 0 the conventional structure factor scaling applies in this limit our model falls in the class of diffusion limited reactions a a inert the 1 cluster size grows diffusively t and the two point correlation function obeys scaling however forp 0 i e for the dynamics of formation of stable phase from unstable phase we find that structure factor scaling breaks down the length scale associated with the size of the growing 1 clusters reflects only the short distance properties of the two point correlations

an introduction to stochastic modeling stochastic modeling probability review the major discrete distributions rtant continuous distributions some elementary exercises useful functions integrals and sums conditional probability and conditional expectation the discrete case the dice game craps random sums conditioning on a continuous random variable markov chains introduction definitions transition probability matrices of a markov chain some markov chain models first step analysis some special markov chains functionals of random walks and success runs another look at first step analysis the long run behavior of markov chains regular transition probability matrices examples the classification of states the basic limit theorem of markov chains reducible markov chains sequential decisions and markov chains poisson processes the poisson distribution and the poisson processes the law of rare events distributions associated with the poisson process the uniform distribution and poisson processes spatial poisson processes compound and marked poisson processes continuous time markov chains pure birth processes ptire death processes birth and death processes the limiting behavior of birth and death processes birth and death processes with absorbing states finite state continuous time markov chains set valued processes renewal phenomena definition of a renewal process and related concepts some examples of renewal processes the poisson process viewed as a renewal process the asymptotic 3ehavior as renewal process

hipsters on networks how a minority group of individuals can lead to an antiestablishment majority the spread of opinions memes diseases and alternative facts in a population depends both on the details of the spreading process and on the structure of the social and communication networks on which they spread one feature that can change spreading dynamics substantially is heterogeneous behavior among different types of individuals in a social network in this paper we explore how antiestablishment nodes e g hipsters influence the spreading dynamics of two competing products we consider a model in which spreading follows a deterministic rule for updating node states which indicate which product has been adopted in which an adjustable probability p hip of the nodes in a network are hipsters who choose to adopt the product that they believe is the less popular of the two the remaining nodes are conformists who choose which product to adopt by considering which products their immediate neighbors have adopted we simulate our model on both synthetic and real networks and we show that the hipsters have a major effect on the final fraction of people who adopt each product even when only one of the two products exists at the beginning of the simulations a small fraction of hipsters in a network can still cause the other product to eventually become the more popular one to account for this behavior we construct an approximation for the steady state adoption fractions of the products on k regular trees in the limit of few hipsters additionally our simulations demonstrate that a time delay in the knowledge of the product distribution in a population as compared to immediate knowledge of product adoption among nearest neighbors can have a large effect on the final distribution of product adoptions using a local tree approximation we derive an analytical estimate of the spreading of products and obtain good agreement if a sufficiently small fraction of the population consists of hipsters in all networks we find that either of the two products can become the more popular one at steady state depending on the fraction of hipsters in the network and on the amount of delay in the knowledge of the product distribution our simple model and analysis may help shed light on the road to success for antiestablishment choices in elections as such success and qualitative differences in final outcomes between competing products political candidates and so on can arise rather generically in our model from a small number of antiestablishment individuals and ordinary processes of social influence on normal individuals

the noisy voter model under the influence of contrarians abstract the influence of contrarians on the noisy voter model is studied at the mean field level the noisy voter model is a variant of the voter model where agents can adopt two opinions optimistic or pessimistic and can change them by means of an imitation herding and an intrinsic noise mechanisms an ensemble of noisy voters undergoes a finite size phase transition upon increasing the relative importance of the noise to the herding form a bimodal phase where most of the agents share the same opinion to a unimodal phase where almost the same fraction of agent are in opposite states by the inclusion of contrarians we allow for some voters to adopt the opposite opinion of other agents anti herding we first consider the case of only contrarians and show that the only possible steady state is the unimodal one more generally when voters and contrarians are present we show that the bimodal unimodal transition of the noisy voter model prevails only if the number of contrarians in the system is smaller than four and their characteristic rates are small enough for the number of contrarians bigger or equal to four the voters and the contrarians can be seen only in the unimodal phase moreover if the number of voters and contrarians as well as the noise and herding rates are of the same order then the probability functions of the steady state are very well approximated by the gaussian distribution

majority vote on directed erdős renyi random graphs abstract through monte carlo simulation the well known majority vote model has been studied with noise on directed random graphs in order to characterize completely the observed order disorder phase transition the critical noise parameter q c as well as the critical exponents and 1 have been calculated as a function of the connectivity z of the random graph

