spectral formulation and wkb approximation for rare event statistics in reaction systems we develop a spectral formulation and a stationary wkb approximation for calculating the probabilities of rare events large deviations from the mean in systems of reacting particles with infinite range interaction describable by a master equation we compare the stationary wkb approximation to a recent time dependent semiclassical approximation developed for the same class of problems by elgart and kamenev phys rev e 70 41106 2004 as a benchmark we use an exactly solvable problem of the binary annihilation reaction 2a 0

how colored environmental noise affects population extinction environmental noise can cause an exponential reduction in the mean time to extinction mte of an isolated population we study this effect on an example of a stochastic birth death process with rates modulated by a colored that is correlated gaussian noise a path integral formulation yields a transparent way of evaluating the mte and finding the optimal realization of the environmental noise that determines the most probable path to extinction the population size dependence of the mte changes from exponential in the absence of the environmental noise to a power law for a short correlated noise and to no dependence for long correlated noise we also establish the validity domains of the white noise limit and adiabatic limit

discreteness induced extinction abstract two simple models based on ecological problems are discussed from the point of view of non equilibrium statistical mechanics it is shown how discrepant may be the results of the models that include spatial distribution with discrete interactions when compared with the continuous analogous models in the continuous case we have under certain circumstances the population explosion when we take into account the finiteness of the population we get the opposite result extinction we will analyze how these results depend on the dimension d of the space and describe the phenomenon of the discreteness inducing extinction die the results are interpreted in the context of the paradox of sex an old problem of evolutionary biology

degree dispersion increases the rate of rare events in population networks there is great interest in predicting rare and extreme events in complex systems and in particular understanding the role of network topology in facilitating such events in this letter we show that degree dispersion the fact that the number of local connections in networks varies broadly increases the probability of large rare fluctuations in population networks generically we perform explicit calculations for two canonical and distinct classes of rare events network extinction and switching when the distance to threshold is held constant and hence stochastic effects are fairly compared among networks we show that there is a universal exponential increase in the rate of rare events proportional to the variance of a network s degree distribution over its mean squared

applications of field theoretic renormalization group methods to reaction diffusion problems we review the application of field theoretic renormalization group rg methods to the study of fluctuations in reaction diffusion problems we first investigate the physical origin of universality in these systems before comparing rg methods to other available analytic techniques including exact solutions and smoluchowski type approximations starting from the microscopic reaction diffusion master equation we then pedagogically detail the mapping to a field theory for the single species reaction ka a k we employ this particularly simple but non trivial system to introduce the field theoretic rg tools including the diagrammatic perturbation expansion renormalization and callan symanzik rg flow equation we demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative dc d expansions with respect to the upper critical dimension dc with these basics established we then provide an overview of more sophisticated applications to multiple species reactions disorder effects l evy flights persistence problems and the influence of spatial boundaries we also analyse field theoretic approaches to non equilibrium phase transitions separating active from absorbing states we focus particularly on the generic directed percolation universality class as well as on the most prominent exception to this class even offspring branching and annihilating random walks finally we summarize the state of the field and present our perspective on outstanding problems for the future

two strain competition in quasineutral stochastic disease dynamics we develop a perturbation method for studying quasineutral competition in a broad class of stochastic competition models and apply it to the analysis of fixation of competing strains in two epidemic models the first model is a two strain generalization of the stochastic susceptible infected susceptible sis model here we extend previous results due to parsons and quince theor popul biol 72 468 2007 parsons et al theor popul biol 74 302 2008 and lin kim and doering j stat phys 148 646 2012 the second model a two strain generalization of the stochastic susceptible infected recovered sir model with population turnover has not been studied previously in each of the two models when the basic reproduction numbers of the two strains are identical a system with an infinite population size approaches a point on the deterministic coexistence line cl a straight line of fixed points in the phase space of subpopulation sizes shot noise drives one of the strain populations to fixation and the other to extinction on a time scale proportional to the total population size our perturbation method explicitly tracks the dynamics of the probability distribution of the subpopulations in the vicinity of the cl we argue that whereas the slow strain has a competitive advantage for mathematically typical initial conditions it is the fast strain that is more likely to win in the important situation when a few infectives of both strains are introduced into a susceptible population

classification of phase transitions in reaction diffusion models equilibrium phase transitions are associated with rearrangements of minima of a lagrangian potential treatment of nonequilibrium systems requires doubling of degrees of freedom which may be often interpreted as a transition from the coordinate to the phase space representation as a result one has to deal with the hamiltonian formulation of the field theory instead of the lagrangian one we suggest a classification scheme of phase transitions in reaction diffusion models based on the topology of the phase portraits of corresponding hamiltonians in models with an absorbing state such a topology is fully determined by intersecting curves of zero energy we identify four families of topologically distinct classes of phase portraits stable upon renormalization group transformations

generalizations of the fkkm theorem and the ky fan minimax inequality with applications to maximal elements price equilibrium and complementarity abstract this paper generalizes the fan knaster kuratowski mazurkiewicz fkkm theorem of ky fan game theory and related topics pp 151 156 north holland amsterdam 1979 and math ann 266 1984 519 537 and the ky fan minimax inequality by introducing a class of the generalized closedness and continuity conditions which are called the transfer closedness and transfer continuities we then apply these results to prove the existence of maximal elements of binary relations under very weak assumptions we also prove the existence of price equilibrium and the complementarity problem without the continuity assumptions thus our results generalize many of the existence theorems in the literature

