two grid method for the two dimensional time dependent schrodinger equation by the finite element method abstract in this paper we construct a backward euler full discrete two grid finite element scheme for the two dimensional time dependent schrodinger equation with this method the solution of the original problem on the fine grid is reduced to the solution of same problem on a much coarser grid together with the solution of two poisson equations on the same fine grid we analyze the error estimate of the standard finite element solution and the two grid solution in the h 1 norm it is shown that the two grid algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h k k 1 finally a numerical experiment indicates that our two grid algorithm is more efficient than the standard finite element method

approximate probabilistic cellular automata for the dynamics of single species populations under discrete logisticlike growth with and without weak allee effects we investigate one dimensional elementary probabilistic cellular automata pca whose dynamics in first order mean field approximation yields discrete logisticlike growth models for a single species unstructured population with nonoverlapping generations beginning with a general six parameter model we find constraints on the transition probabilities of the pca that guarantee that the ensuing approximations make sense in terms of population dynamics and classify the valid combinations thereof several possible models display a negative cubic term that can be interpreted as a weak allee factor we also investigate the conditions under which a one parameter pca derived from the more general six parameter model can generate valid population growth dynamics numerical simulations illustrate the behavior of some of the pca found

urn models replicator processes and random genetic drift to understand the relative importance of natural selection and random genetic drift in finite but growing populations the asymptotic behavior of a class of generalized polya urns is studied using the method of ordinary differential equation ode of particular interest is the replicator process two balls individuals are chosen from an urn the population at random with replacement and balls of the same colors strategies are added or removed according to probabilities that depend only on the colors of the chosen balls under the assumption that the expected number of balls being added always exceeds the expected number of balls being removed whenever balls are in the urn the probability of nonextinction is shown to be positive on the event of nonextinction three results are proven i the number of balls increases asymptotically at a linear rate ii the distribution x n of strategies at the nth update is a noisy cauchy euler approximation to the mean limit ode of the process and iii the limit set of x n is almost surely a connected internally chain recurrent set for the mean limit ode under a stronger set of assumptions it is shown that for any attractor of the mean limit ode there is a positive probability that the limit set for x n lies in this attractor theoretical and numerical estimates for the probabilities of nonextinction and convergence to an attractor suggest that random genetic drift is more likely to overcome natural selection in small populations for which pairwise interactions lead to highly variable outcomes and is less likely to overcome natural selection in large populations with the potential for rapid growth

role of noise in population dynamics cycles noise is an intrinsic feature of population dynamics and plays a crucial role in oscillations called phase forgetting quasicycles by converting damped into sustained oscillations this function of noise becomes evident when considering langevin equations whose deterministic part yields only damped oscillations we formulate here a consistent and systematic approach to population dynamics leading to a fokker planck equation and the associate langevin equations in accordance with this conceptual framework founded on stochastic lattice gas models that describe spatially structured predator prey systems langevin equations in the population densities and predator prey pair density are derived in two stages first a birth and death stochastic process in the space of prey and predator numbers and predator prey pair number is obtained by a contraction method that reduces the degrees of freedom second a van kampen expansion in the inverse of system size is then performed to get the fokker planck equation we also study the time correlation function the asymptotic behavior of which is used to characterize the transition from the cyclic coexistence of species to the ordinary coexistence

predator prey quasicycles from a path integral formalism the existence of beyond mean field quasicycle oscillations in a simple spatial model of predator prey interactions is derived from a path integral formalism the results agree substantially with those obtained from analysis of similar models using system size expansions of the master equation in all of these analyses the discrete nature of predator prey populations and finite size effects lead to persistent oscillations in time but spatial patterns fail to form the path integral formalism goes beyond mean field theory and provides a focus on individual realizations of the stochastic time evolution of population not captured in the standard master equation approach