conformists and contrarians in a kuramoto model with identical natural frequencies we consider a variant of the kuramoto model in which all the oscillators are now assumed to have the same natural frequency but some of them are negatively coupled to the mean field these contrarian oscillators tend to align in antiphase with the mean field whereas the positively coupled conformist oscillators favor an in phase relationship the interplay between these effects can lead to rich dynamics in addition to a splitting of the population into two diametrically opposed factions the system can also display traveling waves complete incoherence and a blurred version of the two faction state exact solutions for these states and their bifurcations are obtained by means of the watanabe strogatz transformation and the ott antonsen ansatz curiously this system of oscillators with identical frequencies turns out to exhibit more complicated dynamics than its counterpart with heterogeneous natural frequencies

diffusive clustering in the two dimensional voter model on donne des resultats de simulation et une enquete concernant les resultats connus sur le modele du votant dans trois cas distincts trois ou plusieurs dimensions une dimension et deux dimensions

disorder induced phase transition in kinetic models of opinion dynamics abstract we propose a model of continuous opinion dynamics where mutual interactions can be both positive and negative different types of distributions for the interactions all characterized by a single parameter p denoting the fraction of negative interactions are considered results from exact calculation of a discrete version and numerical simulations of the continuous version of the model indicate the existence of a universal continuous phase transition at p p c below which a consensus is reached although the order disorder transition is analogous to a ferromagnetic paramagnetic phase transition with comparable critical exponents the model is characterized by some distinctive features relevant to a social system

commitment versus persuasion in the three party constrained voter model in the framework of the three party constrained voter model where voters of two radical parties a and b interact with centrists c and c we study the competition between a persuasive majority and a committed minority in this model a s and b s are incompatible voters that can convince centrists or be swayed by them here radical voters are more persuasive than centrists whose sub population comprises susceptible agents c and a fraction of centrist zealots c whereas c s may adopt the opinions a and b with respective rates 1 a and 1 b with a b 0 c s are committed individuals that always remain centrists furthermore a and b voters can become susceptible centrists c with a rate 1 the resulting competition between commitment and persuasion is studied in the mean field limit and for a finite population on a complete graph at mean field level there is a continuous transition from a coexistence phase when b whereas all species coexist when a b when c and the initial density of centrists is low one finds lnn when n 1 our analytical findings are corroborated by stochastic simulations

voter model dynamics in complex networks role of dimensionality disorder and degree distribution we analyze the ordering dynamics of the voter model in different classes of complex networks we observe that whether the voter dynamics orders the system depends on the effective dimensionality of the interaction networks we also find that when there is no ordering in the system the average survival time of metastable states in finite networks decreases with network disorder and degree heterogeneity the existence of hubs i e highly connected nodes in the network modifies the linear system size scaling law of the survival time the size of an ordered domain is sensitive to the network disorder and the average degree decreasing with both however it seems not to depend on network size and on the heterogeneity of the degree distribution

splitting the voter potts model critical point recently some two dimensional models with double symmetric absorbing states were shown to share the same critical behavior that was called the voter universality class we show that for an absorbing states potts model with finite but further than nearest neighbor range of interactions the critical point is split into two critical points one of the ising type and the other of the directed percolation universality class similar splitting takes place in the three dimensional nearest neighbor model

nonequilibrium transitions in complex networks a model of social interaction we analyze the nonequilibrium order disorder transition of axelrod s model of social interaction in several complex networks in a small world network we find a transition between an ordered homogeneous state and a disordered state the transition point is shifted by the degree of spatial disorder of the underlying network the network disorder favoring ordered configurations in random scale free networks the transition is only observed for finite size systems showing system size scaling while in the thermodynamic limit only ordered configurations are always obtained thus in the thermodynamic limit the transition disappears however in structured scale free networks the phase transition between an ordered and a disordered phase is restored

dynamics of a stochastic holling tanner predator prey model abstract in the paper we study a stochastic holling tanner predator prey system first by constructing a suitable stochastic lyapunov function we establish sufficient conditions for the existence of a unique ergodic stationary distribution of the positive solutions to the system after that sufficient causes for extinction of the predator population is obtained meanwhile we also obtain the threshold between persistence and extinction of the predator population at last some examples and numerical simulations are provided to illustrate our theoretical results