extinction rates for fluctuation induced metastabilities a real space wkb approach the extinction of a single species due to demographic stochasticity is analyzed the discrete nature of the individual agents and the poissonian noise related to the birth death processes result in local extinction of a metastable population as the system hits the absorbing state the fokker planck formulation of that problem fails to capture the statistics of large deviations from the metastable state while approximations appropriate close to the absorbing state become in general invalid as the population becomes large to connect these two regimes a real space wkb method based on the master equation is presented and is shown to yield an excellent approximation for the decay rate and the extreme events statistics all the way down to the absorbing state the details of the underlying microscopic process smeared out in a mean field treatment are shown to be crucial for an exact determination of the extinction exponent this general scheme is shown to reproduce the known results in the field to yield new corollaries and to fit quite precisely the numerical solutions moreover it allows for systematic improvement via a series expansion where the small parameter is the inverse of the number of individuals in the metastable state

fluctuations of current in nonstationary diffusive lattice gases we employ the macroscopic fluctuation theory to study fluctuations of integrated current in one dimensional lattice gases with a steplike initial density profile we analytically determine the variance of the current fluctuations for a class of diffusive processes with a density independent diffusion coefficient our calculations rely on a perturbation theory around the noiseless hydrodynamic solution we consider both quenched and annealed types of averaging the initial condition is allowed to fluctuate in the latter situation the general results for the variance are specialized to a few interesting models including the symmetric exclusion process and the kipnis marchioro presutti model kipnis marchioro and presutti j stat phys 27 65 1982 we also probe large deviations of the current for the symmetric exclusion process this is done by numerically solving the governing equations of the macroscopic fluctuation theory using an efficient iteration algorithm

demographic stochasticity and extinction in populations with allee effect we study simple stochastic scenarios based on birth and death markovian processes that describe populations with the allee effect to account for the role of demographic stochasticity in the mean field deterministic limit we recover well known deterministic evolution equations widely employed in population ecology the mean time to extinction is in general obtained by the wentzel kramers brillouin wkb approximation for populations with the strong and weak allee effects an exact solution for the mean time to extinction can be found via a recursive equation for special cases of the stochastic dynamics we study the conditions for the validity of the wkb solution and analyze the boundary between the weak and strong allee effect by comparing exact solutions with numerical simulations

spatiotemporal dynamics of continuum neural fields we survey recent analytical approaches to studying the spatiotemporal dynamics of continuum neural fields neural fields model the large scale dynamics of spatially structured biological neural networks in terms of nonlinear integrodifferential equations whose associated integral kernels represent the spatial distribution of neuronal synaptic connections they provide an important example of spatially extended excitable systems with nonlocal interactions and exhibit a wide range of spatially coherent dynamics including traveling waves oscillations and turing like patterns

epidemic models and percolation the authors argue that local epidemic models with immunisation are in the same universality class as percolation cluster growth models and show that the static exponents are equal to all orders in the epsilon expansion they calculate the dynamic exponent nu t 1 1 28 epsilon o epsilon 2 and relate this to other exponents involving the chemical distance

some random approximation theorems with applications fan 8 proved a very interesting theorem which provides a tool to study xed point theory in connection with best approximation various aspects of that theorem have been studied by fan 9 ha 10 singh and watson 22 lin 14 lin and yen 16 and many others fan s theorem has been of great importance in nonlinear analysis approximation theory game theory and minimax theorems the study of random approximations and random xed points have been a very active area of research in probabilistic functional analysis in the last decade see sehgal and waters 21 sehgal and singh 20 papageorgiou 18 lin 15 xu 25 tan and yuan 23 24 and beg and shahzad 1 4 most random xed point theorems in banach spaces deal with condensing or nonexpansive random operators what about the random operators which are neither of the above cases the interesting case would be a 1 set contractive random operator the class of 1 set contractive random operators includes condensing and nonexpansive random operators besides it also includes other important random operators such as semicontractive random operators and lane locally almost nonexpansive random operators recently beg and shahzad 3 studied these random operators in details and gave many results regarding random approximations and random

an iterative action minimizing method for computing optimal paths in stochastic dynamical systems abstract we present a numerical method for computing optimal transition pathways and transition rates in systems of stochastic differential equations sdes in particular we compute the most probable transition path of stochastic equations by minimizing the effective action in a corresponding deterministic hamiltonian system the numerical method presented here involves using an iterative scheme for solving a two point boundary value problem for the hamiltonian system we validate our method by applying it to both continuous stochastic systems such as nonlinear oscillators governed by the duffing equation and finite discrete systems such as epidemic problems which are governed by a set of master equations furthermore we demonstrate that this method is capable of dealing with stochastic systems of delay differential equations

some deterministic and random vector equilibrium problems in this paper some generalized concepts about semicontinuity and convexity for vector valued bifunctions are introduced a new existence theorem for vector equilibrium problems is proved further a random version of this result is considered applications to cone saddle points random vector optimization random vector variational inequalities and random vector best approximation problems are finally given

dimensional reduction of the master equation for stochastic chemical networks the reduced multiplane method chemical reaction networks which exhibit strong fluctuations are common in microscopic systems in which reactants appear in low copy numbers the analysis of these networks requires stochastic methods which come in two forms direct integration of the master equation and monte carlo simulations the master equation becomes infeasible for large networks because the number of equations increases exponentially with the number of reactive species monte carlo methods which are more efficient in integrating over the exponentially large phase space also become impractical due to the large amounts of noisy data that need to be stored and analyzed the recently introduced multiplane method a lipshtat and o biham phys rev lett 93 170601 2004 is an efficient framework for the stochastic analysis of large reaction networks it is a dimensional reduction method based on the master equation which provides a dramatic reduction in the number of equations without compromising the accuracy of the results the reduction is achieved by breaking the network into a set of maximal fully connected subnetworks maximal cliques a separate master equation is written for the reduced probability distribution associated with each clique with suitable coupling terms between them this method is highly efficient in the case of sparse networks in which the maximal cliques tend to be small however in dense networks some of the cliques may be rather large and the dimensional reduction is not as effective furthermore the derivation of the multiplane equations from the master equation is tedious and difficult here we present the reduced multiplane method in which the maximal cliques are broken down to the fundamental two vertex cliques the number of equations is further reduced making the method highly efficient even for dense networks moreover the equations take a simpler form which can be easily constructed using a diagrammatic procedure for any desired network architecture it is shown that the method provides accurate results for the population sizes of the reactive species and their reaction rates