mean field stochastic theory for species rich assembled communities a dynamical model of an ecological community is analyzed within a mean field approximation in which one of the species interacts with the combination of all of the other species in the community within this approximation the model may be formulated as a master equation describing a one step stochastic process the stationary distribution is obtained in closed form and is shown to reduce to a log series or log normal distribution depending on the values that the parameters describing the model take on a hyperbolic relationship between the connectance of the matrix of interspecies interactions and the average number of species exists for a range of parameter values the time evolution of the model at short and intermediate times is analyzed using van kampen s approximation which is valid when the number of individuals in the community is large good agreement with numerical simulations is found the large time behavior and the approach to the stationary state is obtained by solving the equation for the generating function of the probability distribution the analytical results which follow from the analysis are also in good agreement with direct simulations of the model

viewpoint invariant recovery of visual surfaces from sparse data an algorithm for the reconstruction of visual surfaces from sparse data is proposed an important aspect of this algorithm is that the surface estimated from the sparse data is approximately invariant with respect to rigid transformation of the surface in 3d space the algorithm is based on casting the problem as an ill posed inverse problem that must be stabilized using a priori information related to the image and constraint formation to form a surface estimate that is approximately invariant with respect to viewpoint the stabilizing information is based on invariant surface characteristics with appropriate approximations this results in a convex functional to minimize which is then solved using finite element analysis the relationship of this algorithm to several previously proposed reconstruction algorithms is discussed and several examples that demonstrate its effectiveness in reconstructing viewpoint invariant surface estimates are given

phase transitions and oscillations in a lattice prey predator model a coarse grained description of a two dimensional prey predator system is given in terms of a simple three state lattice model containing two control parameters the spreading rates of prey and predator the properties of the model are investigated by dynamical mean field approximations and extensive numerical simulations it is shown that the stationary state phase diagram is divided into two phases a pure prey phase and a coexistence phase of prey and predator in which temporal and spatial oscillations can be present besides the usual directed percolationlike transition the system exhibits an unexpected different type of transition to the prey absorbing phase the passage from the oscillatory domain to the nonoscillatory domain of the coexistence phase is described as a crossover phenomena which persists even in the infinite size limit the importance of finite size effects are discussed and scaling relations between different quantities are established finally physical arguments based on the spatial structure of the model are given to explain the underlying mechanism leading to local and global oscillations

finite difference schemes for the parabolic equation in a variable depth environment with a rigid bottom boundary condition we consider a linear schrodinger type partial differential equation the parabolic equation of underwater acoustics in a layer of water bounded below by a rigid bottom of variable topography using a change of depth variable technique we transform the problem into one with horizontal bottom for which we establish an a priori h1 estimate and prove an optimal order error bound in the maximum norm for a crank nicolson type finite difference approximation of its solution we also consider the same problem with an alternative rigid bottom boundary condition due to abrahamsson and kreiss and prove again a priori h1 estimates and optimal order error bounds for a crank nicolson scheme

the computation of visible surface representations a computational theory of visible surface representations is developed the visible surface reconstruction process that computes these quantitative representations unifies formal solutions to the key problems of 1 integrating multiscale constraints on surface depth and orientation from multiple visual sources 2 interpolating dense piecewise smooth surfaces from these constraints 3 detecting surface depth and orientation discontinuities to apply boundary conditions on interpolation and 4 structuring large scale distributed surface representations to achieve computational efficiency visible surface reconstruction is an inverse problem a well posed variational formulation results from the use of a controlled continuity surface model discontinuity detection amounts to the identification of this generic model s distributed parameters from the data finite element shape primitives yield a local discretization of the variational principle the result is an efficient algorithm for visible surface reconstruction

rising relative fluctuation as a warning indicator of discontinuous transitions in symbiotic metapopulations abstract the long time limit behavior of the stochastic lotka volterra model of a symbiotic metapopulation subjected to generalized verhulst self regulation is considered the influence of a time variable environment on the carrying capacities of subpopulations is modeled as a periodic deterministic part and a symmetric dichotomous noise relying on the mean field approach it is established that at certain parameter regimes the mean field average subpopulations size exhibits hysteresis in respect to the noise correlation time manifested in the appearance of colored noise induced discontinuous transitions especially it is shown that the relative fluctuation of the subpopulation sizes exhibits accelerated increase prior to abrupt transitions of the metapopulation state moreover in certain cases the autocorrelation function of the population sizes demonstrates anticorrelation at some values of the lag time