universality of weak selection weak selection which means a phenotype is slightly advantageous over another is an important limiting case in evolutionary biology recently it has been introduced into evolutionary game theory in evolutionary game dynamics the probability to be imitated or to reproduce depends on the performance in a game the influence of the game on the stochastic dynamics in finite populations is governed by the intensity of selection in many models of both unstructured and structured populations a key assumption allowing analytical calculations is weak selection which means that all individuals perform approximately equally well in the weak selection limit many different microscopic evolutionary models have the same or similar properties how universal is weak selection for those microscopic evolutionary processes we answer this question by investigating the fixation probability and the average fixation time not only up to linear but also up to higher orders in selection intensity we find universal higher order expansions which allow a rescaling of the selection intensity with this we can identify specific models which violate linear weak selection results such as the one third rule of coordination games in finite but large populations

analytic treatment of consensus achievement in the single type zealotry voter model abstract we introduce zealots of one opinion in the voter model on a complete graph and examine how they affect consensus achievement using first step analysis for markov chains to obtain an exact solution we find that the mean consensus time scales with the population size n increasing the number of zealots z will decrease the consensus time in a power law fashion for large z the mean magnetization was also analytically obtained and was found to contain an exponential dependence on z the dynamics for the complete graph are qualitatively similar to those obtained in another study for the barabasi albert network in general the existence of zealots serves to hasten consensus except in the case where only a few zealots oppose the vast majority

bifurcations in models of a society of reasonable contrarians and conformists we study models of a society composed of a mixture of conformist and reasonable contrarian agents that at any instant hold one of two opinions conformists tend to agree with the average opinion of their neighbors and reasonable contrarians tend to disagree but revert to a conformist behavior in the presence of an overwhelming majority in line with psychological experiments the model is studied in the mean field approximation and on small world and scale free networks in the mean field approximation a large fraction of conformists triggers a polarization of the opinions a pitchfork bifurcation while a majority of reasonable contrarians leads to coherent oscillations with an alternation of period doubling and pitchfork bifurcations up to chaos similar scenarios are obtained by changing the fraction of long range rewiring and the parameter of scale free networks related to the average connectivity

algebraic relaxation of an order parameter in randomly coupled limit cycle oscillators in their recent paper phys rev e 58 1789 1998 stiller and radons sr study following our earlier work phys rev lett 68 1073 1992 the behavior of globally and randomly coupled phase oscillators with distributed intrinsic frequencies they claim that their simulation results do not confirm the power law behavior of an order parameter found numerically by the author attributing its cause to the poor precision of the author s integration scheme here demonstrated is that the power law survives even for a scheme better than sr s provided that finite size effects are properly taken into account as was done in our previous work

ordering dynamics in the voter model with aging abstract we study theoretically and numerically the voter model with memory dependent dynamics at the mean field level the internal age related to the time an individual spends holding the same state is added to the set of binary states of the population such that the activation probability p i for attempting a change of state has an explicit dependence on the internal age i we derive a closed set of integro differential equations describing the time evolution of the fraction of individuals with a given state and internal age and from it we obtain analytical results characterizing the behavior of the system close to the absorbing states in general different internal age dependent activation probabilities p i have different effects on the dynamics in the case of aging i e p i being a decreasing function of its argument i either the system reaches consensus or it gets trapped in a frozen state depending on whether the limiting value p is zero or positive and on the rate at which p i approaches p moreover when the system reaches consensus the fraction x t of nodes not holding the consensus state may decay exponentially with time or as a power law depending again on the specific details of the functional form of p i for the case of anti aging when the activation probability p i is an increasing function of i the system always reaches a steady state with coexistence of opinions exact conditions for having one or another behavior together with the equations and explicit expressions for the power law exponents are provided

clusters and the entropy in opinion dynamics on complex networks abstract in this work we investigate a heterogeneous population in the modified hegselmann krause opinion model on complex networks we introduce the shannon information entropy about all relative opinion clusters to characterize the cluster profile in the final configuration independent of network structures there exists the optimal stubbornness of one subpopulation for the largest number of clusters and the highest entropy besides there is the optimal bounded confidence or subpopulation ratio of one subpopulation for the smallest number of clusters and the lowest entropy however network structures affect cluster profiles indeed a large average degree favors consensus for making different networks more similar with complete graphs the network size has limited impact on cluster profiles of heterogeneous populations on scale free networks but has significant effects upon those on small world networks