n fold supersymmetry and quasi solvability associated with x2 laguerre polynomials we construct a new family of quasi solvable and n fold supersymmetric quantum systems where each hamiltonian preserves an exceptional polynomial subspace of codimension 2 we show that the family includes as a particular case the recently reported rational radial oscillator potential whose eigenfunctions are expressed in terms of the x2 laguerre polynomials of the second kind in addition we find that the two kinds of the x2 laguerre polynomials are ingeniously connected with each other by the n fold supercharge

stochastic dynamics and logistic population growth the verhulst model is probably the best known macroscopic rate equation in population ecology it depends on two parameters the intrinsic growth rate and the carrying capacity these parameters can be estimated for different populations and are related to the reproductive fitness and the competition for limited resources respectively we investigate analytically and numerically the simplest possible microscopic scenarios that give rise to the logistic equation in the deterministic mean field limit we provide a definition of the two parameters of the verhulst equation in terms of microscopic parameters in addition we derive the conditions for extinction or persistence of the population by employing either the momentum space spectral theory or the real space wentzel kramers brillouin approximation to determine the probability distribution function and the mean time to extinction of the population our analytical results agree well with numerical simulations

intrinsic noise in systems with switching environments we study individual based dynamics in finite populations subject to randomly switching environmental conditions these are inspired by models in which genes transition between on and off states regulating underlying protein dynamics similarly switches between environmental states are relevant in bacterial populations and in models of epidemic spread existing piecewise deterministic markov process approaches focus on the deterministic limit of the population dynamics while retaining the randomness of the switching here we go beyond this approximation and explicitly include effects of intrinsic stochasticity at the level of the linear noise approximation specifically we derive the stationary distributions of a number of model systems in good agreement with simulations this improves existing approaches which are limited to the regimes of fast and slow switching

taming the diffusion approximation through a controlling factor wkb method the diffusion approximation da is widely used in the analysis of stochastic population dynamics from population genetics to ecology and evolution the da is an uncontrolled approximation that assumes the smoothness of the calculated quantity over the relevant state space and fails when this property is not satisfied this failure becomes severe in situations where the direction of selection switches sign here we employ the wkb wentzel kramers brillouin large deviations method which requires only the logarithm of a given quantity to be smooth over its state space combining the wkb scheme with asymptotic matching techniques we show how to derive the diffusion approximation in a controlled manner and how to produce better approximations applicable for much wider regimes of parameters we also introduce a scalable independent of population size wkb based numerical technique the method is applied to a central problem in population genetics and evolution finding the chance of ultimate fixation in a zero sum two types competition

quasi exact solvability and the direct approach to invariant subspaces we propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of this approach is used here to construct new exactly solvable and quasi exactly solvable quantum hamiltonians on the line which are not lie algebraic it is also applied to generate potentials with multiple algebraic sectors we discuss two illustrative examples of these two applications we show that the generalized lame potential possesses four algebraic sectors and describe a quasi exactly solvable deformation of the morse potential which is not lie algebraic

extinction of an infectious disease a large fluctuation in a nonequilibrium system we develop a theory of first passage processes in stochastic nonequilibrium systems of birth death type using two closely related epidemiological models as examples our method employs the probability generating function technique in conjunction with the eikonal approximation in this way the problem is reduced to finding the optimal path to extinction a heteroclinic trajectory of an effective multidimensional classical hamiltonian system we compute this trajectory and mean extinction time of the disease numerically and uncover a nonmonotone spiral path to extinction of a disease we also obtain analytical results close to a bifurcation point where the problem is described by a hamiltonian previously identified in one species population models

evolutionary dynamics with fluctuating population sizes and strong mutualism game theory ideas provide a useful framework for studying evolutionary dynamics in a well mixed environment this approach however typically enforces a strictly fixed overall population size deemphasizing natural growth processes we study a competitive lotka volterra model with number fluctuations that accounts for natural population growth and encompasses interaction scenarios typical of evolutionary games we show that in an appropriate limit the model describes standard evolutionary games with both genetic drift and overall population size fluctuations however there are also regimes where a varying population size can strongly influence the evolutionary dynamics we focus on the strong mutualism scenario and demonstrate that standard evolutionary game theory fails to describe our simulation results we then analytically and numerically determine fixation probabilities as well as mean fixation times using matched asymptotic expansions taking into account the population size degree of freedom these results elucidate the interplay between population dynamics and evolutionary dynamics in well mixed systems

type b 3 fold supersymmetry and non polynomial invariant subspaces we obtain the most general type b 3 fold supersymmetry by solving directly the intertwining relation we then show that it is a necessary and sufficient condition for a second order linear differential operator to have three linearly independent local analytic solutions we find that there are eight linearly independent non trivial linear differential operators of this kind as a by product we find new quasi solvable second order operators preserving a monomial or polynomial subspace one in type b two in type c and four in type x2 all of which have been missed in the existing literature in addition we show that type a type b and type c 3 fold supersymmetries are connected continuously via one parameter a few new quasi solvable models are also presented

disease extinction in the presence of random vaccination we investigate disease extinction in an epidemic model described by a birth death process we show that in the absence of vaccination the effective entropic barrier for extinction displays scaling with the distance to the bifurcation point with an unusual critical exponent even a comparatively weak poisson distributed random vaccination leads to an exponential increase in the extinction rate with the exponent that strongly depends on the vaccination parameters