conservative initial boundary value problems for the wide angle pe in waveguides with variable bottoms we consider the third order claerbout type wide angle parabolic equation pe in the context of underwater acoustics in a cylindrically symmetric medium consisting of water over a soft bottom b of range dependent topography there are strong indications that the initial boundary value problem for this equation with just a homogeneous dirichlet boundary condition on b may not be well posed for example when b is downsloping in previous work we proposed an additional bottom boundary condition that together with the zero field condition on b yields a well posed problem in the present paper we continue our investigation of additional bottom boundary conditions that yield well posed physically correct problems motivated by the fact that the solution of the wide angle pe in a domain with horizontal layers conserves its l2 norm in the absence of attenuation we seek additional boundary conditions on a variable topography bottom that yield l2 conservative solutions of the problem we identify a family of such boundary conditions after a range dependent change of the depth variable that makes b horizontal we discretize the continuous problems by second order accurate crank nicolson type finite difference schemes and show by means of numerical experiments that the new models yield accurate simulations of the acoustic field in standard wedge type domains with upsloping and downsloping bottoms

phase transitions and spatio temporal fluctuations in stochastic lattice lotka volterra models we study the general properties of stochastic two species models for predator prey competition and coexistence with lotka volterra type interactions defined on a d dimensional lattice introducing spatial degrees of freedom and allowing for stochastic fluctuations generically invalidates the classical deterministic mean field picture already within mean field theory however spatial constraints modeling locally limited resources lead to the emergence of a continuous active to absorbing state phase transition field theoretic arguments supported by monte carlo simulation results indicate that this transition which represents an extinction threshold for the predator population is governed by the directed percolation universality class in the active state where predators and prey coexist the classical center singularities with associated population cycles are replaced by either nodes or foci in the vicinity of the stable nodes the system is characterized by essentially stationary localized clusters of predators in a sea of prey near the stable foci however the stochastic lattice lotka volterra system displays complex correlated spatio temporal patterns of competing activity fronts correspondingly the population densities in our numerical simulations turn out to oscillate irregularly in time with amplitudes that tend to zero in the thermodynamic limit yet in finite systems these oscillatory fluctuations are quite persistent and their features are determined by the intrinsic interaction rates rather than the initial conditions we emphasize the robustness of this scenario with respect to various model perturbations

discontinuous regression surfaces fitting we suggest a three stage procedure to recover discontinuous regression surfaces when noisy data are present in the first stage jump candidate points are detected using a jump detection criterion a local principal component line is then fitted through these points in a neighborhood of a design point this line provides a first order approximation to the true jump location curve in that neighborhood in the third stage observations on the same side of the line as the given point are combined using a weighted average procedure to fit the surface at that point if there are no jump candidate points in the neighborhood then all observations in that neighborhood are used in the surface fitting if however the center of the neighborhood is on a jump location curve only those observations on one side of the line are used thus blurring is automatically avoided around the jump locations this methodology requires o n k 2 computation where n is the sample size and k is the window width its assumptions on the model are flexible some numerical results are presented to evaluate the surface fit and to discuss the selection of the window widths

fluctuations and correlations in an individual based model of biological coevolution we extend our study of a simple model of biological coevolution to its statistical properties starting with a complete description in terms of a master equation we provide its relation to the deterministic evolution equations used in previous investigations the stationary states of the mutationless model are generally well approximated by gaussian distributions so that the fluctuations and correlations of the populations can be computed analytically several specific cases are studied by monte carlo simulations and there is an excellent agreement between the data and the theoretical predictions