two spin majority cellular automaton as a model of 2d cluster and interface growth a new probabilistic cellular automaton model is introduced to simulate cluster and interface growth in two dimensions the dynamics of this model is an extension to higher dimensions of the compact directed percolation studied by essam numerical results indicate that the two dimensional cluster coarsening and growth can be described only approximately by the conventional cluster size scaling due to a crossover in the growth mode the spreading of the initially flat interface follows a purely diffusional t1 2 law

dynamics of majority rule in two state interacting spin systems we introduce a two state opinion dynamics model where agents evolve by majority rule in each update a group of agents is specified whose members then all adopt the local majority state in the mean field limit where a group consists of randomly selected agents consensus is reached in a time that scales ln n where n is the number of agents on finite dimensional lattices where a group is a contiguous cluster the consensus time fluctuates strongly between realizations and grows as a dimension dependent power of n the upper critical dimension appears to be larger than 4 the final opinion always equals that of the initial majority except in one dimension

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

stationary distribution of a stochastic predator prey system with stage structure for prey abstract in this paper a predator prey model with stage structure is proposed and analyzed in which preys are divided into juvenile and mature preys from a deterministic framework to a stochastic one we first prove that the system which we investigate has a unique global positive solution furthermore we obtain the sufficient criteria for the existence of stationary distribution and ergodicity by constructing an appropriate lyapunov function which means the species are permanent finally we give the sufficient conditions for extinction of preys

nature of phase transitions in a probabilistic cellular automaton with two absorbing states we present a probabilistic cellular automaton with two absorbing states which can be considered a natural extension of the domany kinzel model despite its simplicity it shows a very rich phase diagram with two second order and one first order transition lines that meet at a bicritical point we study the phase transitions and the critical behavior of the model using mean field approximations direct numerical simulations and field theory the second order critical curves and the kink critical dynamics are found to be in the directed percolation and parity conservation universality classes respectively the first char21 order phase transition is put in evidence by examining the hysteresis cycle we also study the chaotic phase in which two replicas evolving with the same noise diverge using mean field and numerical techniques finally we show how the shape of the potential of the field theoretic formulation of the problem can be obtained by direct numerical simulations

hydrodynamics of the voter model we study the voter model on mathbb z d d geqq 3 for a sequence mu varepsilon of initial states which have a gradient in the mean magnetization of the order varepsilon varepsilon rightarrow 0 we prove that the magnetization field m varepsilon f t varepsilon d sum f varepsilon x eta x varepsilon 2 t tends to a deterministic field m f t int dqf q m q t as varepsilon rightarrow 0 m q t is the solution of the diffusion equation the fluctuations of m varepsilon f t around its mean are given by an infinite dimensional non homogeneous ornstein uhlenbeck process in the limit as varepsilon rightarrow 0 locally i e around varepsilon 1 q varepsilon 2 t the voter model is in equilibrium with parameter m q t

analysis of a stochastic hybrid population model with allee effect abstract this research formulates and studies a stochastic population model with allee effect and markovian switching sufficient criteria for extinction stochastic permanence and the existence of a unique ergodic invariant measure of the model are established the result indicates that these long term behaviors of the model are intimately associated with the random perturbations

phase transitions of extended range probabilistic cellular automata with two absorbing states we study phase transitions in a long range one dimensional cellular automaton with two symmetric absorbing states it includes and extends several other models like the ising and domany kinzel ones it is characterized by competing ferromagnetic linear and antiferromagnetic nonlinear couplings despite its simplicity this model exhibits an extremely rich phase diagram we present numerical results and mean field approximations

exact results for kinetics of catalytic reactions the kinetics of an irreversible catalytic reaction on substrate of arbitrary dimension is examined in the limit of infinitesimal reaction rate reaction controlled limit we solve the dimer dimer surface reaction model or voter model exactly in arbitrary dimension d the density of reactive interfaces is found to exhibit a power law decay for d 2 and a slow logarithmic decay in two dimensions we discuss the relevance of these results for the monomer monomer surface reaction model

voter models on heterogeneous networks we study simple interacting particle systems on heterogeneous networks including the voter model and the invasion process these are both two state models in which in an update event an individual changes state to agree with a neighbor for the voter model an individual imports its state from a randomly chosen neighbor here the average time t n to reach consensus for a network of n nodes with an uncorrelated degree distribution scales as n ensuremath mu 1 2 ensuremath mu 2 where ensuremath mu k is the k text th moment of the degree distribution quick consensus thus arises on networks with broad degree distributions we also identify the conservation law that characterizes the route by which consensus is reached parallel results are derived for the invasion process in which the state of an agent is exported to a random neighbor we further generalize to biased dynamics in which one state is favored the probability for a single fitter mutant located at a node of degree k to overspread the population char22 the fixation probability char22 is proportional to k for the voter model and to 1 k for the invasion process