novel exponents control the quasi deterministic limit of the extinction transition the quasi deterministic limit of the generic extinction transition is considered within the framework of standard epidemiological models the susceptible infected susceptible sis model is known to exhibit a transition from extinction to spreading as the infectivity is increased described by the directed percolation equivalence class we find that the distance from the transition point and the prefactor controlling the divergence of the perpendicular correlation length both scale with the local population size n with two novel universal exponents different exponents characterize the large n behavior of the susceptible infected recovered sir model which belongs to the dynamic percolation class extensive numerical studies in a range of systems lead to the conjecture that these characteristics are generic and may be used in order to classify the high density limit of any stochastic process on the edge of extinction

nonlinear pseudo supersymmetry in the framework of fold supersymmetry we recall the importance of recognizing the different mathematical nature of various concepts relating to symmetric quantum theories after clarifying the relation between supersymmetry and pseudo supersymmetry we prove generically that nonlinear pseudo supersymmetry recently proposed by sinha and roy is just a special case of fold supersymmetry in particular we show that all the models constructed by these authors have type a two fold supersymmetry furthermore we prove that an arbitrary one body quantum hamiltonian which admits two local solutions in closed form belongs to type a two fold supersymmetry irrespective of whether or not it is hermitian symmetric pseudo hermitian and so on

on the time to extinction in recurrent epidemics summary an approximation is derived for the expected time to extinction in a stochastic model for recurrent epidemics numerical illustrations indicate that the approximation is crude but that it has the correct order of magnitude the quasi stationary distribution plays an important role in the derivation approximations for the critical community size and for the persistence threshold are derived comments are made on the classical study by bartlett 1956 1960

dynamics of the chemical master equation a strip of chains of equations in d dimensional space we investigate the multichain version of the chemical master equation when there are transitions between different states inside the long chains as well as transitions between a few different chains in the discrete version such a model can describe the connected diffusion processes with jumps between different types we apply the hamilton jacobi equation to solve some aspects of the model we derive exact in the limit of infinite number of particles results for the dynamic of the maximum of the distribution and the variance of distribution

rare event statistics in reaction diffusion systems we present an efficient method to calculate probabilities of large deviations from the typical behavior rare events in reaction diffusion systems this method is based on a semiclassical treatment of an underlying quantum hamiltonian encoding the system s evolution to this end we formulate the corresponding canonical dynamical system and investigate its phase portrait this method is presented for a number of pedagogical examples

population extinction risk in the aftermath of a catastrophic event we investigate how a catastrophic event modeled as a temporary fall of the reproduction rate increases the extinction probability of an isolated self regulated stochastic population using a variant of the verhulst logistic model as an example we combine the probability generating function technique with an eikonal approximation to evaluate the exponentially large increase in the extinction probability caused by the catastrophe this quantity is given by the eikonal action computed over the optimal path instanton of an effective classical hamiltonian system with a time dependent hamiltonian for a general catastrophe the eikonal equations can be solved numerically for simple models of catastrophic events analytic solutions can be obtained one such solution becomes quite simple close to the bifurcation point of the verhulst model the eikonal results for the increase in the extinction probability caused by a catastrophe agree well with numerical solutions of the master equation

immigration extinction dynamics of stochastic populations how high should be the rate of immigration into a stochastic population in order to significantly reduce the probability of observing the population become extinct is there any relation between the population size distributions with and without immigration under what conditions can one justify the simple patch occupancy models which ignore the population distribution and its dynamics in a patch and treat a patch simply as either occupied or empty we answer these questions by exactly solving a simple stochastic model obtained by adding a steady immigration to a variant of the verhulst model a prototypical model of an isolated stochastic population

invariant modules and the reduction of nonlinear partial differential equations to dynamical systems abstract we completely characterize all nonlinear partial differential equations leaving a given finite dimensional vector space of analytic functions invariant existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinary differential equations and provides a nonlinear counterpart to quasi exactly solvable quantum hamiltonians these results rely on a useful extension of the classical wronskian determinant condition for linear independence of functions in addition new approaches to the characterization of the annihilating differential operators for spaces of analytic functions are presented

stochastic analysis of complex reaction networks using binomial moment equations the stochastic analysis of complex reaction networks is a difficult problem because the number of microscopic states in such systems increases exponentially with the number of reactive species direct integration of the master equation is thus infeasible and is most often replaced by monte carlo simulations while monte carlo simulations are a highly effective tool equation based formulations are more amenable to analytical treatment and may provide deeper insight into the dynamics of the network here we present a highly efficient equation based method for the analysis of stochastic reaction networks the method is based on the recently introduced binomial moment equations barzel and biham phys rev lett 106 150602 2011 the binomial moments are linear combinations of the ordinary moments of the probability distribution function of the population sizes of the interacting species they capture the essential combinatorics of the reaction processes reflecting their stoichiometric structure this leads to a simple and transparent form of the equations and allows a highly efficient and surprisingly simple truncation scheme unlike ordinary moment equations in which the inclusion of high order moments is prohibitively complicated the binomial moment equations can be easily constructed up to any desired order the result is a set of equations that enables the stochastic analysis of complex reaction networks under a broad range of conditions the number of equations is dramatically reduced from the exponential proliferation of the master equation to a polynomial and often quadratic dependence on the number of reactive species in the binomial moment equations the aim of this paper is twofold to present a complete derivation of the binomial moment equations to demonstrate the applicability of the moment equations for a representative set of example networks in which stochastic effects play an important role

extinction in the lotka volterra model birth death processes often exhibit an oscillatory behavior we investigate a particular case where the oscillation cycles are marginally stable on the mean field level an iconic example of such a system is the lotka volterra model of predator prey interaction fluctuation effects due to discreteness of the populations destroy the mean field stability and eventually drive the system toward extinction of one or both species we show that the corresponding extinction time scales as a certain power law of the population sizes this behavior should be contrasted with the extinction of models stable in the mean field approximation in the latter case the extinction time scales exponentially with size