a new class of automata networks a new class of automata networks is defined their evolution rules are determined by a probability measure p on the set of all integers z and an indicator function ia on the interval 0 1 it is shown that any cellular automaton rule can be represented by a nonunique rule formulated in terms of a pair p ia this new class of automata networks contains discrete systems which are not cellular automata for a given p a metric can be defined on the space of all rules which induces a metric on the space of all cellular automata rules

host parasite models on graphs the behavior of two interacting populations hosts and parasites is investigated on cayley trees and scale free networks in the former case analytical and numerical arguments elucidate a phase diagram for the susceptible infected susceptible model whose most interesting feature is the absence of a tricritical point as a function of the two independent spreading parameters for scale free graphs the parasite population can be described effectively by its dynamics in a host background this is shown both by considering the appropriate dynamical equations and by numerical simulations on barab asi albert networks with the major implication that in the thermodynamic limit the critical parasite spreading parameter vanishes some implications and generalizations are discussed

role of the noise on the transient dynamics of an ecosystem of interacting species abstract we analyze the transient dynamics of an ecosystem described by generalized lotka volterra equations in the presence of a multiplicative noise and a random interaction parameter between the species we consider specifically three cases i two competing species ii three interacting species one predator two preys iii n interacting species the interaction parameter in case i is a stochastic process which obeys a stochastic differential equation we find noise delayed extinction of one of two species which is akin to the noise enhanced stability phenomenon other two noise induced effects found are temporal oscillations and spatial patterns of the two competing species in case ii the noise induces correlated spatial patterns of the predator and of the two preys concentrations finally in case iii we find the asymptotic behavior of the time average of the ith population when the ecosystem is composed of a great number of interacting species

fluctuation theory for moran s genetics model abstract a genetics model of moran which includes the bernoulli laplace urn as a special case is examined to give a picture of the model s fluctuation theory and asymptotics as the number of states grows large with particular attention to the time of passage between extreme states these passage times are shown to have means which grow exponentially with the number of states and when normalised converge weakly to an exponential distribution

discontinuity preserving and viewpoint invariant reconstruction of visible surfaces using a first order regularization this paper describes the application of a first order regularization technique to the reconstruction of visible surfaces our approach is a computationally efficient first order method that simultaneously achieves approximate invariance and preservation of discontinuities it is also robust with respect to the smoothing parameter spl lambda the robustness property to spl lambda allows a free choice of spl lambda without struggling to determine an optimal spl lambda that provides the best reconstruction a new approximately invariant first order stabilizing function for surface reconstruction is obtained by employing a first order taylor expansion of a nonconvex invariant stabilizing function that is expanded at the estimated value of the squared gradient instead of at zero the data compatibility measure is the squared perpendicular distance between the reconstructed surface and the constraint surface this combination of stabilizing function and data compatibility measure is necessary to achieve invariance with respect to rotations and translations sharp preservation of discontinuities is achieved by a weighted sum of adjacent pixels the results indicate that the proposed methods perform well on sparse noisy range data in addition the volume between two surfaces normalized by the surface area interpreted as average distance between two surfaces is proposed as an invariant measure for the comparison of reconstruction results

predator prey cycles from resonant amplification of demographic stochasticity we present the simplest individual level model of predator prey dynamics and show via direct calculation that it exhibits cycling behavior the deterministic analogue of our model recovered when the number of individuals is infinitely large is the volterra system with density dependent prey reproduction which is well known to fail to predict cycles this difference in behavior can be traced to a resonant amplification of demographic fluctuations which disappears only when the number of individuals is strictly infinite our results indicate that additional biological mechanisms such as predator satiation may not be necessary to explain observed predator prey cycles in real finite populations