voter dynamics on an ising ladder coarsening and persistence coarsening and persistence of ising spins on a ladder is examined under voter dynamics the density of domain walls decreases algebraically with time as t 1 2 for sequential as well as parallel dynamics the persistence probability decreases as under sequential dynamics and as under parallel dynamics where p 2 s 0 88 numerical values of the exponents are explained the results are compared with the voter model on one and two dimensional lattices as well as ising model on a ladder under zero temperature glauber dynamics

averaging of globally coupled oscillators abstract we study a specific system of symmetrically coupled oscillators using the method of averaging the equations describe a series array of josephson junctions we concentrate on the dynamics near the splay phase state also known as the antiphase state ponies on a merry go round or rotating wave we calculate the floquet exponents of the splay phase periodic orbit in the weak coupling limit and find that all of the floquet exponents are purely imaginary in fact all the floquet exponents are zero except for a single complex conjugate pair thus nested two tori of doubly periodic solutions surround the splay phase state in the linearized averaged equations we numerically integrate the original system and find startling agreement with the averaging results on two counts the observed ratio of frequencies is very close to the prediction and the solutions of the full equations appear to be either periodic or doubly periodic as they are in the averaged equations such behavior is quite surprising from the point of view of generic dynamical systems theory one expects higher dimensional tori and chaotic solutions we show that the functional form of the equations and not just their symmetry is responsible for this nongeneric behavior

social consensus through the influence of committed minorities we show how the prevailing majority opinion in a population can be rapidly reversed by a small fraction p of randomly distributed committed agents who consistently proselytize the opposing opinion and are immune to influence specifically we show that when the committed fraction grows beyond a critical value p c 10 there is a dramatic decrease in the time t c taken for the entire population to adopt the committed opinion in particular for complete graphs we show that when p p c t c ln n we conclude with simulation results for erd s renyi random graphs and scale free networks which show qualitatively similar behavior

topological bifurcations in a model society of reasonable contrarians people are often divided into conformists and contrarians the former tending to align to the majority opinion in their neighborhood and the latter tending to disagree with that majority in practice however the contrarian tendency is rarely followed when there is an overwhelming majority with a given opinion which denotes a social norm such reasonable contrarian behavior is often considered a mark of independent thought and can be a useful strategy in financial markets we present the opinion dynamics of a society of reasonable contrarian agents the model is a cellular automaton of ising type with antiferromagnetic pair interactions modeling contrarianism and plaquette terms modeling social norms we introduce the entropy of the collective variable as a way of comparing deterministic mean field and probabilistic simulations bifurcation diagrams in the mean field approximation the model exhibits bifurcations and a chaotic phase interpreted as coherent oscillations of the whole society however in a one dimensional spatial arrangement one observes incoherent oscillations and a constant average in simulations on watts strogatz networks with a small world effect the mean field behavior is recovered with a bifurcation diagram that resembles the mean field one but where the rewiring probability is used as the control parameter similar bifurcation diagrams are found for scale free networks and we are able to compute an effective connectivity for such networks

competition and cooperation in one dimensional stepping stone models mutualism is a major force driving evolution and sustaining ecosystems although the importance of spatial degrees of freedom and number fluctuations is well known their effects on mutualism are not fully understood with range expansions of microbes in mind we show that even when mutualism confers a selective advantage it persists only in populations with high density and frequent migrations when these parameters are reduced mutualism is generically lost via a directed percolation dp process with a phase diagram strongly influenced by an exceptional symmetric dp dp2 transition

stationary distribution of a stochastic predator prey model with distributed delay and higher order perturbations abstract in this paper we study a stochastic predator prey model with distributed delay and higher order perturbations by the linear chain technique we transform the stochastic system with weak kernel case into an equivalent degenerate system which contains three equations by constructing a suitable stochastic lyapunov function we establish sufficient conditions for the existence of a stationary distribution of the positive solutions to the model

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved

a soluble kinetic model for spinodal decomposition we compare the two dimensional voter model with approximate theories for spinodal decomposition the cluster size distribution and the short time dynamics of the voter model are studied by means of a monte carlo simulation the time dependent structure factor and the long time scaling of the voter dynamics are known analytically