spatial structure in diffusion limited two particle reactions we analyze the limiting behavior of the densities a t and b t and the random spatial structure r a t b t for the diffusion controlled chemical reaction a b inert for equal initial densities b 0 b 0 there is a change in behavior fromd 4 where a t b t c td 4 tod 4 where a t b t c t ast the termc depends on the initial densities and changes withd there is a corresponding change in the spatial structure ind 4 both particle types are after large times present locally in concentrations not depending on type or location ind 4 both particle types are present locally but with random concentrations and the process tends to a limit

colonization of a territory by a stochastic population under a strong allee effect and a low immigration pressure we study the dynamics of colonization of a territory by a stochastic population at low immigration pressure we assume a sufficiently strong allee effect that introduces in deterministic theory a large critical population size for colonization at low immigration rates the average precolonization population size is small thus invalidating the wkb approximation to the master equation we circumvent this difficulty by deriving an exact zero flux solution of the master equation and matching it with an approximate nonzero flux solution of the pertinent fokker planck equation in a small region around the critical population size this procedure provides an accurate evaluation of the quasistationary probability distribution of population sizes in the precolonization state and of the mean time to colonization for a wide range of immigration rates at sufficiently high immigration rates our results agree with wkb results obtained previously at low immigration rates the results can be very different

extinction of metastable stochastic populations we investigate the phenomenon of extinction of a long lived self regulating stochastic population caused by intrinsic demographic noise extinction typically occurs via one of two scenarios depending on whether the absorbing state n 0 is a repelling scenario a or attracting scenario b point of the deterministic rate equation in scenario a the metastable stochastic population resides in the vicinity of an attracting fixed point next to the repelling point n 0 in scenario b there is an intermediate repelling point n n 1 between the attracting point n 0 and another attracting point n n 2 in the vicinity of which the metastable population resides the crux of the theory is a dissipative variant of wkb wentzel kramers brillouin approximation which assumes that the typical population size in the metastable state is large starting from the master equation we calculate the quasistationary probability distribution of the population sizes and the exponentially long mean time to extinction for each of the two scenarios when necessary the wkb approximation is complemented i by a recursive solution of the quasistationary master equation at small n and ii by the van kampen system size expansion valid near the fixed points of the deterministic rate equation the theory yields both entropic barriers to extinction and pre exponential factors and holds for a general set of multistep processes when detailed balance is broken the results simplify considerably for single step processes and near the characteristic bifurcations of scenarios a and b

chemical langevin equation a path integral view of gillespie s derivation in 2000 gillespie rehabilitated the chemical langevin equation cle by describing two conditions that must be satisfied for it to yield a valid approximation of the chemical master equation cme in this work we construct an original path integral description of the cme and show how applying gillespie s two conditions to it directly leads to a path integral equivalent to the cle we compare this approach to the path integral equivalent of a large system size derivation and show that they are qualitatively different in particular both approaches involve converting many sums into many integrals and the difference between the two methods is essentially the difference between using the euler maclaurin formula and using riemann sums our results shed light on how path integrals can be used to conceptualize coarse graining biochemical systems and are readily generalizable

self consistent formulations for stochastic nonlinear neuronal dynamics neural dynamics is often investigated with tools from bifurcation theory however many neuron models are stochastic mimicking fluctuations in the input from unknown parts of the brain or the spiking nature of signals noise changes the dynamics with respect to the deterministic model in particular classical bifurcation theory cannot be applied we formulate the stochastic neuron dynamics in the martin siggia rose de dominicis janssen msrdj formalism and present the fluctuation expansion of the effective action and the functional renormalization group frg as two systematic ways to incorporate corrections to the mean dynamics and time dependent statistics due to fluctuations in the presence of nonlinear neuronal gain to formulate self consistency equations we derive a fundamental link between the effective action in the onsager machlup om formalism which allows the study of phase transitions and the msrdj effective action which is computationally advantageous these results in particular allow the derivation of an om effective action for systems with non gaussian noise this approach naturally leads to effective deterministic equations for the first moment of the stochastic system they explain how nonlinearities and noise cooperate to produce memory effects moreover the msrdj formulation yields an effective linear system that has identical power spectra and linear response starting from the better known loopwise approximation we then discuss the use of the frg as a method to obtain self consistency beyond the mean we present a new efficient truncation scheme for the hierarchy of flow equations for the vertex functions by adapting the blaizot m endez and wschebor approximation from the derivative expansion to the vertex expansion the methods are presented by means of the simplest possible example of a stochastic differential equation that has generic features of neuronal dynamics

normalizability of one dimensional quasi exactly solvable schrodinger operators we completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasi exactly solvable schrodinger operator on the line methods from classical invariant theory are employed to provide a complete list of canonical forms for normalizable quasi exactly solvable hamiltonians and explicit normalizability conditions in general coordinate systems

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

persistence of instanton connections in chemical reactions with time dependent rates the evolution of a system of chemical reactions can be studied in the eikonal approximation by means of a hamiltonian dynamical system the fixed points of this dynamical system represent the different states in which the chemical system can be found and the connections among them represent instantons or optimal paths linking these states we study the relation between the phase portrait of the hamiltonian system representing a set of chemical reactions with constant rates and the corresponding system when these rates vary in time we show that the topology of the phase space is robust for small time dependent perturbations in concrete examples and state general results when possible this robustness allows us to apply some of the conclusions on the qualitative behavior of the autonomous system to the time dependent situation

sharp asymptotics for the quasi stationary distribution of birth and death processes we study a general class of birth and death processes with state space mathbb n that describes the size of a population going to extinction with probability one this class contains the logistic case the scale of the population is measured in terms of a carrying capacity k when k is large the process is expected to stay close to its deterministic equilibrium during a long time but ultimately goes extinct our aim is to quantify the behavior of the process and the mean time to extinction in the quasi stationary distribution as a function of k for large k we also give a quantitative description of this quasi stationary distribution it turns out to be close to a gaussian distribution centered about the deterministic long time equilibrium when k is large our analysis relies on precise estimates of the maximal eigenvalue of the corresponding eigenvector and of the spectral gap of a self adjoint operator associated with the semigroup of the process