self organized packs selection in predator prey ecosystems received 19 july 2005 revised manuscript received 3 october 2005 published 23 february 2006we present a lattice model of a system of predators of ve kinds competing for prey the predators aregrouped in packs and characterized by two parameters the energy spent on hunting and energy gained by thekill the success of hunting depends on the actual competition among predators found near a prey wedetermine via monte carlo simulations the numbers of predators of each kind as a function of time and thedistribution of the size of their packs we show that the ratio of the energy spent by the competing predatorsdetermines their fate the energy gain plays only a secondary role we show also that the system self organizesitself into groups of predators living in well de ned packs which size depends on the energy spent the mostpreferred size dependence on the energy spent follows a very simple power law we present also a mean eld type approach to the problem and we discuss the differences in the results obtained by the two methods showing in particular that the simulation approach produces more reliable results doi 10 1103 physreve 73 021913 pacs number s 87 23 cc 87 23 kg 05 10 ln 87 23 n

error estimates for finite difference methods for a wide angle parabolic equation we consider a model initial and boundary value problem for a third order partial differential equation pde a wide angle parabolic equation frequently used in underwater acoustics with depth and range dependent coefficients in the presence of horizontal interfaces and dissipation after commenting on the existence uniqueness theory of solution of the equation we discretize the problem by a second order finite difference method of crank nicolson type for which we prove stability and optimal order error estimates in suitable discrete l2 h1 and maximum norms we also prove under certain conditions that the forward euler scheme is also stable and convergent for the problem at hand

scaling in a network model of a multispecies ecosystem a new model ecosystem of many interacting species is introduced in which the species are connected through a random matrix with a given connectivity the model is studied both analytically and by numerical simulations a probability distribution derived from the model is in good agreement with simulations and field data it is also shown that the connectivity c and the number of species s are linked through the scaling relation s k c 1 e which is observed in real ecosystems our approach suggests a natural link between log normal and power law distributions of species abundances

boundary conditions for the parabolic equation in a range dependent duct the parabolic wave equation in a range dependent duct with a rigid bottom is considered the range dependence will introduce amplitude errors in addition to the usual phase error of the local modes it is shown that the amplitude error increases by local mode angle and slope of the bottom and is inversely proportional to the depth imposing the vanishing of the normal derivative of the pressure at the bottom which is the physically correct boundary condition might also lead to ill conditioning of the parabolic model various ways to replace the boundary condition u n 0 are discussed and their accuracy is assessed both numerically and theoretically

boundary effects on population dynamics in stochastic lattice lotka volterra models abstract we investigate spatially inhomogeneous versions of the stochastic lotka volterra model for predator prey competition and coexistence by means of monte carlo simulations on a two dimensional lattice with periodic boundary conditions to study boundary effects for this paradigmatic population dynamics system we employ a simulation domain split into two patches upon setting the predation rates at two distinct values one half of the system resides in an absorbing state where only the prey survives while the other half attains a stable coexistence state wherein both species remain active at the domain boundary we observe a marked enhancement of the predator population density the predator correlation length displays a minimum at the boundary before reaching its asymptotic constant value deep in the active region the frequency of the population oscillations appears only very weakly affected by the existence of two distinct domains in contrast to their attenuation rate which assumes its largest value there we also observe that boundary effects become less prominent as the system is successively divided into subdomains in a checkerboard pattern with two different reaction rates assigned to neighboring patches when the domain size becomes reduced to the scale of the correlation length the mean population densities attain values that are very similar to those in a disordered system with randomly assigned reaction rates drawn from a bimodal distribution

galerkin methods for parabolic and schrodinger equations with dynamical boundary conditions and applications to underwater acoustics in this paper we consider galerkin finite element methods that approximate the solutions of initial boundary value problems in one space dimension for parabolic and schrodinger evolution equations with dynamical boundary conditions error estimates of optimal rates of convergence in l 2 and h 1 are proved for the associated semidiscrete and fully discrete crank nicolson galerkin approximations the problem involving the schrodinger equation is motivated by considering the standard parabolic paraxial approximation to the helmholtz equation used in underwater acoustics to model long range sound propagation in the sea in the specific case of a domain with a rigid bottom of variable topography this model is contrasted with alternative ones that avoid the dynamical bottom boundary condition and are shown to yield qualitatively better approximations in the real parabolic case numerical approximations are considered for dynamical boundary conditions of reactive and dissipative type