stochastic neural field theory and the system size expansion we analyze a master equation formulation of stochastic neurodynamics for a network of synaptically coupled homogeneous neuronal populations each consisting of n identical neurons the state of the network is specified by the fraction of active or spiking neurons in each population and transition rates are chosen so that in the thermodynamic or deterministic limit n rightarrow infty we recover standard activity based or voltage based rate models we derive the lowest order corrections to these rate equations for large but finite n using two different approximation schemes one based on the van kampen system size expansion and the other based on path integral methods both methods yield the same series expansion of the moment equations which at mathcal o 1 n can be truncated to form a closed system of equations for the first and second order moments taking a continuum limit of the moment equations while keeping the system size n fixed generates a system of integrodifferential equations for the

switching path distribution in multidimensional systems we explore the distribution of paths followed in fluctuation induced switching between coexisting stable states we introduce a quantitative characteristic of the path distribution in phase space that does not require a priori knowledge of system dynamics the theory of the distribution is developed and its direct measurement is performed in a micromechanical oscillator driven into parametric resonance the experimental and theoretical results on the shape and position of the distribution are in excellent agreement with no adjustable parameters in addition the experiment provides the first demonstration of the lack of time reversal symmetry in switching of systems far from thermal equilibrium the results open the possibility of efficient control of the switching probability based on the measured narrow path distribution

giant disparity and a dynamical phase transition in large deviations of the time averaged size of stochastic populations we study large deviations of the time averaged size of stochastic populations described by a continuous time markov jump process when the expected population size n in the steady state is large the large deviation function ldf of the time averaged population size can be evaluated by using a wentzel kramers brillouin wkb method applied directly to the master equation for the markov process for a class of models that we identify the direct wkb method predicts a giant disparity between the probabilities of observing an unusually small and an unusually large values of the time averaged population size the disparity results from a qualitative change in the optimal trajectory of the underlying classical mechanics problem the direct wkb method also predicts in the limit of n a singularity of the ldf which can be interpreted as a second order dynamical phase transition the transition is smoothed at finite n but the giant disparity remains the smoothing effect is captured by the van kampen system size expansion of the exact master equation near the attracting fixed point of the underlying deterministic model we describe the giant disparity at finite n by developing a different variant of wkb method which is applied in conjunction with the donsker varadhan large deviation formalism and involves subleading order calculations in 1 n

exact results in the large system size limit for the dynamics of the chemical master equation a one dimensional chain of equations cme we found exact analytical expressions in large system size limit for the probability distribution including explicit expression for the dynamics of variance of distribution we also give the solution for some simple cases of the model with time dependent rates we derived the results of the van kampen method from the hje approach using a special ansatz using the van kampen method we give a system of ordinary differential equations odes to define the variance in a two dimensional case we performed numerics for the cme with stationary noise we give analytical criteria for the disappearance of bistability in the case of stationary noise in one dimensional cmes

phase diagram for logistic systems under bounded stochasticity extinction is the ultimate absorbing state of any stochastic birth death process hence the time to extinction is an important characteristic of any natural population here we consider logistic and logisticlike systems under the combined effect of demographic and bounded environmental stochasticity three phases are identified an inactive phase where the mean time to extinction t increases logarithmically with the initial population size an active phase where t grows exponentially with the carrying capacity n and a temporal griffiths phase with a power law relationship between t and n the system supports an exponential phase only when the noise is bounded in which case the continuum diffusion approximation breaks down within the griffiths phase this breakdown is associated with a crossover between qualitatively different survival statistics and decline modes to study the power law phase we present a new wkb scheme which is applicable both in the diffusive and in the nondiffusive regime

wilson renormalization of a reaction diffusion process abstract healthy and sick individuals a and b particles diffuse independently with diffusion constants da and db sick individuals upon encounter infect healthy ones at rate k but may also spontaneously recover at rate 1 the propagation of the epidemic therefore couples to the fluctuations in the total population density global extinction occurs below a critical value c of the spatially averaged total density the epidemic evolves as the diffusion reaction decay process a b 2b b a for which we write down the field theory the stationary state properties of this theory when da db were obtained by kree et al the critical behavior for da db remains unsolved

finite genome length corrections for the mean fitness and gene probabilities in evolution models using the hamilton jacobi equation approach to study genomes of length l we obtain 1 l corrections for the steady state population distributions and mean fitness functions for horizontal gene transfer model as well as for the diploid evolution model with general fitness landscapes our numerical solutions confirm the obtained analytic equations our method could be applied to the general case of nonlinear markov models

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

population dynamics in a changing environment random versus periodic switching environmental changes greatly influence the evolution of populations here we study the dynamics of a population of two strains one growing slightly faster than the other competing for resources in a time varying binary environment modeled by a carrying capacity switching either randomly or periodically between states of abundance and scarcity the population dynamics is characterized by demographic noise birth and death events coupled to a varying environment we elucidate the similarities and differences of the evolution subject to a stochastically and periodically varying environment importantly the population size distribution is generally found to be broader under intermediate and fast random switching than under periodic variations which results in markedly different asymptotic behaviors between the fixation probability of random and periodic switching we also determine the detailed conditions under which the fixation probability of the slow strain is maximal

population extinction in a time modulated environment the extinction time of an isolated population can be exponentially reduced by a periodic modulation of its environment we investigate this effect using as an example a stochastic branching annihilation process with a time dependent branching rate the population extinction is treated in eikonal approximation where it is described as an instanton trajectory of a proper reaction hamiltonian the modulation of the environment perturbs this trajectory and synchronizes it with the modulation phase we calculate the corresponding change in the action along the instanton using perturbation techniques supported by numerical calculations the techniques include a first order theory with respect to the modulation amplitude a second order theory in the spirit of the kapitsa pendulum effect and adiabatic theory valid for low modulation frequencies

population switching under a time varying environment we study the switching dynamics of a stochastic population subjected to a deterministically time varying environment our approach is demonstrated on a problem of population establishment which is important in ecology at the deterministic level the model we study gives rise to a critical population size beyond which the system experiences establishment notably the latter has been shown to be strongly influenced by the interplay between demographic and environmental variations here we consider two prototypical examples of a time varying environment a temporary change in the environment and a periodically varying environment by employing a semiclassical approximation we compute within exponential accuracy the change in the establishment probability and mean establishment time of the population due to the environmental variability our analytical results are verified by using a modified gillespie algorithm which accounts for explicitly time dependent reaction rates importantly our theoretical approach can also be useful in studying switching dynamics in gene regulatory networks under external variations

the analytical dynamics of the finite population evolution games abstract we study the dynamics of a finite number of replicators with different strategies in evolutionary games using the moment closure approximation for the master equation and the hamilton jacobi equation approach these methods give finite population corrections to the results of the replicator equation the model under investigation has two strategies n overall replicators with constant payoff matrix and the moran process as the update mechanism our results are compared with the results of the deterministic replicator equation and the results of numerical stochastic calculations

fermionic field theory for directed percolation in 1 1 dimensions we formulate directed percolation in 1 1 dimensions in the language of a reaction diffusion process with exclusion taking place in one space dimension we map the master equation that describes the dynamics of the system onto a quantum spin chain problem from there we build an interacting fermionic field theory of a new type we study the resulting theory using renormalization group techniques this yields numerical estimates for the critical exponents and provides a new alternative analytic systematic procedure to study low dimensional directed percolation

fluctuation statistics in networks a stochastic path integral approach we investigate the statistics of fluctuations in a classical stochastic network of nodes joined by connectors the nodes carry generalized charge that may be randomly transferred from one node to another our goal is to find the time evolution of the probability distribution of charges in the network the building blocks of our theoretical approach are 1 known probability distributions for the connector currents 2 physical constraints such as local charge conservation and 3 a time scale separation between the slow charge dynamics of the nodes and the fast current fluctuations of the connectors we integrate out fast current fluctuations and derive a stochastic path integral representation of the evolution operator for the slow charges the statistics of charge fluctuations may be found from the saddle point approximation of the action once the probability distributions on the discrete network have been studied the continuum limit is taken to obtain a statistical field theory we find a correspondence between the diffusive field theory and a langevin equation with gaussian noise sources leading nevertheless to nontrivial fluctuation statistics to complete our theory we demonstrate that the cascade diagrammatics recently introduced by nagaev naturally follows from the stochastic path integral by generalizing the principle of minimal correlations we extend the diagrammatics to calculate current correlation functions for an arbitrary network one primary application of this formalism is that of full counting statistics fcs the motivation for why it was developed in the first place we stress however that the formalism is suitable for general classical stochastic problems as an alternative approach to the traditional master equation or doi peliti technique the formalism is illustrated with several examples both instantaneous and time averaged charge fluctuation statistics in a mesoscopic chaotic cavity as well as the fcs and new results for a generalized diffusive wire

spectral theory of metastability and extinction in a branching annihilation reaction we apply the spectral method recently developed by the authors to calculate the statistics of a reaction limited multistep birth death process or chemical reaction that includes as elementary steps branching a 2a and annihilation 2a 0 the spectral method employs the generating function technique in conjunction with the sturm liouville theory of linear differential operators we focus on the limit when the branching rate is much higher than the annihilation rate and obtain accurate analytical results for the complete probability distribution including large deviations of the metastable long lived state and for the extinction time statistics the analytical results are in very good agreement with numerical calculations furthermore we use this example to settle the issue of the lacking boundary condition in the spectral formulation

local scale invariances in the bosonic contact and pair contact processes local scale invariance for ageing systems without detailed balance is tested through studying the dynamical symmetries of the critical bosonic contact process and the critical bosonic pair contact process their field theoretical actions can be split into a schrodinger invariant term and a pure noise term it is shown that the two time response and correlation functions are reducible to certain multipoint response functions which depend only on the schrodinger invariant part of the action for the bosonic contact process the representation of the schrodinger group can be derived from the free diffusion equation whereas for the bosonic pair contact process a new representation of the schrodinger group related to a nonlinear schrodinger equation with dimensionful couplings is constructed the resulting predictions of local scale invariance for the two time responses and correlators are completely consistent with the exactly known results in both models

theory of time averaged neutral dynamics with environmental stochasticity competition is the main driver of population dynamics which shapes the genetic composition of populations and the assembly of ecological communities neutral models assume that all the individuals are equivalent and that the dynamics is governed by demographic shot noise with a steady state species abundance distribution sad that reflects a mutation extinction equilibrium recently many empirical and theoretical studies emphasized the importance of environmental variations that affect coherently the relative fitness of entire populations here we consider two generic time averaged neutral models in both the relative fitness of each species fluctuates independently in time but its mean is zero the first model a describes a system with local competition and linear fitness dependence of the birth death rates while in the second model b the competition is global and the fitness dependence is nonlinear due to this nonlinearity model b admits a noise induced stabilization mechanism that facilitates the invasion of new mutants a self consistent mean field approach is used to reduce the multispecies problem to two species dynamics and the large n asymptotics of the emerging set of fokker planck equations is presented and solved our analytic expressions are shown to fit the sads obtained from extensive monte carlo simulations and from numerical solutions of the corresponding master equations