critical and oscillatory behavior of a system of smart preys and predators it is shown that a system of smart preys and predators exhibits irreversible phase transitions between a regime of prey predator coexistence and an state where predator extinction is observed within the coexistence regime the system exhibits a transition between a regime where the densities of species remain constant and another with self sustained oscillations respectively this transition is located by means of a combined treatment involving finite size scaling and fourier transforms furthermore it is shown that the transition can be rationalized in terms of the standard percolation theory the existence of an oscillatory regime in the thermodynamic limit which is in contrast to previous findings of boccara et al phys rev e 50 4531 1994 may be due to subtle differences between the studied models

critical behavior of the susceptible infected recovered model on a square lattice by means of numerical simulations and epidemic analysis the transition point of the stochastic asynchronous susceptible infected recovered model on a square lattice is found to be c0 0 176 500 510 where c is the probability a chosen infected site spontaneously recovers rather than tries to infect one neighbor this point corresponds to an infection recovery rate of c 1 c0 c0 4 665 713 and a net transmissibility of 1 c0 1 3 c0 0 538 4102 which falls between the rigorous bounds of the site and bond thresholds the critical behavior of the model is consistent with the two dimensional percolation universality class but local growth probabilities differ from those of dynamic percolation cluster growth as is demonstrated explicitly

discontinuous galerkin methods for the linear schrodinger equation in non cylindrical domains the convergence of a discontinuous galerkin method for the linear schrodinger equation in non cylindrical domains of mathbb r m m 1 is analyzed in this paper we show the existence of the resulting approximations and prove stability and error estimates in finite element spaces of general type when m 1 the resulting problem is related to the standard narrow angle parabolic approximation of the helmholtz equation as it appears in underwater acoustics in this case we investigate theoretically and numerically the order of convergence using finite element spaces of piecewise polynomial functions

population dynamics with or without evolution a physicist s approach abstract modeling the dynamics of interacting species or populations is a long standing problem in sciences which in the recent years has attracted a lot of physicists working in statistical physics the similarities and differences between models of dynamics of population and usual statistical mechanics problems on a lattice are discussed first the question of the appropriate level of description ordinary differential equations reaction diffusion equations patches models or individual based models for extended systems is considered second the role of the internal degrees of freedom associated to the phenotype of the individuals on the dynamics is discussed

stable oscillations of a predator prey probabilistic cellular automaton a mean field approach we analyze a probabilistic cellular automaton describing the dynamics of coexistence of a predator prey system the individuals of each species are localized over the sites of a lattice and the local stochastic updating rules are inspired by the processes of the lotka volterra model two levels of mean field approximations are set up the simple approximation is equivalent to an extended patch model a simple metapopulation model with patches colonized by prey patches colonized by predators and empty patches this approximation is capable of describing the limited available space for species occupancy the pair approximation is moreover able to describe two types of coexistence of prey and predators one where population densities are constant in time and another displaying self sustained time oscillations of the population densities the oscillations are associated with limit cycles and arise through a hopf bifurcation they are stable against changes in the initial conditions and in this sense they differ from the lotka volterra cycles which depend on initial conditions in this respect the present model is biologically more realistic than the lotka volterra model

robust ecological pattern formation induced by demographic noise we demonstrate that demographic noise can induce persistent spatial pattern formation and temporal oscillations in the levin segel predator prey model for plankton herbivore population dynamics although the model exhibits a turing instability in mean field theory demographic noise greatly enlarges the region of parameter space where pattern formation occurs to distinguish between patterns generated by fluctuations and those present at the mean field level in real ecosystems we calculate the power spectrum in the noise driven case and predict the presence of fat tails not present in the mean field case these results may account for the prevalence of large scale ecological patterns beyond that expected from traditional nonstochastic approaches