interfacial velocity corrections due to multiplicative noise the problem of velocity selection for reaction fronts has been intensively investigated leading to the successful marginal stability ms approach for propagation into an unstable state because the front velocity is controlled by the leading edge which perforce has low density it is interesting to study the role that finite particle number fluctuations have on this picture here we use the well known mapping of discrete markov processes to stochastic differential equations and focus on the front velocity in the simple a a ensuremath leftrightarrows a system our results are consistent with a recent heuristic proposal that v mathrm ms ensuremath v ensuremath sim 1 mathrm ln 2 n

noise driven unlimited population growth demographic noise causes unlimited population growth in a broad class of models which without noise would predict a stable finite population we study this effect on the example of a stochastic birth death model which includes immigration binary reproduction and death the unlimited population growth proceeds as an exponentially slow decay of a metastable probability distribution mpd of the population we develop a systematic wkb theory complemented by the van kampen system size expansion for the mpd and for the decay time important signatures of the mpd are a power law tail such that all the distribution moments except the zeroth one diverge and the presence in the solution of two different wkb modes

enhancement of large fluctuations to extinction in adaptive networks during an epidemic individual nodes in a network may adapt their connections to reduce the chance of infection a common form of adaption is avoidance rewiring where a noninfected node breaks a connection to an infected neighbor and forms a new connection to another noninfected node here we explore the effects of such adaptivity on stochastic fluctuations in the susceptible infected susceptible model focusing on the largest fluctuations that result in extinction of infection using techniques from large deviation theory combined with a measurement of heterogeneity in the susceptible degree distribution at the endemic state we are able to predict and analyze large fluctuations and extinction in adaptive networks we find that in the limit of small rewiring there is a sharp exponential reduction in mean extinction times compared to the case of zero adaption furthermore we find an exponential enhancement in the probability of large fluctuations with increased rewiring rate even when holding the average number of infected nodes constant

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

gl 3 c invariance of type b 3 fold supersymmetric systems type b 3 fold supersymmetry is a necessary and sufficient condition for a quantum hamiltonian to admit three linearly independent local solutions in closed form we show that any such a system is invariant under gl 3 c homogeneous linear transformations in particular we prove explicitly that the parameter space is transformed as an adjoint representation of it and that every coefficient of the characteristic polynomial appeared in 3 fold superalgebra is algebraic invariants in the type a case it includes as a subgroup the gl 2 c projective transformation studied in the literature we argue that any n fold supersymmetric system has a gl n c invariance for an arbitrary integral n

population extinction under bursty reproduction in a time modulated environment in recent years nondemographic variability has been shown to greatly affect dynamics of stochastic populations for example nondemographic noise in the form of a bursty reproduction process with an a priori unknown burst size or environmental variability in the form of time varying reaction rates have been separately found to dramatically impact the extinction risk of isolated populations in this work we investigate the extinction risk of an isolated population under the combined influence of these two types of nondemographic variation using the so called momentum space wentzel kramers brillouin wkb approach and accounting for the explicit time dependence in the reaction rates we arrive at a set of time dependent hamilton equations to this end we evaluate the population s extinction risk by finding the instanton of the time perturbed hamiltonian numerically whereas analytical expressions are presented in particular limits using various perturbation techniques we focus on two classes of time varying environments periodically varying rates corresponding to seasonal effects and a sudden decrease in the birth rate corresponding to a catastrophe all our theoretical results are tested against numerical monte carlo simulations with time dependent rates and also against a numerical solution of the corresponding time dependent hamilton equations

field theory of branching and annihilating random walks we develop a systematic analytic approach to the problem of branching and annihilating random walks equivalent to the diffusion limited reaction processes 2a and a m 1 a where m 1 starting from the master equation a field theoretic representation of the problem is derived and fluctuation effects are taken into account via diagrammatic and renormalization group methods for d 2 the mean field rate equation which predicts an active phase as soon as the branching process is switched on applies qualitatively for both even and odd m but the behavior in lower dimensions is shown to be quite different for these two cases for even m and d near 2 the active phase still appears immediately but with nontrivial crossover exponents which we compute in an expansion in 2 d and with logarithmic corrections in d 2 however there exists a second critical dimension d c 4 3 below which a nontrivial inactive phase emerges with asymptotic behavior characteristic of the pure annihilation process this is confirmed by an exact calculation in d 1 the subsequent transition to the active phase which represents a new nontrivial dynamic universality class is then investigated within a truncated loop expansion which appears to give a correct qualitative picture the model with m 2 is also generalized to n species of particles which provides yet another universality class and which is exactly solvable in the limit n for odd m we show that the fluctuations of the annihilation process are strong enough to create a nontrivial inactive phase for all d 2 in this case the transition to the active phase is in the directed percolation universality class finally we study the modification when the annihilation reaction is 3a when m 0 mod 3 the system is always in its active phase but with logarithmic crossover corrections for d 1 while the other cases should exhibit a directed percolation transition out of a fluctuation driven inactive phase

reducing the extinction risk of stochastic populations via nondemographic noise we consider nondemographic noise in the form of uncertainty in the reaction step size and reveal a dramatic effect this noise may have on the stability of self regulating populations employing the reaction scheme ma ka but allowing e g the product number k to be a priori unknown and sampled from a given distribution we show that such nondemographic noise can greatly reduce the population s extinction risk compared to the fixed k case our analysis is tested against numerical simulations and by using empirical data of different species we argue that certain distributions may be more evolutionary beneficial than others

stochastic foundations in nonlinear density regulation growth in this work we construct individual based models that give rise to the generalized logistic model at the mean field deterministic level and that allow us to interpret the parameters of these models in terms of individual interactions we also study the effect of internal fluctuations on the long time dynamics for the different models that have been widely used in the literature such as the theta logistic and savageau models in particular we determine the conditions for population extinction and calculate the mean time to extinction if the population does not become extinct we obtain analytical expressions for the population abundance distribution our theoretical results are based on wkb theory and the probability generating function formalism and are verified by numerical simulations