parabolic wave equation approximations in heterogenous media the properties of different variants of parabolic approximations of scalar wave equations are analyzed these equations are of general form which includes those used in seismology underwater acoustics and other applications a new version of the parabolic approximation is derived for heterogeneous media it has optimal properties with respect to wave reflection at material interfaces the amplitudes of the reflected and transmitted waves depend continuously on the interface existence uniqueness and energy estimates are proved

stochastic spatial models in the models we will consider space is represented by a grid of sites that can be in one of a finite number of states and that change at rates that depend on the states of a finite number of sites our main aim here is to explain an idea of durrett and levin 1994 the behavior of these models can be predicted from the properties of the mean field ode i e the equations for the densities of the various types that result from pretending that all sites are always independent we will illustrate this picture through a discussion of eight families of examples from statistical mechanics genetics population biology epidemiology and ecology some of our findings are only conjectures based on simulation but in a number of cases we are able to prove results for systems with fast stirring by exploiting connections between the spatial model and an associated reaction diffusion equation

crank nicolson finite element discretizations for a two dimensional linear schrodinger type equation posed in a noncylindrical domain first published in mathematics of computation online 2014 84 2015 1571 1598 published by the american mathematical society

the threshold of coexistence and critical behaviour of a predator prey cellular automaton we study a probabilistic cellular automaton to describe two population biology problems the threshold of species coexistence in a predator prey system and the spreading of an epidemic in a population by carrying out mean field approximations and numerical simulations we obtain the phase boundaries thresholds related to the transition between an active state where prey and predators present a stable coexistence and a prey absorbing state the numerical estimates for the critical exponents show that the transition belongs to the directed percolation universality class in the limit where the cellular automaton maps into a model for the spreading of an epidemic with immunization we observe a crossover from directed percolation class to the dynamic percolation class patterns of growing clusters related to species coexistence and spreading of epidemic are shown and discussed

stochastic models in population biology and their deterministic analogs we introduce a class of stochastic population models based on patch dynamics the size of the patch may be varied and this allows one to quantify the departures of these stochastic models from various mean field theories which are generally valid as the patch size becomes very large these models may be used to formulate a broad range of biological processes in both spatial and nonspatial contexts here we concentrate on two species competition we present both a mathematical analysis of the patch model in which we derive the precise form of the competition mean field equations and their first order corrections in the nonspatial case and simulation results these mean field equations differ in some important ways from those which are normally written down on phenomenological grounds our general conclusion is that mean field theory is more robust for spatial models than for a single isolated patch this is due to the dilution of stochastic effects in a spatial setting resulting from repeated rescue events mediated by interpatch diffusion however discrete effects due to modest patch sizes lead to striking deviations from mean field theory even in a spatial setting

finite size scaling analysis of the critical behavior of a general epidemic process in 2d abstract we investigate the critical behavior of a stochastic lattice model describing a general epidemic process by means of a monte carlo procedure we simulate the model on a regular square lattice and follow the spreading of an epidemic process with immunization a finite size scaling analysis is employed to determine the critical point as well as some critical exponents we show that the usual scaling analysis of the order parameter moment ratio does not provide an accurate estimate of the critical point precise estimates of the critical quantities are obtained from data of the order parameter variation rate and its fluctuations our numerical results corroborate that this model belongs to the dynamic isotropic percolation universality class we also check the validity of the hyperscaling relation and present data collapse curves which reinforce the accuracy of the estimated critical parameters

fluctuations and correlations in lattice models for predator prey interaction including spatial structure and stochastic noise invalidates the classical lotka volterra picture of stable regular population cycles emerging in models for predator prey interactions growth limiting terms for the prey induce a continuous extinction threshold for the predator population whose critical properties are in the directed percolation universality class we discuss the robustness of this scenario by considering an ecologically inspired stochastic lattice predator prey model variant where the predation process includes next nearest neighbor interactions we find that the corresponding stochastic model reproduces the above scenario in dimensions 1 d or 4 in contrast with the mean field theory which predicts a first order phase transition however the mean field features are recovered upon allowing for nearest neighbor particle exchange processes provided these are sufficiently fast

stochastic analysis of a pulse type prey predator model a stochastic lotka volterra model a so called pulse type model for the interaction between two species and their random natural environment is investigated the effect of a random environment is modeled as random pulse trains in the birth rate of the prey and the death rate of the predator the generalized cell mapping method is applied to calculate the probability distributions of the species populations at a state of statistical quasistationarity the time evolution of the population densities is studied and the probability of the near extinction time from an initial state to a critical state is obtained the effects on the ecosystem behaviors of the prey self competition term and of the pulse mean arrival rate are also discussed our results indicate that the proposed pulse type model shows obviously distinguishable characteristics from a gaussian type model and may confer a significant advantage for modeling the prey predator system under discrete environmental fluctuations

study of interacting particle systems the transition to the oscillatory behavior of a prey predator model abstract a prey predator system with intelligent pursuit and evasion is studied by means of computer simulations it is found that the model exhibits a coexistence state between prey and predators in this state and depending on the parameters the system may evolve toward two different regimes characterized by the behavior of the prey predator population whitin the oscillatory fixed point regime the specie s density changes periodically remains constant respectively the transition between such regimes is rationalized in terms of a dynamic percolation process

risk perception in epidemic modeling we investigate the effects of risk perception in a simple model of epidemic spreading we assume that the perception of the risk of being infected depends on the fraction of neighbors that are ill the effect of this factor is to decrease the infectivity that therefore becomes a dynamical component of the model we study the problem in the mean field approximation and by numerical simulations for regular random and scale free networks we show that for homogeneous and random networks there is always a value of perception that stops the epidemics in the worst case scenario of a scale free network with diverging input connectivity a linear perception cannot stop the epidemics however we show that a nonlinear increase of the perception risk may lead to the extinction of the disease this transition is discontinuous and is not predicted by the mean field analysis

edge preserving image denoising and estimation of discontinuous surfaces in this paper we are interested in the problem of estimating a discontinuous surface from noisy data a novel procedure for this problem is proposed based on local linear kernel smoothing in which local neighborhoods are adapted to the local smoothness of the surface measured by the observed data the procedure can therefore remove noise correctly in continuity regions of the surface and preserve discontinuities at the same time since an image can be regarded as a surface of the image intensity function and such a surface has discontinuities at the outlines of objects this procedure can be applied directly to image denoising numerical studies show that it works well in applications compared to some existing procedures

quasicycles in a spatial predator prey model we use spatial models of simple predator prey interactions to predict that predator and prey numbers oscillate in time and space these oscillations are not seen in the deterministic versions of the models but are due to stochastic fluctuations about the time independent solutions of the deterministic equations which are amplified due to the existence of a resonance we calculate the power spectra of the fluctuations analytically and show that they agree well with results obtained from stochastic simulations this work extends the analysis of these quasicycles from that previously developed for well mixed systems to spatial systems and shows that the ideas and methods used for nonspatial models naturally generalize to the spatial case

colored noise induced discontinuous transitions in symbiotic ecosystems a symbiotic ecosystem is studied by means of the lotka volterra stochastic model using the generalized verhulst self regulation the effect of fluctuating environment on the carrying capacity of a population is taken into account as dichotomous noise the study is a follow up of our investigation of symbiotic ecosystems subjected to three level trichotomous noise r mankin a ainsaar a haljas and e reiter phys rev e 65 051108 2002 relying on the mean field theory an exact self consistency equation for stationary states is derived in some cases the mean field exhibits hysteresis as a function of noise parameters it is established that random interactions with the environment can cause discontinuous transitions the dependence of the critical coupling strengths on the noise parameters is found and illustrated by phase diagrams predictions from the mean field theory are compared with the results of numerical simulations our results provide a possible scenario for catastrophic shifts of population sizes observed in nature